analytic surgery and the eta invariant

G e o m e t r i c And Func t iona l Analys i s

Vol. 5, No. 1 (1995)

1016-443X/95/010014-6251.50-{-0.20/0

�9 1995 Bi rkh~use r Verlag, Basel

A N A L Y T I C S U R G E R Y A N D T H E E T A I N V A R I A N T

R . R . MAZZEO AND R . B . MELROSE

Abstract

Let (M, h) be an odd-dimensional compact spin manifold in which H is an embedded hypersurface with quadratic defining function x 2 E C~176 Let ~ be the Dirac operator associated to the metric

lY__L + h g~ x 2 + e 2

where e > 0 is a parameter. The limiting metric, go, is an exact b-metric on the compact manifold with boundary M obtained by cutting M along H and compactifying as a manifold with boundary, i.e. it gives M \ H asymptotically cylindrical ends with cross-section OM, a double cover of H. Under the assumption that the induced Dirac operator on this double cover is invertible we show that

~ ( ~ ) = ~b(a-~.) + Tl(d + T2(d log ~ + ~(~)

where 7/b(~-~M) is the 'b' version of the eta invariant introduced in [Me1], rl (~) and r2(e) are smooth, vanish at e = 0 and are integrals of local geometric data, and where ~(e) is the finite dimensional eta invariant, or signature, for the small eigenvalues of ~ . If ~ is invertible then ~(e) -- 0 and

lim ~/(~r = ~/b(~D-M) �9 E -""* 0

Even if ~ is not invertible this holds in R/Z. In fact the discussion takes place more naturally in the context of the generalized Dirac operators associated to Hermitian Clifford modules.

These results are proved by analyzing the resolvent family of ~ , uniformly away from the spectrum and near zero. This leads to a precise description of the behaviour of the small eigenvalues. The corresponding 'heat calculus' is also constructed. It contains, and hence describes rather precisely, the heat kernel for ~ uniformly as e ~ 0. This calculus is related to, but different from, the surgery pseudodifferential operator calculus of McDonald ([Mc]).

The research of the first author was supported in part by NSF grant DMS-9001702 and a Sloan Foundation Fellowship and that of the second author by NSF grant DMS-8907710.

Vol.5, 1995 A N A L Y T I C SURGERY AND THE ETA I N V A R I A N T 15

1. I n t r o d u c t i o n

The eta invariant of a twisted Dirac operator was introduced by Atiyah, Patodi and Singer ([APSil]) as the non-local correction term in their index formula on a compact manifold with boundary. The index formula can be interpreted as computing the (extended) L 2 index of the Dirac operator on a noncompact manifold with cylindrical ends, or in the terminology of [Mel], for an exact b-spin structure on a compact manifold with boundary. Ex- tensions of the APS index theorem have been obtained by Cheeger ([Chl]), Mfiller ([Mill], [Mil2]), Stern ([Stl], [St2]) and Brilning-Seeley ([nrS]). From the point of view of b-metrics, the most natural extension is to compact manifolds with corners, which will be discussed elsewhere. Such a generalization suggests that it should be interesting to examine the behaviour of the eta invariant on the boundary of an even-dimensional spin manifold under a singular deformation as the manifold with boundary is deformed to a manifold with corners. On the boundary this corresponds to a degeneration of the metric to an exact b-metric across an embedded hypersurface, as described below; this can be visualized as stretching the boundary in the direction normal to the hypersurface until it has two infinite cylindrical ends in place of the hypersurface, the resulting metric on the complement of the hypersurface is complete. This type o f deformation is discussed geometrically by Singer in [Si]. In this paper we analyze the behaviour of the eta invariant under degenerations of this type, which we call 'analytic surgery'. The null space of the Laplacian and Dirac operator have already been studied for the geometrically similar degenerations where the limiting metric is an incomplete conic metric; see Seeley (IS1], [$2]), Seeley and Singer ([SSi]) and McDonald ([Mc]). McDonald's analysis of this incomplete limit is closely related to the approach used here and is discussed further below. Cheeger [Ch2] studied degenerations to an incomplete conical limit somewhat earlier by a process he terms the singular continuity method. He then related this to the s tudy of the adiabatic limit of the eta invariant.

Let M be a compact, odd-dimensional spin manifold, i.e. an oriented Riemannian manifold with a nontrivial fibre double covering of the bundle of oriented orthonormal frames by a principal Spin(n) bundle, where n = dim M. If H is an embedded hypersurface in M there exists a local defining function x E C~176 for H in some small neighbourhood around H, i.e. H = {x = 0} and dx # 0 on H, if and only if H has orientable normal bundle (or equivalently separates M locally). The defining function can be extended to a global function on M with the same properties if and only if H is globally separating. However the function x 2 always exists locally and extends globally to M so as to vanish only on H, and to do so

16 R . R . M A Z Z E O A N D R . B . M E L R O S E G A F A

exactly quadratically. In fact such a global function exists whether or not H is orientable. As will be evident below, all s tatements and results may be phrased in terms of the function x 2, but for simplicity we shall usually regard x as well-defined and unless a special point needs to be made H will be thought of as locally and globally separating in M.

Consider the family of metrics on M

Idxl 2 Idx2l g ~ - x 2 + e 2 + h - 4x2(x2+e2 ) + h (1)

where h is a fixed nondegenerate metric on M (it could also be allowed to vary smoothly with e, though again for simplicity we shall assume it does not). In the terminology of [Mel] the singular metric go is an exact b-metric on the compact manifold with boundary M obtained by cutting M along H. In more traditional terminology go gives M\H the structure of a complete Riemannian manifold with asymptotically cylindrical ends. The metric on these ends need not be exactly of product type, however it converges at an exponential rate to a product metric. In [Mel] the standard Riemannian structures are examined systematically from the 'b' perspective, for example the frame bundle is shown to have structure group reducing to SO(n - 1) over the boundary. A similar analysis shows that the structure group of the frame bundle for the metric g~ has a natural refinement to a spin structure degenerating in the same way as the metric. We phrase this geometrically in w below, showing that there is a natural resolution, Xs, (already to be found in [Mc]) of M x [0, 1L, obtained by blowing up H x {0}, to which the frame bundle of g~ lifts as a smooth principal SO(n) bundle with structure group reducing to S O ( n - 1 ) over the new boundary hypersurface introduced by the blow-up. The spin structure lifts to a covering of this frame bundle.

This geometric resolution allows us to understand the degeneration of the twisted Dirac operator ~iE,~ associated to an Hermitian bundle E over M and the metric g~ from the general point of view of a 'boundary fibration structure ' elaborated in [Me2], although the only results we use from [Me2] are those described in [Me3]. The surgery calculus of McDonald ([Mc]) quantizes this surgery boundary fibration structure on X~ and allows the resolvent of 3E,~ to be described uniformly as e ~ 0. Let H be the oriented double cover of H. Using the hypothesis

the induced Dirac operator on ~r is invertible, (2)

which is assumed throughout this paper, we show that the spectrum of ~JE,e remains discrete near 0 as e ---* 0 with eigenvalues and eigenfunctions collectively classical conormal. This allows us to control the behaviour of the heat kernel exp( - tb~ , , ) uniformly in e as t ---* r

Vol.5, 1995 A N A L Y T I C S U R G E R Y AND T H E ETA I N V A R I A N T 17

For e > 0 the eta invariant can be expressed in terms of the heat kernel of the Dirac operator by

# 1 t -1/2 Tr(~IE,r e -tn~,') dt (3) =

That this integral converges absolutely near t = 0 was shown originally by ra tod i ([P]) and Gilkey ([Gi]), but is now more readily seen using Getzler's rescaling ([G]) (see for example [Mel, Chapter 8]) or the closely related argument of Bismut and Freed ([BiF1]). For an exact b-metric such as g0 this definition is extended in [Mel, Chapter 9] to

Z 1 t -1/2 b-Tr~(~E,O e -tn~ o) dt (4) =

where the b-trace is introduced as a regularization of the trace of the heat kernel, which in this case is not trace class. In particular the integral in t is again absolutely convergent; in fact under the assumption (2) the integrand converges exponentially to 0 as t ~ c~.

As already noted, the assumption (2) implies that spectrum of 5E,0 is discrete near 0. Let H, be the ge-orthogonal projector onto the eigenspaces for 5E,e in (--6, 6) where 6 > 0 is chosen so small that 0 is the only eigenvalue of BE,0 in (--6,6). Then IIe is a finite rank self-adjoint projector. Let n_, no and n+ be the dimensions of the subspaces of the range of He spanned by the negative, zero and positive eigenspaces of SE,e, respectively, and set

= - n _ + n + . ( 5 )

This is the signature of 3E,e on the range of Fie. The possibility of repeated vanishing of the small eigenvalues means that (/(e) may not be continuous at e = 0; however it is bounded and takes values in Y.

MAIN T H E O R E M . For the surgery deformation (1) of an odd-dimensional compact spin manifold with embedded hypersurface, and for an Hermitian bundle E satisfying (2), the eta invariant satisfies

~(~E,e) = ~b(~E,0) + r l ( �9 ~- r2(E)log e q- 7)(e) in e < e0 (6)

for eo > 0 sumcient lysmall where r l , r2 e C~176 e0]) and rl(0) = r2(0) = 0.

That , for small positive e, r/(SE,e) -- O(e) is the sum of a smooth term and one with a logarithmic factor follows from a standard deformation argument (see w and w and the description of the heat kernel given below. In particular, if BE,0 is invertible then ~)(e) = 0 for �9 sufficiently small, and it follows from (6) that

! i ra 0 = (7)

18 R.R. MAZZEO AND R.B. M ELROSE GAFA

In any case this equality holds modulo l . This is similar to the behaviour of the eta invariant in the adiabatic limit as studied by Bismut and Cheeger ([BiCh]), cf. also Cheeger ([Ch2]), and by Dal ([D]). The small eigenvalues of ~hE,~ fall naturally into two classes, those with power law behaviour in e as e ~ 0 and those vanishing rapidly with e. This gives rise to a similar decomposition

= phg + rap( ) for < c0 �9 (8)

The first term on the right is eventually constant, while the second might have infinitely many jumps as e -* 0. The small eigenvalues of degenerating operators such as 8E,~ were estimated, using a mini-max argument by Cappell, Lee and Miller ([CLeMil]). They show that the small eigenvalues vanish exponentially as the length of the neck of the manifold. In terms of the parameter e this length scale is - l o g e, so our results are more precise and show the existence of a complete asymptotic expansion for the eigenvalues, determining them up to errors decreasing faster than any exponential in the length (i.e. any power of e). Only the eigenvalues with trivial expansion, i.e. those decreasing faster than any power of e can contribute to the jumps in ~/(e) as e ~ 0.

Results closely related to the limiting formula for the ?/-invariant in (6) have been obtained by Bunke ([BUD, Mfiller ([M/i3], [M/i4]) and Woj- ciechowski ([W]). These authors discuss the r/-invariant for an elliptic boundary problem for the Dirac operator (defined for example by Douglas and Wojciechowski in [DoW]) and its relation, via a gluing formula, to the r/- invariant for a compact manifold under the direct division of the manifold in two along an embedded hypersurfaze, under the assumption that the metric is a product near the hypersurface. The b-eta invariant is, in case the metric is of product type on the cylindrical end, equal to the ?/-invariant for the Atiyah-Patodi-Singer boundary condition for the manifold with boundary obtained by cutting off the end.

The boundary-fibration structure underlying the problem is discussed in w after preliminary material on manifolds with corners is reviewed in w In w we review McDonald's surgery pseudodifferential calculus, although there we actually discuss the slightly more general 'surgery calculus with bounds.' This is applied in w to find the uniform structure of the resolvent

2 --1 (gE,~--A) for small )~ as e -~ 0. The analysis of the heat kernel exp(-tS~,e) in w again uniformly for small e, is based on the surgery heat calculus described in w which is the main (new) technical component of this paper. The polyhomogeneous and holomorphic refinement of the surgery calculus is described in w and used to analyze the small eigenvalues. Finally, this machinery permits a quick proof of the main theorem in w

Vol.5, 1995 A N A L Y T I C SURGERY AND THE ETA INVARIANT 19

The analysis of the heat kernel holds equally well without the hypothesis (2), except for the discussion of the asymptotic behaviour as t --4 oo. To treat the general case it is therefore necessary to extend the discussion of the spectrum in w to an analysis of the manner in which the discrete spectrum of ~E,r accumulates as e I 0, at least near zero. The continuous spectrum of ~IE,0 in this case is described in [Mel]. The analysis of the limit when (2) is not assumed is examined in a continuation of this paper, see [HMMe].

The first author would like to thank MSRI for providing a room with a view during the preparation of part of this manuscript, and also the Uni- versity of Washington, where he held a position when this paper was being written. The second author is happy to acknowledge helpful discussions with Is Singer and Werner Mfiller.

2. Manifolds wi th Corners

To orient the reader we briefly recall some of the notions and terminology of manifolds with corners, blowing up and b-maps. These results are discussed in [Me3] and a more extensive treatment can be found in the book [Me2].

2.1 T h e Lie a l g e b r a "l)b(X). Let X be a manifold with corners. Thus, near any of its points, X is modeled on a product [0, co) ~ x R n-k, where k depends on the point and is the maximal codimension of the boundary faces containing that point. By local coordinates near a boundary point in X we always mean such product coordinates. Since it is always satisfied in the constructions below we make the simplifying assumption that all boundary faces of X are embedded; therefore they too are manifolds with corners. For each k let M k ( X ) be the set of boundary faces of codimension k (including by convention Mo(X) = {X}) and let M ( X ) be the union over k. For each q E X let Fa(q) be the (unique) element F E M ( X ) containing q in its interior, i.e. the boundary face of maximal codimension containing q.

The space 12(X) of all smooth vector fields on X is a Lie algebra under the usual bracket operation. As such, it has an important Lie subalgebra

12b(X) .= { V e 12(X) ; V is tangent to each F e M ( X ) } . (9)

This is the space of all smooth sections of a C cr vector bundle over X, 12b(X) = g ~ ( X ; bTX). Here, if :/q C g ~ ( X ) is the ideal of all functions vanishing at q E X, the fibre of bTX is

bTqX = ]2b(X)l:T q �9 ]2b(X ) . (10)

There is a natural bundle map tb : bTX ---* T X induced by the inclusion Nb(X) C 12(X) which is an isomorphism over the interior of X and has null

2 0 R . R . M A Z Z E O A N D R . B . M E L R O S E G A F A

space, the b-normal space bNqF, of rank k at each point q in the interior of any F 6 Mk(X). These spaces form a bundle over the interior of each boundary face and this bundle extends by continuity to be a canonically trivial bundle over the whole face. For example, when F 6 MI(X) and x is any defining function for F, then the vector field xOx is independent of the choice of x and spans bNqF for every q in the interior of F. A vector field VF 6 ]2b(X) which restricts to F 6 M~(X) to give the canonical section of bNF is said to be a radial vector field for F.

2.2 B l o w i n g up. An embedded submanifold Y is called a p-submanifold if near each point of Y local (product) coordinates can be chosen so that Y is given by the vanishing of some subset of them. That is, X and Y must have consistent local product decompositions (the 'p-' stands for product). When Y is a p-submanifold of X, a new manifold with corners, [X; Y], the (normal) blow-up of X around Y, may be defined. This is obtained by replacing Y by its inward-pointing spherical normal bundle; the union of this with XkY has a unique minimal differential structure as a manifold with corners with respect to which the lifts from X of smooth functions and of polar coordinates around Y are smooth. There is a natural blow- down map fl = fl[X; Y] : [X; Y] ---* X under which the inverse image of Y is its own inward-pointing spherical normal bundle; this is a boundary hypersurface which is the 'front face' of IX; Y] denoted if[X; Y]. Thus

[x; Y] = ff[x; Y] u (x \ Y). (11)

If X is blown up along a p-submanifold Y we define the lift, denoted fl*Z, of Z to [X; Y] of a closed subset Z (generally a submanifold) under two mutual ly exclusive conditions. If Z C Y then fl*Z = f l - l ( Z ) is the union of the fibres of if[X; Y] over points of Z. On the other hand if Z is the closure in X of Z \ Y then fl*Z is taken to be the closure in [X; Y] of fl-1 (Z \ Y).

In the constructions below we need to consider iterated blow up, where X is first blown up along a p-submanifold Y1 and then [X; Y1] is blown up along some p-submanifold Y2 C [X; Y] producing the new space [[X; Y1]; :V2].

In practice Y2 is often the lift of a submanifold Y2 C X. In that case the notation is simplified to

[[X; ]/'1]; Y2] ---- [X; ]I1; Y2] where Y2 = (fl[X; Y1])*Y2. (12)

The notation for further i terated blow-ups is then fixed inductively. Thus

= [ [ X ; Y l ; ' " ; Y k - , ] ; f k ] (13)

Vol.5, 1995 A N A L Y T I C SURGERY AND T H E ETA INVARIANT 21

where :Vk is the lift of Yk under the iterated blow-down map from [X; I I1 ; ' " ; Yk-1] to X and is assumed (or asserted by the use of the notation) to be a p-submanifold.

When Y1 and ]I2 C X are both imsubmanifolds it may well happen that both i terated blow-ups [X; Y~; ]12] and IX; II2; Y1] are defined, i.e. II1 lifts to a p-submanifold of [Z; 112] via fl[X; Y2] and Y2 lifts to a p-submanifold of [X; YI] via ~[X; Y1]. The blow-down maps to X restricted to the comple- ments of the preimages of Y1 U Y2 give a canonical isomorphism between open dense subsets of these two spaces but in general the blown-up spaces are distinct in the sense that this natural map does not extend to an isomorphism. However in important cases they are the same. The most obvious condition under which [X; Y1; ]I2] -= [X; Y2; Y1] is when Y1 and Y~ axe disjoint; the isomorphism is fixed by the condition that the diagram formed by the blow-down maps:

[x ; Y2] , , [ x ; Y2;

I I IX; Y~I Ix; Y~]

I I Id

X , ~ X

should commute.

(14)

More generally if ]I1 and Y2 intersect transversally then the two blow-ups commute in the sense of (14). Indeed, if Z = II1 oh ]I2 then locally near any point p E Z there is a decomposition X ,,~ ]I[ x Y~ x Z, p = (pt ,P2,P3) with Y1 "~ Y{ x {P2} x Z and Y2 ~ {Pl} x Y~ x Z. Then the blow-up of YI is locally the blow-up of {p~} in Y~ and the blow-up of Y2 is locally the blow-up of {Pl} in Y[, so the two-fold blow-ups both give the space

[YI'; {p,}] • {p2}] • z . The other case of this commutat ion of blow-ups which is used below

(for r = 2, 3) is for a chain of p-submanifolds in the following sense: Yj for j = 1, 2 , . . . , r are p-submanifolds of X with

Y1 C Y2 C Y3-.. C Y~ (15)

and are such that at any point q E Y~ if s is the smallest index such that q E Y~ then there are local product coordinates in X near q with respect to which Ys, Y~+I,- . . , Y~ are each given by the vanishing of some subset of the coordinates. A simple local coordinate computation shows that on blowing up Y1 the remaining elements lift to a chain of p-submanifolds of IX; Y1]. Thus the iterated blow-up in (13) is well defined for such a chain.

22 R.R. M A Z Z E O A N D R.B. M E L R O S E G A F A

Suppose that Yi (k), k = 1 , . . . , L and i = 1 , . . . , r k , are p-submanifolds forming chains for each k. We say such chains meet transversally if, near

any point, there axe local (product) coordinates in which each Yi (k) is given

by the vanishing of some set of the coordinates and Yi (k) r --Yj(k') whenever k ~ k'. On blowing up the smallest element of any chain the remaining submanifolds lift to form new chains meeting transversally. From the commutat ivi ty of blow-up in case of transversal intersection it follows thai the order of blow-up amongst the chains does not matter. In fact the order of blow-up within a chain is also immaterial.

LEMMA 1. / f Y1, ] /2, '" , Yr form a chain of embedded p-submanifolds of a manifold with corners X, then the spaces

[X;Y~o);Y~(z); ...;Yo(~)] (16)

are defined by iterated blow-up for any permutation a E ~.~. They are a11 naturally isomorphic in the sense of (14). The subspace of Vb(X) consisting of those vector fields tangent to all the submanifolds Y~ lifts to span vb([x; Y2;.. .; Y,.]) over C~176 Y,; ;

Proof: In [MeR] a proof is given in the case that X is a manifold without boundary and r = 2. This proof extends easily to the case that X is a manifold with corners proving the lemma for r = 2. To see the general case observe what happens when any one element of a chain of p- submanifolds is blown up. The smaller elements (if any) lift to form a chain of p-submanifolds of the front face produced by the blow up, while the larger elements lift to form a chain of p-submanifolds transversal to this front face. In fact the two chains (of combined length one less than before the blow up) meet transversally in the sense described above. Under the blow-up of any p-submanifold chains which axe transversal to it lift to be transversal chains. This shows that all the spaces in (16) axe defined. To show that they are all naturally isomorphic it suffices to check this for any two permutations which differ by the exchange of two neighbouring elements. This reduces either to the case of two manifolds which are part of a chain or which meet transversally, both of which have already been discussed.

The spanning property of the tangent vector fields follows similarly.

2.3 b - m a p s . For each F E M ( X ) let Z(F) C C~176 be the ideal of functions vanishing on F. If H E MI (X) with defining function PH then Z(H) = PH Coo(X) �9 A map between C ~ manifolds with corners, f : X1 X2, is C ~162 if f*C~(X2) C C~(Xa). It is said to be a b-map if for each

Vol.5, 1995 A N A L Y T I C S U R G E R Y A N D T H E ETA I N V A R I A N T 23

H 6 MI(X2) either

f*(Z(H)) = {0} or f*(Z(H)) = H [Z(G)] e'(c'H) ' (17) G6MI(XI)

where the e l (G, H) are necessaxily nonnegative integers. These numbers constitute the exponent matrix of f . If the first case in (17) holds for some boundary face H then the image f(X1) is contained entirely within that face; if this case does not occur then the b-map f is said to be an interior b- map. Clearly a b-map is an interior b-map if and only if f-a(OX2) C OX1. If Y C X is a p-submanifold of X the blow-down map 13[X; Y] is an interior b-map.

When f : X1 ~ X2 is an interior b-map its differential extends by continuity from the interior to define the b-differential

b f , : bTqX1 ~ b T f ( q ) X 2 V q E X l �9 (18)

For any q E X, the b-differential of a b-map necessarily satisfies

bf,: bNq F_+ bNf(q)G ' F = Fa(q) , C = Fa(f(q)) . (19)

An interior b-map is called a b-submersion if the b-differentiM is surjective for all q E X1. It is said to be b-normal if b f, is surjective as a map (19) for each q E X. It is called a b-fibration if it is both b-normai and a b-submersion. There is an alternate, more combinatorial characterization of b-fibrations. Namely a b-fibration is just a b-submersion such that no boundary hypersurfaee of X1 has image in a boundary face of )(2 of codimension greater than one. Hence blow-down maps are essentially never b-fibrations. Note as a special case that a b-submersion into a manifold with only codimension one boundaries (i.e. a manifold with boundary) is automatically a b-fibration. The blow-down map/3[X; Y] where Y E M(X) is always a b-submersion.

2.4 C o n o r m a l f u n c t i o n s . Conormal distributions with respect to a submanifold, especially polyhomogeneous conormal ones, are discussed in [HS]. In [Me2] a t reatment in the context of manifolds with corners is given, with particular emphasis on conormality up to boundary hypersurfaces. The notation here follows that of [Me3].

The space .A ~ (X) of functions on X conormal with respect to all hound- ary hypersurfaces and of order - 6 for M1 6 > 0 is defined by iterative regularity:

A~ = {uEp-~i~(X); Yb(X)kuCp-6L~176 VS>0 and keNo} (20)

24 R.R. M A Z Z E O A N D R.B. M E L R O S E GAFA

where p is the product of defining functions for all the boundary hypersurfaces. A multiweight is a function ~ : MI(X) ~ R U {c~}. For any multiweight s let

P~ = I I p~ F) FEMI(X)

be the product of the corresponding powers of defining functions. The general weighted conormal spaces are defined in terms of these by

.A~_(X) = pS,A~ . (21)

If some ~(H) = ~o then rapid vanishing is required at H. Sometimes we r X = use the notation ,A_( ) for r E R when s(H) T for every H E MI(X) .

Since these spaces (and others below) are all C ~ modules, the corresponding spaces of conormal sections of an arbitrary smooth vector bundle E over X can be defined as the tensor product:

A"_(X; E) = A ~ ( X ) | C~(X; E ) . (22)

They can also be defined directly by iterative regularity with respect to b-differential operators acting on the sections of E, as in (20).

2.5 I n d e x se ts . A polyhomogeneous conormal function on a manifold with corners, X, is one which is conormal at all boundaries and in addition has an expansion in powers of the defining function and its logarithm at each boundary hypersurface. Only non-negative integral powers of the logarithm are admitted. A space of polyhomogeneous conormal functions is fixed by specifying an index set for each boundary hypersurface. An index set, E, is a set of pairs (z,p) E C x IN0 indicating the (possible) presence of a term pZ(logp)P in the expansion, where p is a defining function for the corresponding boundary hypersurface. We require that (z,p) E E, p >_ 1, implies that ( z , p - 1) E E, that (z,p) E E and q E N implies (z + q,p) E E and also that for any sequence (zi,pi) E E, on which [zi I +p~ is unbounded, Re zi is also unbounded. If r E R, we say that Re E > r if Re z > r for all (z,p) E E, and R e E _> r if R e E > r ' for all r ' < r and moreover p = 0 for all (z,p) E E with Rez = r. The leading terms of E are those on which Re z takes its minimum value, which is denoted inf Re E. It is convenient to denote the index set {(z + j , 0); j E [No} simply by z.

There are two basic operations on pairs of index sets, E and F, which will arise below, namely addition and extended union. The sum E + F is the vector sum of subsets of 122 , i.e. elements of the sum are obtained by adding the corresponding components of the summands. The extended union E U F is the union of the two index sets, augmented so that if, for some z E 12, (z,p) E E and (z, q) E F, then (z,p + q + 1) E E U F .

Vol.5, 1995 A N A L Y T I C S U R G E R Y A N D T H E E T A I N V A R I A N T 25

2.6 P o l y h o m o g e n e o u s C o n o r m a l Func t i ons . If f is an index family for X, i.e. an assignment of an index set E(H) = EH to each boundary hypersurface g E MI(X) , then provided ReEH >_ ~(H) for all H E MI(X) (which we write simply as 8 _> s),

X `4~hg(X) C `4_( )

is defined by the existence of appropriate expansions. It is convenient to express this in terms of decay (which should be thought of as regularity) under the action of suitable operators which annihilate terms in the putative expansion. If VH is a radial vector field for the boundary hypersurface H and V E R set

BH(E, 7) = 1-I (Vu- z). (23) (z,k)EE Re z ~ 7

Then the space of conormal functions which are polyhomogeneous up to H with index set EH is defined by:

BE" ̀ 4~ (X) = N (u E ̀ 4~ ( X); BH( EH, 7)u E (pu)~ (p~H)'~`4~ (X) }

for any r < Re EH. Here p~ is the product of defining functions for all boundary hypersurfaces other than H. Full polyhomogeneity is the same as polyhomogeneity up to each boundary hypersurface:

. 4 _ ( ) w h e r e r < R e E u V H e M l ( X ) (24) `4 hg(x)= N Bf- x HEM](X)

There are also spaces of partially polyhomogeneous functions. If ~ > t are two multiweights (so that s (H) > t ( g ) for every H E MI(X)) then the space of conormal functions which are polyhomogeneous up to multiweight

is:

BE/~`4t'-(X)={ uE`4t-(X); IX BH(E'r(H))uE`4r--(X) Vrwith t < r < ~ } . HEM1(X)

(25) The particular case of interest below, especially in w is of conormal functions which have some positive order of vanishing, r up to some collection of boundary faces and are partially smooth, up to the same order T at the others (so the index sets at these faces are just 0). Thus if C C MI(X) is the subset up to which partial smoothness is desired and C' = M1 (X) \ C is the complementary set then the appropriate index family g is given by EH = 0 when H E C, EH = O when H E C r. In this case we shall adopt the shorter notation

]~c`4r_..(X) d ef B~./r`4O(x ) . (26)


Again all spaces are modules over C~176 so the definition extends to sections of general vector bundles by taking tensor products as in (22). For more details, in particular the independence of choices, see [Mel] and [Me2].

2.7 O p e r a t i o n s on C o n o r m a l Func t i ons . It is important to know when the conormal (and polyhomogeneous conormal) regularity of functions is preserved by functorial operations. We consider the four operations of restriction, multiplication, push-forward and pull-back.

Restriction to a boundary hypersurface only makes sense for at least partially polyhomogeneous conormal functions. We only need consider this operation in the context of the spaces (26). Suppose r > 0 and H E C so there is regularity up to order T on H. Since H is also a manifold with corners, the splitting of 341 (X) induces a similar splitting MI(H) - D U D', where F E D if and only if F -- G N H for some G E C. Then restriction to H yields a map

R , : B c A : ( X ) ---* 13DAh(H) . (27)

Various multiplicative properties of conormal functions are also straightforward:

.A~(X). AL(X) = ,~_+t(X) , 7 (28) 4phg(x). ,4phg(x) c Aph, (X),

S c A = ( X ) . 1 3 D A h ( X ) C t CnDA=(X) �9

Thus multiplication of conormal functions corresponds to adding multiweights, and multiplication of polyhomogeneous conormal functions corresponds to adding index families.

The more complicated operations of pull-back and push-forward will only be considered for an interior b-map f : X1 -~ X2 (although restriction is really a form of pull-back for a non-interior b-map).

For pull-back we consider the operation f # on multiweights and index sets. Given a multiweight s on X2,

f#~(H) = ~ ef(C,H).~(G), H e M,(X1) (29) GEM1(X~)

is a multiweight on X1. In particular, f#s (H) = 0 if H e null(e f) where

null(el) d=ef { g e MI(X1);ef(G,H) = 0 V G e MI(X2)} .

Similarly for any index family g on X2 let f # ( g ) be defined by f # ( g ) ( H ) = 0 if H e null(el) and otherwise f # (g) consists of the points (z,p) such that there exists (zi,pi) e C(GI) for each Gi E MI(X2) for which ey(H, Gi) r 0 and

z - Z e f ( H , Gi)z i is a non-negative integer and p = ~_,pi. (30) i i

Vol.5, 1995 ANALYTIC SURGERY AND THE ETA INVARIANT 27

As in Theorem 3 of [Me3] pull-back under an interior b-map is always defined for conormal functions and

.At(x1) t = f # a f* : .A_(X2) ---* g yr

f * : .Aphg(X2) - - , , .T" �9 Aphg ( X l ) ---- f # S f * : C A _ ( X 2 ) --~ ~ JDAr_ (X1)

where D = null(e/) U {H E MI(X1); f (H) N G = 0 V G e C'} .

(31)

As in Theorem 4 of [Me3] push-forward of integrable conormal densities remain conormal under b-submersions between compact manifold with corners. More precisely let bf~ be the b-density bundle (smooth sections of which are just of the form p-1/Z where # is a smooth density in the ordinary sense and p is the product of boundary defining functions) and let s and t be multiweights on X1 and X2 satisfying

s ( H ) > 0 V H E n u l l ( e f ) , s _ > f # t

then, assuming f : X1 ---* X2 to be a b-submersion,

(32)

o f . : A _ ( X 1 , t X . --* (33)

If f is a b-fibration and z is a multiweight o n X 1 satisfying the first part of (32) there is a unique maximal multiweight t with s > f # t, it is denoted f#z. When f is a b-fibration and ~ is an index family for X1 consider the family for X2 defined by

f#(g)(H) = -U GEM1(X1) e:(G,H)#O

{ ( z / ey (G ,H) ,p ) ; ( z , p )eg (G)} . (34)

Then, as in Theorem 5 of [Me3], provided f is a b-fibration between compact manifolds with corners and E is an index family for X1 with

C(H) > 0 V H E null(e/) (35)

push-forward under f preserves polyhomogeneity in the sense that

E z 7 = f # ( c ) f , : A p h g ( X l ; b~'~) --+ .Aphg

f . : BC/tA~(X1; b~))_.~BYlt' A((X2; b~) , ~'----f#S , s t = f # z , tt-----f#t. (36)


3. G e o m e t r i c R e s o l u t i o n

First we introduce the single surgery space Xs, already used in the slightly different context of degeneration to an incomplete conic metric by McDonald ([Mc]), to resolve the geometry of the metric (1). The space X , carries the structure algebra, a boundary-fibration structure in the sense of [Me2].

3.1 S ing le s u r g e r y space , X , . If H C M is an embedded hypersurface in a compact manifold without boundary, consider the space X = M • [0, �9 in which the parameter is included. The single surgery space is

X~ = [ X ; H x {0}] . (37)

0 H

r ~ /

Figure 1. The single surgery space X8

Since Y = H x {0} is an embedded submanifold of the boundary of the manifold with boundary X, it is clearly a p-submanifold, so the space (37) is well defined. The blow-down map will be denoted /3s : Xs ~ X. The original boundary e = 0 in X lifts to a boundary component, which we denote either Bbb (for b-boundary) or M (as in the introduction), of Xs. It is the closure in Xs of /3-1({e = 0} \ H) . The new boundary hypersurface introduced by the blow-up will be denoted Bss (the surgery boundary) or H. There is also a trivial extension boundary �9 = e0 both for X and Xs, but it never really enters the discussion and we cheerfully neglect it. Since H = Bs~ is the radial compactification of the normal bundle to H in X it has the structure of a bundle of semicircles, or intervals, over H. The projection 7r~ : X --* [0, e0] lifts to

[0, �9 = (38)

As the composition of a projection and a blow-down map it is an interior b-map. Since ed/de is, as a vector field on M x [0, e0], tangent to H x {0} it lifts into )2b(X~). Thus ~r~,r is a b-submersion and, the range being a manifold with boundary, a b-fibration.


As noted above the Lie algebra ]2b(X~) is spanned over g~ by the lifts from X of the elements of ]2b(X) which are tangent to H x {0}. It is convenient to have a somewhat stronger form of this result. Namely, the space ];b(X,) is spanned over g~176 by the lifts from X of the four types of vector fields:

{ V � 9 {eW;W �9 I;(M)} , {fo } o (39)

~ee ; f e g ~ 1 7 6 f - - 0 o n H a n d e s � 9

In fact this follows from the weaker statement since over C~176 the vector fields in (39) span all the vector fields in ~b(X) tangent to H • {0}.

3.2 S t r u c t u r e a lgeb ra . The structure algebra, which is a boundary- fibration structure in the sense of [Me2], is the space of smooth vector fields on X~ tangent to the fibres of ~r~,~ :

= {v �9 vb(x ); 0}. (40)

It is a Lie subalgebra of "l;b(X~). Since 7r~,~ is a b-submersion, its b- differential has null space a subbundle of codimension one. This is the structure bundle ~TX~ and

= .

Observe that the passage from X to X~ is necessary in order to resolve the singularity of the vector field v ~ + e 2 0x, which is naturally associated to the metric (1); indeed the lift of this vector field to X~ is a smooth nonvanishing section of ~T(Xs). Over the part of X8 where e > 0, sTX~ is simply the pull-back of TM. From the discussion around (39) it follows that I)~(X,) is spanned over C~176 by the lifts from X of vector fields in

{V e 12(M) ; V is tangent to H} , {eW ; W e 12(M)} {,0 } and Oe ; f � 9 f = 0 ~ .

(41)

In particular restricted to Bss or Bbb the bundle ~TX, is canonically isomorphic to the b- tangent bundles of these as compact manifolds with boundary.

If E and F are Ccr vector bundles over Xs the space Diff , (M; E, F) consists of those differential operators

P: C~ E) -* C~(X~; F) (42)

30 R.R. M A Z Z E O AND R.B. M E L R O S E GAFA

which are given, with respect to local bases of E and F, by sums of up to k-fold products of elements of V,(X,) with coefficients smooth on X~. Despite the fact that they are operators on X8 the notation refers to M since they axe to be thought of as differential operators on M depending on the parameter e with 'surgery degeneration' at H when e = 0. An element of V~(X~) can be identified with an element of Diff , (M), acting on the trivial bundle, and also can be identified with a smooth function on ~T*Xs, the

X dual bundle to T ~, linear on the fibres. The symbol map

~ : V~(X~) -~ P[ ' ] (~T*X~,C) , V , , i V (43)

is defined by these identifications. Here, for any k, P [ k I ( E , F ) denotes the space of smooth bundle maps from E to F which are homogeneous poly- nomials of degree k on the fibres. Since it vanishes on commutators the symbol map extends multiplicatively to

~a : D i f f ~ ( M ; E , F ) ~ P [ k I ( ~ T X ~ , h o m ( E , F ) ) �9 (44)

3.3 L i f t ed m e t r i c . The metric (1) lifts to a smooth, non-degenerate fibre metric g~ on ~TX~. To see this observe that by duality from the lifting properties of the smooth vector fields tangent to a p-submanifold, Y, and to the boundary of X the extended metric

dx 2 de 2 gb -- x2 + e~ + --fi- + h (45)

lifts to a b-metric in the sense of [Mel] on X~, i.e. to a nondegenerate metric on the fibres of bTX~. Hence g~ lifts to a non-degenerate metric, g~, on the annihilator of de~e, namely ~TX~.

As a bundle ~TX~ is isomorphic to the pull-back of T M under the projection r~,b = 7rM o /3s : X~ ~ M; there is no natural isomorphism but there is a natural homotopy class of isomorphisms. The frame bundle of the metric on ~TX~ is therefore isomorphic to the pull-back of the frame bundle of the metric h on M. Spin structures are discrete so it follows that a spin structure (including orientation) on M fixes a surgery spin structure in the sense of a non-trivial double cover of the bundle of oriented orthonormal frames of g~ as a metric on ~TX~.

The spin bundle, S, over X~ is therefore well defined as a bundle associated to this double cover. Directly from the form assumed in (1) the metric g~ restricts to M to an exact b-metric in the sense of [Mel]. The same is true for the restriction to H and moreover the coordinate r = sinh -1 (x /e ) on the fibres of 7r : H --* H reduces the metric on H to the form

d7 -2 + 7r*hH . (46)

V o l . 5 , 1 9 9 5 A N A L Y T I C S U R G E R Y A N D T H E E T A I N V A R I A N T 31

m

Thus H has an R-action by translations (which we will usually regard as the equivalent multiplicative R+-action using exponentials) with respect to which the metric is invariant. This shows that the structure group of the principal bundle associated to *TX, reduces to SO(n - 1) over H.

3.4 C o n n e c t i o n . The Levi-Civita connection of the extended metric gb, in (45), lifts to a b-connection on Xs in the sense of [Mel]. Namely it defines a differential operator

V e Diff ,(M; E, bT *X~ | E) (47)

for any bundle E associated to the orthonormal frame bundle. The fibres of 7r,,~ are totally geodesic for the metric gb, since the additional term in (45) is the square of an exact differential vanishing on the fibres. The restriction of the connection (47) to the fibres of r,,~ over e > 0 therefore reduces to the connection of the metric g~. Hence, by restriction, the Levi-Civita connection defines an s-connection:

V e Di f f~ (M;E,*T*X~| E) (48)

for any associated bundle E. As for any boundary-fibration structure there is an exterior differential

generated by the action of the vector fields, i.e.

d E Diff,(M; *A k, sAk+l) V k (49)

with ~AkX, the kth exterior power of ~T*X~. Since the Hodge star isomorphism is defined algebraically it extends to the s-form bundles:

, : ~AkX~ , , ~A'~-kXs , n = d i m M . (50)

Hence the adjoint ~ and the Laplacian are also s-differential operators:

= ( -1 ) nk+n+l , d , E Diff~ (M; ~A k, ~A k- l ) , A = d~ + ~d e Diff2(M; ~A k, ~Ak). (51)

3.5 D i r a c o p e r a t o r . The Clifford bundle of the metric g, is the bundle of Clifford algebras of the fibres of *TX~. With the conventions of [Mel] this is the full tensor algebra modulo the relations

e . e' + = 2<e, , V e . (52)


If M is a spin manifold there is a natural action of this bundle of algebras on the spin bundle, i.e. S is a Clifford module,

el : ~T*X~ ~ hom(S) . (53)

The Levi-Civita connection extends naturally to S, as a bundle associated to the surgery spin structure, and the Dirac operator can be writ ten .as the composite

$~ = - i c l ' . V e Diff , (M; S) . (54)

Here cl ~ is the bundle map cl ~ : sTX8 | S ~ S induced by (53). More generally if E is any Hermitian bundle on X~ with Hermitian connection then, with V interpreted as the induced connection on S | E, the same formula (54) defines ~E,~ e Diff , (M; S | E).

3.6 H e r m i t i a n Cl i f fo rd m o d u l e s . For a more extensive discussion of Hermitian Clifford modules see [BGV], [LMi] or [Mel]. A bundle, E, over X~ is an Hermitian Clifford module for the surgery structure if there is a smooth bundle action

cl: CI(~T*Xs) ~ hom(E) (55)

of the Clifford bundle of 8T*X~. If E is endowed with an Hermitian metric, then a connection V on E is called an Hermitian Clifford connection if it is compatible with the metric in the usual way, i.e. the maps cl(~) are self- adjoint with respect to this metric for ~ E ~T*X~ and V satisfies Leibniz' rule for Clifford multiplication:

Vv(cl(c~)e) = c l ( V v a ) e + c l ( a ) V v e , V V e ]?~(X~) , e e C ~ ( M ; E ) . (56)

As in (54) a 'generalized Dirac operator ' 8E,~ e Diff~ (M; E) is defined on the sections of E. The signature operator is a basic example. The interested reader will readily see that the techniques and results above, and below, extend directly to this more general setting.

3.7 N o r m a l o p e r a t o r s . As noted earlier, the restriction of the Lie algebra V~(Xs) to the boundary hypersurfaces M = Bbb and H = Bss gives the full algebras "l~b(M) and "12b(H) respectively. The boundaries of M and H are canonically identified with one another and with the orientation bundle of H:

0 M - 0 H = J~ --* H , (57)

which is the trivial double cover if H is orientable or the oriented double cover if it is not. The restriction of ]2~(X~) to H gives the space of all g ~


vector fields, ])(___H), on__H. If V e 12~(X~) let Nb(V) and Ns(V) denote the restrictions to M and H, respectively. Clearly

Nb(Y)u = (Yf i ) t~ , V u e C~ , ft E C~ with firM = u (58)

and similarly for Ns(V). Since these operations are local they extend multiplicatively to define the normal homomorphisms:

Nb: Diff,(M; E, F) --* Diff,(M; E, F) (59)

N~: Diff,(M; E, F) --* Diffbk(H; E, F)

where E and F are arbitrary smooth bundles over Xs, and also stand for the restrictions of these bundles to M and H. There is a similar map given by restriction to J~:

ni l : Diff~(M; E,F) ---* Diffk(H; E , F ) . (60)

This can be factored through the restriction to either M or H. The b-normal bundle, bNH, to a boundary hypersurface H is the canon-

ically trivial line subbundle spanned by the element xO., where x is a defining function for the hypersurface. The b-tangent bundle of the hypersurface is canonically isomorphic to the quotient bTHX/bNH of the restriction to H of the b-tangent bundle of the manifold. The b-normal bundles to as the boundary of M and H are canonically isomorphic because they are spanned by elements of ]J~(X~). The [t+-structure on H means that the radially compactifiednormal bundle to its boundary is canonically isomorphic to the product H x [0, 1]. This is the orientation cover of H and will be denoted l~ :

] y (61)

H ~ H

The indicial operator of an element P E Diff'(M; E, F) is an element

I(P) e Diff~,b(H; E , F ) C Diff,(_0; E , F ) (62)

where the subscript I denotes the subspace of R+-invariant operators. Simi- larly the indicial operator of an element of Diffbk(H; E, F) is also an element of Diff/k b(/t; E, F). The compatibility condition between the normal operators on M and H is then just

NH(P) d~=f I(Nb(P)) = I(N~(P)) V P e Diff~(M;E,F) . (63)

34 R.R. M A Z Z E O A N D R.B. M E L R O S E GAFA

For any pair of operators Pb E Diffbk(M; E, F) , P, E Diffbk(H; E, F ) satisfying I(Pb) = I ( P , ) there exists P e Dif f , (M; E, F) such that Nb(P) = Pb, N , ( P ) = Ps. In the spin case, with M odd-dimensional, the spin bundles on M, H and H are canonically isomorphic to the restrictions to these submanifolds of the spin bundles from Xs. From the natural i ty of the discussion above it follows that the normal operators are always the Dirac operators for the induced spin structure. Thus

Nb(~Jc) = ~ e Diff , (M; S ) ,

N,(~i~) = 5-fill e Diff ,(H; S) , (64)

= Di l(9; S) ,

RH(~J,) = ~ e Diff , (H; S ) .

The same conclusions hold for the twisted Dirac operator, 8E.~, associated to any Hermit ian bundle E.

3.8 D e n s i t i e s . The density, w~, of the metric g~ has the proper ty that w = we | 7r,.~*lde I is a nonvanishing smooth section of f~(X,). Tha t is, if " i t (X,) is the density bundle of 8 T X ,

s i t (X,) | %.,* (it[0, col) - ~ (X , ) . (65)

4. Surgery Calculus

In this section we recall the salient features of the surgery calculus of Mc- Donald ([Mc]) and discuss some extensions. Here we treat the 'calculus with bounds ' since this allows an easy discussion of operators in this calculus depending holomorphically on a parameter. In w the polyhomogeneous form of the calculus, essentially as in [M], is described. The calculus is used in w to analyze the resolvent of the Dirac Laplacian for the resolvent parameter near zero, uniformly as the surgery parameter tends to zero. It is also the basis of the heat calculus introduced in w

4.1 D o u b l e s u r g e r y space , X~. The double surgery space, on which the resolvent kernel takes a relatively simple form, is obtained by i terated blow- up from the product M 2 x [0, e0]. Consider the submanifolds H L = H x M and H n = M x H of M 2 and their intersection H 2. Then set, as in (13),

X 2 = [M 2 x [0, e0]; H 2 x {0};HL x {0} ;Hn x {0}] (66)

with overall blow-down map /~ : X~ --. M 2 x [0, e0]. To see that this i terated blow-up is well defined note first tha t H 2 x {0} is a p-submanifold

Vol.5, 1995 A N A L Y T I C S U R G E R Y A N D T H E E T A I N V A R I A N T 3 5

of M 2 x [0, e0]. As is clear from local coordinates HL x {0} and HR x {0} have independent images in the normal bundle to H 2 x {0} in M 2 x [0, e0] so become disjoint p-submanifolds when lifted to [M 2 x [0, e0]; H 2 x {0}]. Thus everywhere locally (66) is just given by the two iterated blow-ups for the chains of p-submanifolds

H 2 x {0} C HR x {0} C M 2 x [0, eo] (67)

H 2 x { 0 } C H L X { 0 } c M 2 x [ 0 , % 1 .

In particular X~ 2 is well defined. The disjointness of the lifts of the last two spaces in (66) means that the

order in which they are blown up can be reversed. Applying Lemma 1 also allows the first two blow-ups to be interchanged. This gives two alternate descriptions of X, :

X~ 2 = [M 2 x [ 0 , e 0 ] ; H L x { 0 } ; H 2 x { 0 } ; H R x { 0 } ] ~ X , x M (6.8)

Xy= [M2x[O, eo];gR x {0};H2 x {0};HL x {0}] ~ M x X s . (69)

Since the first blow-up in (68) occurs in M x [0, %] for the first factor of M it gives Xs x M; the map in (68) is the partial blow-down map corresponding to the last two blow-ups and similarly in (69). The two projections r~, 7r2: M 2 x [0, e0] --* X, dropping the first (left) or second (right) factor of M, therefore lift to C ~ maps

(70) % , o : X ~ ~ X ~ , O = L , R

obtained by following the blow-down maps in (68) and (69) by projection off the remaining factor of M. This gives a commutat ive diagram:

X~ , X~ , X~ [

(Wl) X , M 2 • [0, 1] ~ X.

It is of considerable importance that both 2 and 2 7r~,L %,R are b-fibrations. That they are b-maps follows directly from their decomposi t ion as products of b-maps, namely blow-down maps and projections. Tha t they are b- submersions follows from the fact tha t the vector fields in (39) lift to X~ to span bTX~. Thus if v E bTqX~ there is a vector field of the form

v +/_0 0 Oe + cede where

c e R , V, W e V ( M ) , V i s t a n g e n t t o H a n d f e C ~ f = 0 o n H (72)

36 R . R . M A Z Z E O A N D R.B. M E L R O S E G A F A

which lifts into "I;b(X~) and takes the value v at q. Then consider the vector field on M ~ x [0, e0]

u = ( v + v ' ) + e w + ( / - f ')alOe + cold,

with V acting on the first factor of M, V r -- V but acting on the second factor of M and similarly f E C ~ ( M ) varying on the first factor and f ' = f but pulled-back from the second factor of M. Now U is tangent to the boundary of M 2 x [0, Co] and to all three manifolds blown up in the definition of X 2. It follows that U lifts successively under each blow up and so lifts

b 2 to U ~ E Yb(X2) . By construction if w E T,.X~ is the value at any point r in the p r e i m a g e o f q under 2 then b 2 7r~, L (Tr~,L).(w) = v. Thus "Ks, L2 is a

b-submersion and so is 7rs, R.2 Finally the b-normality of these maps is the condition that no boundary hypersurface of the domain is mapped into a boundary face of codimension two or higher in the range space; this can be seen directly from (68) and (69).

The projection to [0, e0] also lifts to a b-fibration 2 2 ~,~ : X , ~ [0, e0] 2 2 The density bundles axe which factors as either 7r~,~ o 7r, L or 7r~,~ 0 7r~, R.

related by

Tr ~* Ide[ -1 ,.., ~ , . , , | 1 7 4 (73)

/

Brs

/ /

/

/ /k s

/ /

/ /

Bdb

Figure 2. Boundary faces of X~

Apart from the 'trivial' boundary at e = e0, X 2 has four (sets of) bound- axy hypersurfaces: the three produced by blow-ups and the lift of the original hypersurface e = 0. The latter is connected or consists of four components, depending on whether or not H separates M (we are of course thinking of M itself as connected). It will be labelled Bdb. The boundary hypersurface obtained from blowing up H 2 x {0} will be denoted Bds, while


those arising from HL x {0} and HR • {0} will be denoted Bk and B,~, respectively. The faces Bk and B,s have either two components or one, again according to whether H separates M or not. The structure of these boundary hypersurfaces is discussed further below.

The diagonal A x [0, e0] C M 2 x [0, e0] lifts to a psubmani fo ld

A s ---- c l ( ( / ~ 2 ) - l ( A X ( 0 , ~ 0 ] ) ) C X 2 ( 7 4 )

which is naturally diffeomorphic to X~. Indeed the stretched projections ~r~,o,2 O = L, R, restrict to As to give the same diffeomorphism onto Xs.

4.2 S u r g e r y p s e u d o d i f f e r e n t i a l o p e r a t o r s . The surgery pseudodifferential operators are pseudodifferential operators on M depending parametrically on e, and with precise regularity properties at e = 0. They are characterized in terms of the lifts of their Schwartz kernels from M 2 x [0, e0] to Xy. The calculus (with bounds) is a sum of two parts. The first part is the small calculus, which contains the most general singularities on the lifted diagonal permit ted in the calculus but is essentially trivial in its boundary behaviour. The second part contains all boundary and residual terms but no interior diagonal singularities.

To define the small calculus we use the fact that As C Xs 2 is an interior p-submanifold, i.e. is a p-submanifold and is the closure of its intersection with the interior. As such it can be extended across the boundary in any thickening of X to a manifold without boundary. The space of conormal distributions with respect to it is therefore well defined. The elements of order m in the small calculus correspond to the 1-step polyhomogeneous conor-

1 cf . [Ho] . These form mal distributions with respect to As of order m - ~,

the space I m- �88 (X~; As) consisting of distributions which are smooth away from As. We also demand that the kernels of elements of the small surgery calculus vanish rapidly at the boundary hypersurfaces Bk, Br~, which are the ones not meeting As. The resulting space can be written

~x~ o o r a - - ! X 2 p,,p,,•

where the formal factors of the defining functions indicate rapid vanishing at those boundary hypersurfaces. This space is a local C~(X2)-module (so it consists of the sections of a fine sheaf over X2). Consequently there is a similar space of sections of any C ~ vector bundle over X 2, defined by localization. If E and F are any vector bundles over X~ let

H o m s ( E , F ) 2 �9 f~-�89 2 �9 , 1 = (F | ) | (E | (75)


be the surgery-normalized homomorphism bundle over X 2 of the two bundles. In case E = F = ft�89 it is canonically trivial. Now define the part of order m of the small surgery calculus by

q m ( M ; E , F) = Pl~ Prs~176188 ~, A,; Horns (E, F) | gt�89 (X2)) . (76)

Directly from the definition it follows that

Di f f ' (M) C ~ ( M ) . (77)

Since the boundary terms in the calculus are intermediate between the small and the residual parts we define the residual part of the calculus first. Using the conormal space defined in (21) the residual part of the calculus, corresponding to the boundary order T > 0 (i.e. with multiweight t (H) = T for every H above {e = 0}) is identified with the space of density sections of the normalized homomorphism bundle:

= ( ~ , H o m ~ ( E , F ) | (78) ~ , r e s ( M ; E , F ) Mr_. X 2. 1- 2

The boundary terms are similar, but are required to have kernels lying in the spaces (26) which are partially smooth up to the boundary faces Bds and Bdb, which are the ones meeting A~. Letting dB = {ds, db}, and referring to the notation (26), define for any T > 0

1 2 kO~-~'~(M; E, F) = BdBA[ (X2; Horns(E, F) | f~2 (X~)) ,

~ - ~ ' ~ ( i ' E , F ) . ~Ym,~ ' (M;E,F) = ~ m ( M ; E , F ) + ~ ~ , (79)

These are the surgery operators 'with bounds. ' Notice that the stability of conormal functions under Diff~ (X 2) implies that

r . ff/~-~,r (M; E, F) . 61~,res(M,E,F ) C (80)

The intersection of the two terms in (79) is non-trivial, namely it is equal to the intersection over r of the spaces of boundary terms:

L 9 7 ( M ; E , F ) A q - j + ' * ( M ; E , F ) = ~ - ~ + ' + ( M ; E , F ) . (81)

4.3 S y m b o l m a p . The symbol map for conormal distributions induces a surjective symbol homomorphism (see [HS])

rm-�88 A:) -+ SIn(N'A:; a.b) (82)

Vol.5, 1995 A N A L Y T I C SURGERY AND THE ETA I N V A R I A N T 39

with null space ~'m-l-�88 As). Lifting 1Js(Xs) from left or right gives a canonical identification of NAs with sTXs. The density factors in Horns(E, F), the fibre density bundle in (82) and 12�89 (Xs) combine to give a naturally trivial factor over As = Xs, so the resulting symbol map

~am: ~ ? ( M ; E, F) --+ S ( TX~, E, F)

has null space precisely the operators of order rn - 1 in the small calculus. From (81) the symbol map vanishes identically on the intersection of the small calculus with boundary terms, so extends to the whole calculus to be zero off the small part:

Sam: ~ ' ~ ( M ; E, F) ---+ sm(~TXs; E, F) .

4.4 A c t i o n o n d i s t r i b u t i o n s . So far the surgery operators have only been defined as distributions. Since 7r8, R2 is a b-submersion all extendible distributions can be lifted from Xs to X~ under it. Wavefront set consider- ations, using the transversality of 7r2,R to the lifted diagonal, allow the lifted distributions to be multiplied by the kernel of an s-pseudodifferential operator and the result can be pushed back to X~ under 7r~, L2 to an extendible distribution. Taking into account the bundle factors this gives the first part of:

A: C - ~ ( X ~ ; E ) ~ C-~(X~;F) fit m'~'(M" E, F) 9 A =~ ,45(X~; F) , r < 7-. (83) s , , A : A S ( X s ; E )

The second part is similar using the results of w The action of s-pseudodifferential operators on appropriate Sobolev spaces

also needs to be considered. If L2(Xs; E) is the space of square-integrable sections of E over Xs with respect to a non-vanishing density then it can be shown as in [M] that

V'~'~-(M;E,F) 9 A: L2(Xs;E) ~ L2(X~;F) (84) provided m < 0 ~ T > 0.

For the boundary terms this can be shown using Schur's lemma. Bound- edness for elements of the small calculus can be reduced to the boundary case by use of HSrmander's symbolic technique to construct an approximate square-root.

Commutat ion and duality arguments immediately lead to boundedness on the integral order weighted Sobolev spaces

f {u : e-"uEL2(Xs;E);Diff~(M;E)e-'~uCL2(X~;E)} k>0 erHk(Xs; E)= I, Diff~-k(M; E) . e"L2(Xs; E) , k<0

(ss) as long as ]r I < T.

40 R.R. MAZZEO AND R.B. MELROSE GAFA

The small calculus also preserves C ~ regularity. A general s-pseudodifferential operator does not, but there are partial results. Once again referring to (26):

~ , T t M . E, F) 9 A i t . , A : 13,sA_(X~, E) ~ B~A ~" (X~; F)

13 " " X . A: bb,A (Xs ; E) ~ ~ b b A _ ( s; F ) 0 ~ r ~ T

(86)

4.5 N o r m a l h o m o m o r p h i s m s . The utility of the calculus of s-pseudodifferential operators is based on the existence of certain, noncommutative, analogues of the symbol map called normal homomorphisms. There is a normal homomorphism associated to each boundary face of X2; at Bgf the normal homomorphism Ngf is defined by restricting the kernel of an s- pseudodifferential operator to that face, using (27). Since in the present circumstances kernels are required to vanish at some positive rate at Bl~ and Br~, the normal homomorphisms NI~ and Nrs are trivial on ~ ' T ( M ) , and can be ignored. We shall also ignore restriction to the extension face {e = e0 }, which is isomorphic to M 2 and where the normal homomorphism has range in the pseudodifferential calculus on M. Hence normal homomorphisms are only of interest at Ba~ and Bdb, and these will be denoted N~ and Nb as in (59).

There is an alternate definition of N~(A) and Nb(A) for Aer E, F) using (86) and (27) w___hich allows the normal operators to be regarded as operators on H and M, respectively:

N~(A)u=A~rss, if ueg~ E) , ~EBssA[,bb(X~; E) and ~r~=u

Nb(A)u=Af~rbb, if uEg ( M ; ) ~EBbbAh,~(X~;E) and ~rbb=U. (87)

This is consistent with (58), (59) and (77). It is important that the two definitions, by restriction of the kernel and restriction of the operator in this sense, coincide, so we explain this correspondence more carefully next. This requires a somewhat more detailed discussion of the structure of Bds and Bdb.

4 .6 b-no rma l h o m o m o r p h i s m . The interior of the boundary hypersurface, Bah, of X~ is the product (M\H) 2. The blow-ups used to construct X~, viewed from this face, correspond to the blow-ups needed to construct the 'overblown' b-double product of M with itself. Recall that for any manifold with boundary Y, the b-double product Y~ is the blow-up of y2 along those components of (0Y) 2 intersecting the diagonal (i.e. the components


of the corner meeting the diagonal of the boundary). The overblown b- double product Yo2b is the result of blowing up the full corner including the components disjoint from the diagonal:

y s = (0v) .

These two notions obviously coincide whenever OY is connected. The space of overblown b-pseudodifferential operators tI, ob (Y, E, F) is associated to Yo2b in the same way that the space of b-pseudodifferential operators ~I'~'~(Y; E, F) considered in [Mel] is associated to Yb 2. Of course, it is slightly larger because the additional blow-ups in the definition of Y2 b allow these kernels to have additional singularities at the off-diagonal corners. By definition these singularities are resolved by lifting to Yo2b, and the mapping

*~T �9 r and composit ion properties for ffYob (Y, E, ) are essentially the same as *'~(Y; E, F) , which are explained in [Mel]. those for tI/b

In fact, restricting the kernel of an s-pseudodifferential operator A of order m to Bdb yields an element of

1 m (8s) Plb Prb I (~oob , Ab; �89 2 r - - 2 db(X )) + Bfb,4_ (Mob,

Here Plb and Prb are defining functions for the two non-front boundary hy- - - 2 persurfaces of Mob and Bfb denotes partial smoothness up to the front

faces, those defined by the blow-up. Dividing by the canonical factor [de[�89 reduces the restricted half-density bundle to f~�89 as in w

~T'm'T(M" E) as expected. This agrees with (87). That Nb so Nb(A) 6 Zob ~ , is a homomorph ism in general follows from the composit ion formula for s-pseudodifferential operators below, but the identi ty

Nb(P. A) = Nb(P) " Nb(A) (89)

when P 6 Diff*(X) follows easily from (58), and is the only case actually used here.

4.7 S u r g e r y n o r m a l h o m o m o r p h i s m . The second normal homomorphism, N~ at Bds, is of a very similar nature. First note that Bds is naturally the overblown b-double product of the boundary face H of X~ with itself:

--2 (90) Bds ---- Hob.

To see this consider the diagonal map M 2 • [0, 1] --* (M x [0, 1]) ~ sending (w, w', e) to (w, e, w', e). This lifts to a map X 2 --* (X~)2, the image of which is the fibre-diagonal ~'~ of the map (2.2) (i.e. the set of pairs (p, q) E (X~) 2


such that ~rs,~(p) -- ~rs,,(q)). The boundary of 9v8 is the product of the singular fibre OX~ = Bbb t3 Bss with itself, hence is the union of the four possible two-fold products of these individual faces with each other. The product (Bss) 2 is precisely the 'square' we are looking for: it is the image of Bd8 with respect to the lifted diagonal map. The restriction of this map to Bds is the blow-down map for (64) (and indeed, its restriction to Bdb is the blow-down map for Bdb = (M)2b). It should be noted that ~'s 'is not a manifold with corners. Indeed, it has conic varieties, the images of the overblown b-front faces of Bds and Bdb, at which however the cross-sections are manifolds with corners. Recall that H = B~, (X~) is the compactification of the normal bundle to H in M, hence in particular fibres over H with semi-circular fibres. Its interior has an alternate structure as a principal R+-bundle over H, which is seen as follows. Consider the vector fields in 128(X~) which are required to lie tangent to these fibres of B~s at this face. This set is closed under Lie bracket, and has a natural ideal consisting of all vector fields in V~(Xs) which vanish identically at Bs~. The quotient of these two spaces is one-dimensional on each fibre, and inherits the quotient Lie algebra structure, hence may be identified with the Lie algebra of R +. Exponentiation produces the fibrewise group structure.

The fact that

N , : ~ y ' r ( M ; E, F) m,r - - --* 905 (g ; E , F ) (91)

now follows exactly as before, as does the formula

N~(P. A) = N~(P) . N~(A) (92)

when P e Diff*(M; F).

4.8 Compat ib i l i ty . Directly from the definitions it follows that these normal homomorphisms satisfy a compatibility condition. Since the normal operators Ns(A) and Nb(A) are overblown b-pseudodifferential operators they have normal operators corresponding to restriction of the kernels to

- - 2 ~ 2 the front face(s) of Hob and Mob respectively. The normal operator of a b-pseudodifferential operator at the front face of the b-double product is called (see [Mel], [Me2]) the indicial operator, and we continue to use this terminology here in the overblown context. The compatibility we refer to is that the indicial operator of N~(A), computed at any one of the components of Bds NBdb, is equal to the indicial operator of Nb(A) computed at the same component. There is a consistency between the s-symbol computed along A~ and then restricted to either Bds or Bdb and the b-symbol of Ns(A) or Nb(A). These are the only compatibility conditions amongst the symbol and the two normal homomorphisms.


4.9 T h e t r i p l e s u r g e r y space . The fundamental step in the proof that the composition of two surgery operators is an operator of the same type is the introduction of the surgery triple product X 3. The construction is done in [Mc]; for completeness we recall it here. The triple space is obtained by blowing up a sequence of submanifolds in M 3 x [0, eo]. Three copies of M are needed because the composition of two operators requires three sets of variables. The submanifolds to be blown up are all possible threefold products of H and M, at e = 0, in order of increasing dimension (with M 3 x {0} omitted since blowing it up makes no difference):

X~= [M3x[O, eo];H3x{O};H2xMx{O};HxMxHx{O}; MxH2x{O};HxM2x{O};MxHxMx{O};M2xHx{O}]. (93)

To see that this manifold is well defined we shall use Lemma I repeatedly. The first two blow-ups are certainly sanctioned by Lemma 1 since

H 3 x { 0 } C H 2 x M x { 0 } C M 3x[0,e0] (94)

is a chain of p-submanifolds. There are three chains of this type, the other two being obtained by replacing the middle space by its cyclic images H x M x H x {0} and M x H 2 x {0}, which are the fourth and fifth spaces in (93). In fact under the blow-up of H 3 x {0} these three spaces lift to be disjoint since they only meet, in pairs, at H 3 x {0} and their images in the normal bundle to this space are independent. Thus we can in fact carry out the first four blow-ups in (93), so the manifold

[M3x[O, eo];H3x {0};g2 x M x {O};HxMxHx { 0 } ; M x H 2 x {0}] (95)

is well defined; the last three blow-ups can be done in any order. The chain (94) can be lengthened to the chain of p-submanifolds

H 3 x {0} C U 2 x M x {0} C g x M 2 x {0} C M 3 x [0, e0] (96)

and there are five other chains related cyclically to this one. Locally on the space in (95) the last three blow-ups in (91) just reduce to the space defined by one of these six chains, in fact on lifting to (95) the last three spaces in (91) are disjoint, so these blow-ups can also be performed in any order.

Thus X~ is well defined by (93); l e t / ~ : X 3 --* M 3 x [0,e0] be the overall blow-down map. In (93) consider the fourth, fifth and sixth spaces blown up. In M 3 x [0, e0] the first two of these submanifolds meet at H a x {0} and as soon as it has been blown up they become disjoint. Thus the order of their blow-up can be exchanged. The two spaces M x H 2 x [0, e0] and M x H x M x [0, e0] form a chain, so they too can be reordered giving

X~ : [M3x[O, eo];H3x{O};H2xMx{O};HxMxHx{O}; (97)

H x M 2 x {0}; M x H x M x {0}; M x H 2 x {0}; M 2 x H x {0)] .

44 R.R. M A Z Z E O A N D R.B. M E L R O S E G A F A

The same argument can then be applied to the third, fourth and fifth spaces in (97) so

X3=- [M3x[O, eo];H3x{O};g2xMx{O};HxM2•

M x H x M x { O } ; H x M x H x { O } ; M x H 2 x { O } ; M 2 x H x { O } ] " (98)

The third and fourth spaces here are disjoint after the second blow-up and the first three elements form a chain as do the first two and the fourth. Thus Lemma I allows the first space to be moved to fourth position, giving the alternate characterization of the triple product as

X 3 = [M3x[O, eo];H2xMx{O};HxM2x{O};MxHxMx{O}; (99)

H3• {0}; H x M x H • {0}; g x H 2 x {0}; M 2 x H x {0}] .

The first three blow-ups now take place in the first two factors of M (and [0, e0]) and correspond to the definition of the double space on these factors. Thus (99) shows that there is a stretched projection from X 3 to X 2 obtained by composition of the blow-down map for the last four blow-ups in (99) with projection off the last factor of M. By the cyclic symmetry of the construction there are in fact three such maps

3 3 2 , (100) 7cs, O : X s --~X s , O = g , m , r

covering the three projections 7r~) : m 3 • [0, e0] ---* m 2 • [0, e0], O = g, m, r, where the second suffix stands for left, middle and right corresponding to which of the three factors of M is dropped. Thus the discussion above is for the right projection. Directly from the definition all three of these stretched projections are b-maps; they are in fact b-fibrations. To see this we need to discuss the geometry 9f the double and triple spaces a little more.

In particular it is convenient to introduce a labelling system for the boundary hypersurfaces of X 2 and X 3. Observe from (93) that each boundary hypersurface of X 3 is either the lift of a boundary hypersurface of M 3 x [0, e0] or else is the front face produced by the blow up of the product of one, two or three factors of H inside e = 0. We shall denoted the boundary hypersurfaces introduced by blow-up as Bijk where i, j , k = 0, 1 with 1 indicating a factor of H in the blow-up and 0 a factor of M. Thus B l l l is the boundary face introduced by the first blow-up in (93), i.e. of H 3 x {0} whereas B001 corresponds to the last blow-up. It is then natural to have B 0 0 0 be the lift of the original boundary e = 0. The same systematic notation can be applied to the boundary faces, above e = 0, of X 2. In terms of the previous notation for the boundary hypersurfaces of X 2

Bl l = Bds , B10 = Bls , Bol = Brs and Boo = Bdb �9

Vol.5, 1995 A N A L Y T I C S U R G E R Y AND T H E ETA INVARIANT 4 5

The lifts of the Bij with respect to the maps ~r~, o can then be computed easily:

(~3 ~*/B..~ s,~l ~ z.7] = Boi j U B l i j

71.3 )*(Bij) - = Bijo U Bijl for i , j = 0, 1 (101)

(~'3,m)*(Bij) = Bioj U Bilj ,

From this it follows immediately that each of these stretched projections is b-normM. The exponent matrices may also be calculated from knowledge of their b-differentials and from (101). So, for example

{ l i f i = ~ , j = 7 , eTr~, t ( B i j , B o ~ - ~ ) = 0 otherwise ,

(102)

with analogous results for 7~, m3 and 7r~,~.3 Observe also that null(e,~, o) is

empty for each O. We shall be referring to index families and multiweights for the collections of faces of Xff and X~ using this notation; if s is, for example, an index family for the faces of X~, then we set Eij = s

It remains to see that the stretched projections are b-submersions. For b 2 this it suffices to check that any element v E TqX~ is the value at q of the

projection from X~ • M of an element of Vb(X~ • M) which is tangent to the lifts of the four last submanifolds in (99). Since the second and third last submanifolds are disjoint as soon as the fourth last is blown up and are interchanged by a symmetry preserved by the stretched projection the second last space can be dropped. Thus if suffices to show that v E bTqXs is the value at q of an element of Vb(X~ • M) which is also tangent to the lift to this space of the chain of submanifolds of M 3 • [0, e0]

H 3 x {0} C H x M x H x {0} C M 2 x H • {0}. (103)

This can be checked directly. Thus all three stretched projections are b-fibrations. The projection

3 M 3 [0, e0]--~ 3 3 [0, e0]. 7r~ : • [0, e0] also lifts to a b-fibration 7r~,~ : X~ -*

4.10 C o m p o s i t i o n . If v(2 ) is a fixed nonvanishing smooth half-density on X~, so that the Schwartz kernel of D E ff~*'~(M; E, F ) is associated to

1 2 the section ~(D)v(2) of Hom~ (E, F) | ~ ~ (X~), then the composition of two operators C = A �9 B has Schwartz kernel associated to

~(C)u(2) 3 -- (~,,m)* ((~,e)*(~(A)v(2))(~,~)* (~( B)v(2))(~,r Idel-�89 �9 (104)

The composition formula

�9 y '~ (M;E ,G) . ~ " T ( M ; G , F ) c k ~ (105)


is thus proved by first computing the lifts of the kernels of A and B via 3 7r8,s

and r,,~,3 taking the product of these lifts, and then computing the push forward of the result via rs, m.3 Since kernels of surgery pseudodifferential

1 2 operators are sections of f~ (X~), the product in (104) has an extra factor of the canonical partial density [de[�89 which must then be divided out.

We will prove (105) only when m = mr = - ~ and all bundles are trivial. Removing the latter restriction requires only a change of notation. For the former we observe that the behaviour of the conormal singularities of ~r and ~r along As under the pull-backs, multiplication and push- forwards in (104) is essentially the same as for composition of ordinary pseudodifferential operators. This is explained in [Me2] and the appendix to [EMeMen].

The first step is to transform the density factors in (104) somewhat so that (36) can be applied directly. Multiply both sides of (104) by v(2), so that the left hand side becomes g(C)v~2), a section of the full density bundle on X 2. On the right side this half-density factor is put inside the push-forward, yielding a factor (Tr3,m)*(v(2)) inside the parentheses. The resulting product of partial densities

(71.3 ~* 71.3 3 * 3 * (u(2)) Idel-�89 (106)

is a full density on X~. We now compare (106) with a smooth nonvanishing section u~3 ) of f~(X~)

by transferring the product of half-densities (106) to one on M 3 x [0, e0], computing there and pulling back by f13. If #(2) and #(3) are smooth nonvanishing half densities on M S x [t3, e0] and M 3 x [0, e0], respectively, and Pij and Pijk are defining functions for the faces Bij and Bijk of X~ and X 3 (which will also be regarded as singular functions on the unblown-up spaces) then

2 * (,(2)) = Pll(Pl0 P01)�89 3 1 3 * ~___p~

(f~s) ( ~ ( 3 ) ) 1 1 1 P 1 1 0 P 1 0 1 P 0 1 1 ( P 1 0 0 P 0 1 0 P 0 0 1 ) ~ V(3) �9

(107)

Replace each u(2)in (106) by p l ? ( P l 0 P01) - � 8 9 (~2)*(~(2)), and u s e the identity t32 " %,o3 _- ~r~. fl~ to exchange each factor (~r~,o)*(fl 2) #(2) by (/~)* (7r~)*(~(2)). Finally observe that, up to a nonvanishing smooth mul- tiple,

(r~)*(#(2))(Tr~)*(#(2))(Tr~)*(#(2))(Tr~)*[de[-�89 = l~3 ) . (108)

Vol.5, 1995 A N A L Y T I C S U R G E R Y AND T H E E T A I N V A R I A N T 47

Combining (101), (102), (107) and (108) we see that (106) is simply v~3 ). Hence

ff ) = ( 8,m)* ((Tr3,e)*(n(A))(~r3,,')*(n(B))v~3)) �9 (109)

The final change is to divide both sides by e; this has the simple effect of replacing v~2 ) and v~3 ) by smooth nonvanishing b-densities because ~(Tr 28,~j]*,(e],

is a product of defining functions for the faces of Xy and similarly for the triple product.

Because of the decomposition (79) of ~ ' ~ ' r ( U ) , it suffices to prove that the summands ~ - ~ ( M ) (the small calculus) and ~ - ~ ' r ( U ) may be composed with one another. The simplest instance is when both A and B are in the small calculus. This means that both kernels g(A) and g(B) are polyhomogeneous, with index families E and .~, respectively. Here C = ~" and El l = E00 = 0 and M1 other index sets are empty. We now compute

3 * (7/-3 )* (Try,t) (C) = C~, ~ ~,rJ (~-) = J ' r , the product G of these two lifts, and the

push-forwaxd 7-/ 3 =

According to (31), (101) and (102) we have

# o l l = E f H = E ooo = E foo = F h o = F ; , , = = = 0

and all other E/tjk and Fi'~k empty. Now apply (28) to conclude that Gi l l = G000 = 0 with all other Gijk = 0 (recall that E + O = 0 for any index set E). Finally, using (102), apply (36) to (109) to conclude that the pushed-forward index family 7-/coincides with E and ~ above.

B ~ r / X 2 ~ The opposite case, when n(A), n(B) e dB-~_[ ~) is handled quite similarly. Since these kernels are partially smooth up to order r at Bd~ and Bdb and conormal of order r at all other faces, the index families E = 9 F are exactly the same as above, and the multiweights ~ for n(A) and t and n(B) are equal and take the value r at all faces. The pull-backs of a(A) and g(B) axe partially smooth on X 3. They have index families C t and ~'~ as above and multiweights taking the value r at all faces of X 3. Using (28) their product is partially smooth up to the faces BH1 and B0oo up to order T, hence has index family G as above and has multiweight u taking values greater than or equal to r at all faces (in fact, it takes the value T at all faces except B101 and B010 where it takes the value 2T). Finally, the push-forwaxd is again partially smooth with index family 7-/equal to E -- 5 c and multiweight u = 5 = t. Hence

~ - ~ ' ~ ( M ) . ~ 2 ~ ' ~ ( M ) C k0:~ '~(M) �9 (110)

Finally, a very similar argument shows that

@~-~(M). ~ - ~ ' ~ ( M ) C ~ - ~ ' r ( M )

4 8 R .R . M A Z Z E O A N D R.B. M E L R O S E C, AFA

as desired, and similarly with the factors on the left interchanged. In w below the polyhomogeneous form of the surgery calculus is introduced. In this context there is also a composition formula (190), so that the polyhomogeneous surgery calculus is also closed under composition. This is proved in the same way as the result above; the only difference is that none of the index sets for ~(A) and n(B) are empty.

4.11 U n i f o r m l y f in i te r a n k o p e r a t o r s . The projector onto the small eigenvalues of the Laplacian is an operator depending on e and of uniformly finite rank as e ~ 0. The kernels of such operators in the calculus are the products of lifts of functions from X~. In the case which occurs below uj,

vj E/~dBAS(X~; f~]) for j = 1 , . . . , N vanish at Bse. Then the kernel

N

~-~(Tr ~ ~*u. 2 �9 2 �9 _x ffg_o~,~(M, fl�89 (111)

j = l

defines an operator uniformly of rank (at most) N with vanishing surgery normal operator. The case of general bundles is similar. The determinant of Id - A, when A is of this type, is a conormal function in e. Clearly the operators of rank N form a two-sided ideal in the general calculus (for the same r).

4 .12 S e m i - i d e a l p r o p e r t y . One other property of the usual pseudodifferential calculus that is useful to generMize to the surgery calculus is the fact that on a closed manifold the smoothing operators form a semi-ideal in the algebra of L2-bounded operators. That is,

~ - ~ ( M ) . ~(L2(M;f] �89 . t~-~ c ~ - ~ , (112)

provided M is compact without boundary. (The terminology 'semi-ideal' is used in [Sz]; sometimes the terms 'bi-ideal' or 'corner' are used to de- scribe the same property.) The generalization here concerns the algebra of bounded operators on L2(X~), with respect to a nonvanishing density on X~, which depend parametrically and conormally on e:

B �9 ~(L~(X~)) , [e,B] = O, and

\ /

Clearly ~r,r~8(M ) C s is a subalgebra, but more is true.

PROPOSITION 1. I / ' r > O, ~,res(M) is a semi-ideal in the space s

V o l . 5 , 1 9 9 5 A N A L Y T I C S U R G E R Y A N D T H E E T A I N V A R I A N T 49

Proof: To simplify notation let us prove the result for operators on half- densities. Fix 0 < v E C~ f~ �89 and let /5~ e C-~176 be the Dirac delta measure at z E M. Using v to convert this to a half-density, lifting to M x (0, eo] and multiplying by the canonical factor Idel�89 we obtain the family of e-independent half-densities

M 9 z, , ,%v-lldel �89

which we regard as a map. Consider its lift to X~. Then for any T > 0 we shall show that

Mgz~T(z )=%,b*( (x2+e2) �89189189189 , 1 dim M + 1 k > ~

(114)

is continuous. This is certainly the case locally in e > 0, and also away from the surgery face of X~. Near Bs~ the question is essentially two dimensional, since the manifold is locally a product there. Writing z = (x', y') and w = (x, y) and using the polar coordinates where x = r cos0, e = r sin0 one finds directly that

2 1 l 1 T(z)(w) e~-�89 + e )~6(x x')5(y = - _ y )ldxdedy]~

= ar~+l(sina)~-�89 cos8 - x')6(y - y')v

where a is smooth. This demonstrates (114) including continuity. The same regularity is true after applying any power of e. d/de, as T depends on e only

through the factors (x2+e2)�89 and e~-�89 and these are certainly conormal in T �9 1 e. Thus i f A 9 ~ , , e s ( M , ~t~) for T > 0 then M 9 z ~ AT(z ) E L2(X4 ~ �89

(in fact, it is in H~~ f~�89 If I ( E s ~�89 and B E qY~,res(M; f~�89 it follows that

z , , e - 2 " B K A T ( z ) E H~~189 V T' < T . (115)

Indeed, it certainly takes values in e2"'L2(X~; f~�89 and remains in this space

after the action of Dlff~ (M; f~ ~ ) and any power of e. d/de. However the space

on the right in (115) consists of half-densities of the form e-�89 where v is continuous on Xs and # is a non-vanishing smooth half-density on Xs. It therefore follows that the Schwartz kernel of B K A is of the form

| ld l-�89


where K is continuous on X8 x M and T' < Z. Lifting to X~ shows that the kernel is the product of e 2r' and a continuous section of the kernel density bundle. Since this regularity is clearly stable under the repeated action of e. d/de, and of ])s(Xs) lifted from either the right or the left, and taken together these vector fields span ]?b(X 2) it follows that

r . 1 r . 1 2 r . O,, ,e,(M, a ~ ) . / : c ( M ) . ~ , , r e s (M,n~) C ~ , r r E) . (116)

This implies the semi-ideal property. Since the argument is local in nature it applies equally well in the case of operators on sections of a general smooth vector bundle E over X, .

COROLLARY i. If R E @~,rr E) for v > 0 then if Co > 0 is taken small enough Id - R is invertible on L2(Xs; E) and the inverse is of the form I d - S, with S ~ ~2,~(M; E).

4.13 T r a c e f u n c t i o n a l . Since they are smoothing operators on M for e > 0, the elements of f f l / ~ , ' ( M ; E) are of trace class. By Lidsky's theorem the trace is the integral over the diagonal of the point wise trace of the kernel, which can be interpreted as a density. The bundle over X 2 in (76) restricted to the lifted diagonal reduces to:

Hom~(E) | a �89 ~ - hom(E) @ a(X~) . (117)

Thus the trace of A E q~-~ ' r (M; E) is, as a function of e, the push-forward to [0, col of a density

(tr A) W,, E C ~ (X~; f~) + Bss,bb ,,4 ~" (Zs; ~) ( 1 1 8 )

corresponding to (79). Here tr is the trace functional on horn(E). The general push-forward results of [Me2] or Theorem 5 of [Me3] as

noted in (36) above therefore show that

T r : ~ - ~ 1 7 6 --+ g~([0 , e0]) + logeg~ e0])+ A : ( [ 0 , e0]) . (119)

Since T > 0 the first two terms dominate and (119) shows tha t

Tr(A) = rA(e) + loger'A(e) (120)

where rA, r~4 are each the sum of a C ~176 te rm and an element of ,4=([0, e0]). The logari thmic term comes from the fact that the two boundary hypersurfaces, M and H, bo th map into {e = 0} under r , . Recall from [Mel]


that the b-trace functional is defined on the space of b-pseudodifferential operators of order - c~ . This functional depends on the choice of a trivialization, L,, of the normal bundle to the boundary of the compact manifold with boundary on which the operators act, i.e. a choice of boundary defining function up to quadratic terms. In this case the function e is a product of defining functions of the two boundary hypersurfaces, so a trivialization of the normal bundle to either gives a trivialization of the normal bundle to the other. Using such consistent trivializations allows us to deduce, from the general results of [Me2], a simple formula for the leading terms in (120):

LEMMA 2. For any T > 0 the trace functional (119) satis~es (120) with

r~4(0 ) = f ~ ( t r A ) ~

rA(O) = b-Tr~(N~(A)) + b-Tr,(Nb(A)) . (121)

In the main case of interest here only the last of these three terms is non-zero, as is shown in w

5. U n i f o r m S t r u c t u r e of t h e R e s o l v e n t

Next we use the surgery calculus to analyze the resolvent family for the operator AE,~ = ~2 uniformly as the parameter e I 0. The parametrix construction in this section is closely related to that in [Mc], although the case treated there involves degeneration to an incomplete conic metric whereas here we are concerned with degeneration to a complete metric. Although AE,~ is a singular perturbation of AE,0 the spectrum does not move far, although it changes in character.

5.1 R e s o l v e n t s o f t h e n o r m a l o p e r a t o r s . Recall from [Mel] the structure of the spectrum of the two normal operators, AE,0 -- 23~M and ~-__, which are both b-differential operators associated to exact b-metrics. T~e compatibility condition (63) shows that they have the same indicial family s 2 + 32. Since they are formally positive the spectrum of these operators

H

consists, in principle, of finite discrete spectrum in [0, #2) and continuous spectrum occupying [p02, ~ ) where #5 is the least eigenvalue of ~i~, and pos-

sibly containing embedded discrete spectrum. The surgery normal operator is R+-invariant on H so this operator has no discrete spectrum at all. These facts are established as follows. First note that, by virtue of its

•+-invariance, ~ is canonically identified with I(AE,0) when _~ is dis- connected, and to I(AE,O) acting on certain Z2-invariant functions if not.


In either case, its behaviour is determined directly by that of I ( A ] E , 0 ) , and this indicia] operator can be analyzed simply by use of the Mellin transform. This analysis shows that, because of the hypothesis (2), I (AE,o) -- ~ is an isomorphism on L2(R • H; E) for )~ E C\[A 2, oo), where A0 is the element of the spectrum of the induced Dirac operator on H with smallest norm. The analysis of AE, 0 -- ~ itself is based on the invertibility of I(AE,0) -- )~ using the calculus of b-pseudodifferential operators in [Me2], [Mel], [MeMen], cf. also [M]. The conclusion is that A~E, 0 -- A is Fredholm on L2(dgo) whenever I ( A E , o ) - ~ is invertible on L2(H). The (generalized)inverse for AE,o --~ is constructed starting from the expectation that the inverse for I (AE,o) -- )~ should be its indicial family. There might be a finite number of nonzero eigenvalues of AE,O in [0, #02). We therefore choose 6 < #2 so that the only (possible) eigenvalue in [-5,~] is 0.

- - 2 The kernels of these resolvents are conormal distributions on Hob and Mo2b, corresponding to the (overblown) b-pseudodifferential calculus, and depend holomorphically on A. Thus, for A in any bounded open set ~ C C\[#02, oo), there exists a T > 0 such that

,T,--2,T r E) : , , , e , - , , ( 1 2 2 )

- - 2 , r - - f ~ 3 A , , (AE,0--A) -1 E~ob (M;E)

are holomorphic. This is discussed in some detail and proved in [Mel] and [Me2]. There is a refinement of (122), namely these resolvents actually lie in the polyhomogeneous conormal b-calculus. This is discussed in w below.

5.2 A w a y f r o m t h e s p e c t r u m . First we consider a compact region of the resolvent set of the limiting operator AE,0 82 =" E , 0 "

PROPOSITION 2. If ~ C C is an open bounded set with closure disjoint from the spectrum of A E,O then for some 7- :> 0 and eo > 0 the resolvent of A E,c is a holomorphic map

f~ 3 A, , Res(A) E ~-2 ' r E ) . (123)

Proof: If Res(~) is the putative resolvent then

(AE,~ -- A) Res(A) = Id . (124)

Applying the symbol map and the normM homomorphism to this resolvent identity it follows that if (124) is to hold then

"a2(AE,~). 'a_2 (Res()~)) = I d ,

N~(AE,~). N~ (Res(s = I d , (125)

N b ( A E r Nb(Res(,k)) = Id .


The first of these has the solution

--~ O'2(AE,e) --~ [~]-2Id (126)

since the principal symbol of AE,e is given by the metric. Here [. ]2 is the metric on the fibres of sT*X~. The other two equations are simply the resolvent equations for the normal operators of AE, e.

By assumption the spectral parameter is away from the spectrum of AE,O = Nb(AE,~) . Using (122)

Nb(aes(A)) = (AE,o -- A ) - ' e ~b-2'~ (M; E) �9 (127)

Furthermore the symbol of (AE,0 -- A) -1 is just the restriction of (126) to M. Similarly the other equation in (125) is just the resolvent identity on

i .e . for N s ( n . , o ) = so

~ A I ~ N s (Res(A)) = ( ~ H - A) - I e l I / [2 'V(H; E ) (128)

is also holomorphic with the same T. The indicial operator of this family is the resolvent family for 8 2. Since this is also true of (127) the two normal

H operators are compatible in the sense of (63) and are compatible with the symbol homomorphism.

Thus there exists a family of operators, which can be taken holomorphic in f~:

~ A, , E ' ( A ) ~ ~ : 2 ' T ( M ; E )

satisfying

~a(E ' (A) ) = ]~l-2Id, N b ( E ' ( A ) ) = (AE,0--A)-I , N ~ ( E ' ( A ) ) = (l~ti-t~-A)-l . (129)

This means that E'(A) is already a parametrix for the resolvent family. However it is useful to modify it and so get a more precise parametrix. By standard symbolic arguments there is a holomorphic (in fact entire) family Go(A) E ~I/~-2(M; E) of the small calculus which inverts AE,e -- A modulo a holomorphic family:

(AE,~ -- A). G0(A) = I d - R0(A) , R0(A) E ~ ' ~ 1 7 6 (130)

This family is unique modulo a family in ~ - ~ 1 7 6 E), as follows from the construction. In fact the normal operators of G0(A) must be equal to the resolvents modulo terms of order -oo . It follows that there is a correction term eG~(A) E ~-2(M; E) such that the modification of the parametrix to E = E ~ - eGto still satisfies (129) and is a parametrix in the strong sense that

T . ' �9 (AE,~ -- A). E(A) = Id - R(A) , R(A) e kgs,res(M, E) A e ~ (131)


The final step in the proof is to use Corollary 1 to show that Id - R() 0 is invertible, if eo is taken small enough. Then the resolvent family itself is

Res(A) = E ( A ) ( I d - R ( A ) ) - 1 . (132)

5.3 N e a r t h e d i s c r e t e s p e c t r u m . Next we consider the modifications needed to treat the resolvent near the point spectrum of the limiting operator AE,0. In particular we are most interested in the case that 0 is an eigenvalue of AE,o.

In this case the resolvent has a simple pole at )~ = 0:

(AE, 0 -- )~) = Reso(A) - )~-lH0 near ~ = 0 (133)

where H0 is the orthogonal projection onto the null space of AE,O, which is of rank N > 0 and Res0(A) is a parametrix up to finite rank. Thus

N

(134) j = l

The basis, r E A~_(M; E), of the null space consists of conormal sections

and ~ is the metric density on M. In fact these elements are polyhomogeneous conormal, as we discuss in w but in any case v > 0 because of (2). This means that each r has an extension to 0j E /~bbAr--(Xs; E). Let n ' be the uniformly finite rank operator

N

H ' = E C J "~J~" (135) j = l

has normal operators Nb(n') = no T h u s I I ' E s ~ , The consistency condition between the indicial families of (A 9 - A)-I

and Reso(s still holds (cf. [Me1]) so we can find an element Res'(A) E ~-2'r(M" E) with them as its normal operators. Then

(AE, e -- A)Res'(A) = I d - I I ' - R'(A) (136)

where R'(:~) has zero normal operators, i.e. is residual. Hence Id - R' is invertible if e0 > 0 is small enough and has inverse Id - S with S similarly residual. From (136)

(AE,~ -- A)Res"(A) = Id - H,(A) , (137) aes"(A) = R e s ' ( A ) ( I d - S(A)) , III(A) = H ' ( I d - S(A)) .

In particular the remainder term Hi(A) is, for A near 0, uniformly of rank exactly N and is holomorphic in A.


The null space of I d - H 1 as an operator on L2(X~; E) is contained in the span of the Cj. The determinant of the N x N matrix of conormal functions

(hjk - aj,~) where IIlCk = ~'-~ajkr J

is therefore holomorphic in A near )~ = 0 and vanishes exactly at the eigenvalues of AE,~. Thus the eigenvalues are the zeros of a function

q(e, a) , where q(O,.X) = A N , (138)

with q(e, ~) holomorphic in A for e fixed and the sum of a smooth term and a term in AL([O, eo]) as a function of e. When e E [0, e0] is fixed, the fimction q(e, )~) has exactly N zeros counted with multiplicity, and these are precisely the small eigenvalues of AE,e. As such they are nonnegative. The orthogonal projector H~ onto the sum of the corresponding eigenspaces can be recovered by integrating the resolvent Res()~) around a small loop r disjoint from the spectrum of AE,~ and containing all zeros of q for e < e0 which tend to zero as e --~ 0 but no others. The existence and structure of the resolvent on a neighbourhood of such a loop follows from the results above, so there exists some r > 0 such that Res()~) E ~22'~(M; E) on F. The contour integral preserves this space, and removes the singularity along A~. It follows that

II e ~s -~ '~,(M, E) . (139)

The final step is to meromorphically extend the resolvent inside the loop P. Of course this must be interpreted suitably. The difficulty is that the poles of Res(A) vary with e, but the meromorphic structure of the resolvent for each e is the same as that of ( I - II1) -1, if this inverse exists. By our previous observations, since HI is uniformly of rank N and de t ( I - II1) = p(A, e) is not identically zero, the inverse ( I -111) -1 = 1 _ T is meromorphic in A and smooth in e. More specifically, for each value of e this inverse exists as a meromorphic function of A with poles at the eigenvalues of AE,e. It is also smooth in e in the sense that p(A; e)T has only removable singularities and is a uniformly finite rank surgery operator, as follows from the algebraic construction of this inverse. The resolvent family is

Res(A) = ae s " (a ) - Res"(A)T. (140)

The second term on the right is uniformly finite rank and meromorphic in )~ with values in ff2j~176 E) in the same sense as T, while the first term is holomorphic in )~.


6. T h e Heat Surgery Calcu lus

In the next section the fundamental solution, exp(--tAE,~), to the heat equation is analyzed uniformly as e ~ 0. As in w and w the me thod is to define a class of operators, in this case the heat surgery calculus, and using its properties show, quite constructively, tha t the heat kernel lies in it. The calculus is fixed by specifying the Schwartz kernels of its elements in terms of their lifts to a blown-up version of the space M 2 x [0, 1]~ x [0, oc)t.

6.1 D o u b l e h e a t s u r g e r y space , X~l. The space X2s is defined in two stages. First M 2 x [0, 1], is replaced by the surgery double space, X~ 2. Then in X~ 2 x [0, c~)t the submanifold A~ x {0} is blown up, but parabolically. The notion of parabolic blow up of a submanifold Y with respect to a subbundle S C N*Y is described in general in [EMeMen] and [EMe]; see [Mel, Chapter 7] for the analogous construction for the b-calculus. If, as in this case, the submanifold is contained in the boundary then conditions are placed on S but these are satisfied by S = Span{tit}, the conormal bundle to the boundary. Thus, in the notat ion of [EMe],

X2s= [X 2 x [0, oc); As x {0},S] (141)

is a well-defined manifold with corners, equipped with a blow-down map

/~h : X~s --~ X~ x [0, cx~). (142)

The overall blow-down map is then

/3hS: X~s --* M 2 x [0, 1] x [0, oo) . (143)

6.2 B o u n d a r y h y p e r s u r f a c e s . There are two types of boundary hypersurfaces of X 2 �9 those lying above {t = 0} and those arising from the lifts hs~ of the boundary hypersurfaces of X 2.

The two boundary hypersurfaces lying above {t = 0} are the temporal front face, Btf, produced by the blow up and the lift, Btb, of {t = 0} which we call the temporal boundary. These two faces can be described diffeomorphically by

~TX~ r B t f ( X 2 s ) fib8 /k s ~- X s (144) Btb(X ) 2. =

Thus Btf(Xt~s) is a closed ball bundle over X~, with interior canonically isomorphic to ~TXs; in other words it is a natural compactification of ~TX~, the structure bundle.

The four boundary hypersurfaces arising from the boundary hypersurfaces, Bg~(X2~), of X 2 will be denoted Bgf(X~s ) (or simply Bgr if no confu- sion is likely). Two of them involve parabolic blow-up but the other two, which do not mee t /k~ , are simply products.


B~(X~) = [B~(X~) • [0 ,~) ; A ~ • (0}, S]

B~b(X~s) = [B~b(X~) • [ 0 , ~ ) ; A ~ • {o} , s ] (145)

Bls(X2s)---- Bls(X 2) • [0, oc)

Br~(X~s) = B~s(X~) • [o, ~) where S -- Span{dt} and Agf = A s N Bgf(X~). It is important to note that Btf(X~s ) only meets B t b ( X ~ ) , B d b ( X ~ ) and Bas (X~) amongst the boundary hypersurfaces. This is directly reflected in the structure of the calculus.

6.3 H e a t s u r g e r y o p e r a t o r s . The calculus of heat surgery operators is filtered by three orders which, for present purposes, can be taken to be integral (the first negative the second two non-negative). The kernels are normalized with respect to half-densities for which the 'kernel density bundle' over X ~ is

-�89 KDh~ = Ptf ~ : (X~) . (146)

If E is a C ~ vector bundle over X~ the normalized homomorphism bundle, over X 2 is defined to be hs ,

Homhs(E) =/3~Homs(g; F) (147)

where Homs(E ,F ) , is given by (75), has been lifted to X~ x [0, cx~). If E = f/�89 then HOmhs(E) is canonically trivial. For - k E N, l, m E No we set

f f t k , l , m ( l i / f . E ) -k I m ~ ~ oo ~ 2. KDhs) (148) = Ptf PdsPdbPtbPls Prs C (Xhs , HOmhs(E) | - - h s k ~'* ~

In case 1 = m = 0 the shorter notation

~hks(M;E) = ~k '~176 E) (149) hs ~ ;

will also be used.

6.4 H e a t h o m o m o r p h i s m . The normal operator at Btf, which we call the heat homomorphism, is common to all heat calculi. It is described in some detail in slightly different contexts in [Mel] and [DMe]. In the present case it is defined by division by t - W 2 and restriction to Btf. Manipulation of the density bundles reduces it to

ff, k,t,mf/t~r.E) c~ I m~'~ "hom(E) | ~ f i b r e ) (150) g h s . h k : =hs ~,"* : �9 , ~ PtbPdsPdb (, I tf, �9

In terms of Btf as the compactification of s T X ~ the rapid vanishing at Btb becomes rapid decay at infinity, on the fibres. Using S to denote the Schwartz space of a bundle, (150) can be rewritten

Nhs:h,k : ~:hsfftk'l'mtTt/f'E)\"*, ~ PdsPdbl m S / s T X ( s ; horn(E) | ~ f l b r e ) . (151)


This map is surjective. In fact Nhs:h,k just picks out the leading coefficient at Btf so

k - - l , l , m . k , l , m O ~ tYhs ( i , E ) ~ kVh, ( M , E )

g a . : h , k l m " ~ T X horn(E) | ~-~fibre) 0 P d s P d b O ( s ;

is a short exact sequence for each k, l, m. By repeated use of the heat homomorphism error terms can be arranged to lie in the residual space for the heat homomorphism:

- -oo, l m ~h~ ' (M; E) " d~176 ([0, cx~);~-~ �9

6.5 S u r g e r y a n d b - n o r m a l h o m o m o r p h i s m s . The other two normal homomorphisms, corresponding to restriction of kernels to Bds(X2s) and Bdb (X2s) are quite similar and will therefore be treated together. From the discussion of the geometry of X 2 in w both these spaces can be realized as overblown b-stretched double products. They therefore blow down onto the ordinary b-stretched double products, with only the off-diagonal front faces changed:

B d ~ ( X ~ ) - ~ - ~ - =Hob~H b , H = B ~ ( X , ) (152)

Bdb(X~) - - 2 - - 2 - - =Mob--*M b , M = B b b ( X ~ ) .

Since only faces away from the diagonal are involved here there are induced blow-down maps

Bds(XL) ~ [-g~ • [0, ~) ; a f , S] = --~H,

Bdb(X~s) ~ [M~ x [0, c~); AM, s] = --2M, (153)

where the spaces on the right, defined as the parabolic blow ups, are the b-heat spaces of [Mel, Chapter 7]. The definition, (146), shows that all kernels vanish to infinite order at the faces involved in the blow-down maps (153) so the normal operators become

Nh~:s : VL(M; E) -~ V,~(Y; E)

Nhs:b: ~hks(M;E) --* il~k(M; E) �9 (154)

~Tik,l,0 ( a/r. E) and ~jk,o,1 ( ~/r- E). The null spaces are, respectively, Xhs ~,,,, --hs x '" ,


6.6 C o m p a t i b i l i t y . Although the three maps in (149) and (154) are sep- axately surjective there axe compatibility conditions between their images. Since the kernels are expressed in terms of C ~ sections of a vector bundle these conditions just arise from the restrictions to the common boundary faces, of codimension two in X~s. Altogether then six maps are involved.

Two of these maps are the heat homomorphisms from the two b-heat calculi, for H and M

N,~:h,k: ~ ( H ; E) --~ S(bTH; horn(E) • ~fib) (155)

: E) S (bT ; h o r n ( E ) | �9

Since ~T-~Xs ~- bT'-H, ST-~Xs ~- bT-M the spaces on the right, for 1 -- m = 0, can be identified with the restriction of the image space in (149) to the two boundary hypersurfaces of Xs. Thus the first two compatibility conditions are just

N, vh , k (A) t . d . = N,l:h,k(Nhs:8(A)) V A e ffYhk~(M;E). (156)

g , : h , k ( A ) r'db = N,:h ,k(Nhs:b(A))

The other compatibility condition arises from the indicial homomor- - - 2 - - 2

phisms of the two heat calculi. The front faces of M b and H b are canonically identified and in w are further identified with one of the front faces of ~2. Thus the two indicial homomorphism are

I : ^ --, ~ , j ( H ; E)

I : V~(M,E) --~ V~, , (H;E) . (157)

The third compatibility condition is then

I ( N h s : s ( A ) ) = I ( N h s : b ( A ) ) V A e ~ s ( M ; E ) . (158)

Moreover these are all the constraints on the range, so if

Ah �9 S ( ~ T X ~ ; h o m ( E ) | ~nbr~) , As e ~ k ( y ; E ) and Abe ~ ( M ; E )

satisfy

AhrBa. = N,7:h(As) , AhrBdb = N~:h(Ab) , I ( A s ) = I (Ab) (159)

there exists A �9 ~hks(M; E) with

Nhs:h,k(A) = Ah , Nhs:s(A) = As , Nhs:b(A) = Ab �9 (160)

It is also important to note that

A �9 ~ s ( M ; E ) , Nhs:8(A) = O, Nhs:b(A) = 0 -', ;- (161)

A �9 --h~r E) "" .'. A = e B , B �9 ~ h ~ ( M ; E ) .

This indeed is the way that the surgery and b-normal homomorphisms are used.


6.7 A c t i o n o f t h e o p e r a t o r s . So far the heat calculus has been defined abstractly. Under the blow-down of X~= to M 2 x [0, 1] x [0, c~) the kernels define operators from X, to X~ x [0, c~). These actually have the regularity property

--+ t - ~ C~([O,e~)�89 x X~;E) V A e ~ = ( M ; E ) (162) A : C ~ ( X s ; E )

�9 l

where [0, co)�89 is the closed half-line with t~ replacing t as the linear variable. The mapping property (162) suggests an explicit way in which the normal operators arise. Namely if u E g~ E) and A E ~ = ( M ; E)

Autt=o = ~(A) . (utt=o ) k = - 2

Aut- H = Nh~:~(A)(ut- ~) (163)

Au t u = Nh,:b(A)(u tM) '

where ~(A) is actually a mult iplication operator given in terms of the heat normal operator by

r = f Nh~:h,-2(A) e go~ (X~; horn(E)) . (164) J f i bre

6.8 C o m p o s i t i o n . The surgery heat calculus is a filtered algebra�9 The full force of this is not used here, as usual only the action of differential operators is needed. Suppose P E Diff2(M; E). Its symbol can be regarded as a translation-invariant differential operator on the fibres of ~TX, . Then

t ( O t + P ) . A e ~ = ( M ; E ) V A e ~ = ( M ; E ) (165)

and the normal operators of the composi te are easily computed (cf. [Me1, Chapter 7]):

Nh,:h,k(t(Ot + P) " A) = (*a2(P) 1 7(R + n + k + 2))Nh~:h,k(A)

Nh,:, (t(O, + P) . A) = t(Ot + N, (P) ) . Yh=:,(A) (166)

Nh,:b(t(Ot + P) - A) = t(Ot + Nb(P)) . Nh,:b(d)

where R is the radial vector field on ~TX, . The residual space, q - ~ 1 7 6 consists, by (162), of operators from C~(X~;E) into hs t C~176 c~) x X~; E). They can then easily be extended as convolution operators in t, so act on the space • E) . As such they behave as Volterra operators and

-o~,o~,~ . ,T,-~176176176176 r M" E) . Id - S , S E tX/hs (M, E) has inverse Id - S', S ' E --hs t ,

(167)

Vol.5, 1995 ANALYTIC SURGERY AND T H E ETA INVARIANT 61

7. U n i f o r m S t r u c t u r e o f t h e H e a t K e r n e l

In this section the structure of the heat kernel is analyzed uniformly as the parameter e I 0. First it is analyzed for finite times, and shown to lie in the calculus just described. Scaling arguments originating with Getzler [G] and developed in the context of b-metrics are then applied to analyze the special properties of the kernel near t = 0 for the Dirac operator. The finite-time analysis of the heat kernel is next used to extend the analysis of the resolvent to include its uniform behaviour as ]A I --~ cx~ away from the spectrum. Combining this with the structure of the resolvent away from the continuous spectrum of AE,0, already discussed in w allows the heat kernel to be analyzed uniformly as t ~ c~ and e ~ 0. This is where critical use is made of the hypothesis (2).

7.1 T h e h e a t kerne l for f in i te t i m e .

PROPOSITION 3. If P E Diff2(M; E) has diagonal symbol given by a metric (1) then there is a unique Hp E ~h~(M; E) satisfying

t(a, + P)Hp = 0 in ~ - 2 ( X ; E) (168) Nh,:h,-2(Hp) = Id .

Proof: The two conditions in (168) translate to conditions on the three normal operators of the putative solution Hp. Thus from (166), (163) and (164) we must have

f (*a2(P) + n))Nh,:h,-2(H) = 0 I -�89 Nh~:h,-2(H)

dfi bre

t(Ot + N,(P))Nh,: ,(H) = O, t(O, + Nb(P))Nh,:b(H) = O .

= Id

(169)

Since P is assumed to have diagonal principal symbol given by the metric, the first of these conditions, which is a fibre-by-fibre differential equation, reduces to the same problem for the Euclidean heat operator and hence can be solved uniquely, subject to the integral condition, exactly as in [Me1, Chapter 7]. Furthermore, because of the compatibility condition (156), this fixes the initial conditions for the other two equations in (169):

os N,7:h,_u(Nh,:,(H)) = I d , I N,7:h,_2(Nh,:b(H)) = I d . (170) / ,

bre d f i b r e

Thus the two normal operators Nhs:,(H) and Nh,:b(H) are necessarily the heat kernels for the two elliptic b-differential operators N~(P) and Nb(P).

62 R.R . M A Z Z E O A N D R.B. M E L R O S E G A F A

As such they are unique and, as shown in [Mel], they are elements of the corresponding b-heat calculus. These two operators have the same indicial family, so using the existence part of the compatibility conditions, (159) and (160), it follows that there is an element H ' E ~h2(M; E) satisfying the symbolic conditions (169).

This first approximation to the heat kernel therefore satisfies

t(Ot+ P ) H ' = - e R 1 , R1 e ~ - 3 ( X ; E ) . (171)

Proceeding inductively we observe that the equation

t(COtTP)Hj=Rj-�9 H j e ~ h 2 - J ( x ; E ) ~ " - m - 2 - j - I ( X ; E ) , , z ~ j + l ~ h s ,

(172) always has a solution, for given Rj E ~ - ~ - J ( X ; E). Indeed this follows from the same symbolic argument. Namely (172) is equivalent to

(~a2 (P ) - �89 + n))Nh~:h,-2-j(Hj) = Nh~:h,-2-j(Rj) , t(a, + N, (P))Nh, : , (Hj) = Nh,:,(Rj) , t(CO, + Nb(P))Nh,:b(Hj) = Nh,:b(nj) .

(173)

From the discussion of the b-calculus in [Me1] these equations have a unique solution and this solution satisfies the compatibility conditions, so successive Rj can be constructed. Summing this resulting series as a power series in �9 and at the front face gives a paxametrix, H ' , in the sense that

t(cO, + P)H" = - R ~ E ~hs (X; E) Nh~:h,-2(H") = I d .

(174)

The error term here is a Volterra operator, with smooth kernel vanishing to all orders at t = 0 and �9 = 0. If H" is interpreted as a convolution operator on the line then, as an equation on the kernels of such operators, (175) becomes

(cOt -F P)H" = Id - t - l R ~ . (175)

Since Id - t - lRoo is invertible, with inverse, Id - S, of the same type,

Hp = H"( Id - S) e ~h2(X; E) (176)

differs from H" by a Volterra operator and satisfies (168). The uniqueness follows as usual by observing that this is also a right inverse.


7.2 R e s c a l e d h e a t ke rne l . Having shown that the heat kernel is an element of the heat calculus defined above we proceed to refine this s ta tement , first near t = 0. Recall from [Mel, Chapter 8] the interpretat ion of Get- zler's rescaling in terms of rescaling of the homomorph ism bundle over the heat space. This discussion carries over, essentially without change, to the present si tuation and shows that the heat kernel is a smooth section of the bundle rescaled at the temporal front face. In particular on the diagonal

1 oo tr(~J~exp(--tAE,,)) E (x, x (177) o

whereas without using the special properties of the Dirac operator it only follows tha t this trace lies in t- �89176176 x [0, oo)]; fl).

7.3 T h e r e s o l v e n t n e a r inf ini ty . To est imate the resolvent as the spectral parameter tends to infinity outside a sector containing the spec t rum we use the discussion of the heat kernel above. This gives a uniform parametr ix in the small calculus. The asymptot ic behaviour of the resolvent for the normal operators, described in [Mel], together with the semi-ideal property allows the remainder te rm to be es t imated in the calculus.

Choose r E C~(R) with r = 1 in [t[ < 1 and r = 0 in t > 2. Then the Laplace transform

f 0 ~ 1 7 6 Res'(A) = eXtr exp(-tA)dt (178)

is certainly a bounded operator o n L 2 which is entire as a function of )~. Moreover, from (168),

( A - = m -

R(,~) = e~r exp ( - t a ) e r E). (179)

The error term here is in the small calculus, i.e. has Coo kernel on X 2 vanishing to all orders at the side boundaries, and since r has compact support in (0, oo), vanishes rapidly as IAI ~ oo in any closed sector in Re)~ < 0.

The behaviour of the resolvents for the normal operators follows from the results of [Mel]. In particular the normal operators of the error term in (179) can be removed using these resolvent families, giving holomorphic families Gb(A) and Gs(A) decreasing rapidly as [A[ --* oo and taking values in ~ b ~ 1 7 6 E) and ~b~176 E) where 7 can be chosen arbitrarily large by taking [,~[ > C. These families satisfy the compatibil i ty condition that


their indicial families are the same, so there exists a correction term to the resolvent Resl(),) E ~ [ ~ 1 7 6 which is holomorphic and rapidly decreasing in A and such that

(A -- A)ReSl(A) = R(A) - eR'()~) (180)

where R ~ E ~ - ~ 1 7 6 as for Resl(A). For small e, Id - eR'(A) is certainly invertible in L 2, with inverse Id - S(A) where S(A) has norm rapidly decreasing as IA[ ~ c~. From the Neumann series it follows that S(A) is conormal in e, so is an element of the space Ec(M; E). Since S(A) satisfies the identity:

S(A) = eR'(A) + e2R'(A) �9 R'(A) + e2R'(A) �9 S(A). R'(A) (181)

it follows from Proposition 1 that S(A) is also in the surgery calculus, and is rapidly decreasing in A. Thus the resolvent can be written

Res(A) = Res'(A) + Res"(A) , Res"(A) = Resx. (Id - S(A)) (182)

with Res"(A) being holomorphic in A, rapidly decreasing as [A[ --* c~ and taking values in t ~ ' ~ ( M ; E).

So far the spectral parameter, A, is restricted to a proper sector in the left half-plane. In fact this argument can be extended to any closed sector not containing the positive real axis by replacing P by eiap where 0 E (- 7.4 L o n g - t i m e b e h a v i o u r o f t h e h e a t ke rne l . The behaviour as t --* of H is now easy to determine when AE, 0 has no null space. There is then a uniform lower bound to the spectrum of AE,~, independent of e. The heat kernel can be written as a contour integral in terms of the resolvent:

1 f e_~Res(A)dA (183) H = 2~i

where V is a contour as in Figure 3. Since we have already analyzed the heat kernel for finite t ime we only

need consider the second part, Res", of the resolvent in (182) since combining (183) with (178) gives

( 1 - r ~

With AE,0 assumed invertible the contour can be moved from 7 to 7 t, which lies in the right half-plane but still below the spectrum. Then (184) shows that H(t) is exponentially decreasing, with all t-derivatives, as t --* c~ with values in ~-~ E) for some T > 0; namely the largest T for which the resolvent takes values in ~:o~,r (M; E) along ,ft.

Vol.5, 1995 A N A L Y T I C S U R G E R Y AND T H E E T A I N V A R I A N T 65

Im~

7 t c o n t }

k Re A

Figure 3. The contour in (183).

If we only assume (2), so that 0 might be an eigenvalue of finite multiplicity of AE,O then the same argument applies except that the small eigenvalues of AE, e contributes poles between 7 and 7'. Thus for some a > 0 and 7" > 0, and with H the projector of w

eat(1-r163 �9 ~ - ~ ' ~ ( M ; E ) (185)

uniformly with all t-derivatives. The behaviour of the small eigenvalues was already discussed in w and will be considered further in w

7.5 V a r i a t i o n o f t h e e t a inva r i an t . The proof of the main theorem in w below follows rather directly from the results and machinery already in place. The intervening section on polyhomogeneity is intended to clarify the regularity of the small eigenvalues. However, we first remark on some facts required in showing that the error term in the main theorem has the stated regularity.

As was shown in [APSil] the variation of eta with respect to a parameter is the integral of a locally defined quantity, as long as no eigenvalues cross zero as the parameter varies. The eta invariant itself is not local in this sense. More precisely, for any value of e > 0 near which the rank of the null space of 3E,~ is constant ~(3E,~) is differentiable and

_~ (d (3E,~)e_tnE,~) d~(3E ,~ ) - - LII0M t�89 tr ~ee (186)

where the notation LIM~0 means the coefficient of t o in the asymptotic expansion as t 1 0.

66

C O R O L L A R Y 2 .

R.R. MAZZEO AND R.B. MELROSE

The variation of ~](~JE,e) is given by

= ( . . ) . ( V )

GAFA

where V E C+(X:; 12) is the pointwise trace of the appropriate term in the expansion o[ (186) restricted to the s-diagonal A8 =- Xs.

Applying Lemma 2 we find:

C O R O L L A R Y 3. The variation Off(BE,e) c a n be written as f l ( e )+loge fu(e ) where both f l and f2 axe C ~ on [0, co].

8. P o l y h o m o g e n e i t y

There is a considerable refinement of the bounded conormal surgery calculus which yields more precise regularity statements, arising from the strength- ening of conormality of the kernels to polyhomogeneous conormality. This is the form of the calculus developed by McDonald ([Mc]). It was avoided above because of our emphasis on the holomorphy of the resolvent family which is easier to express in terms of the calculus with bounds and this in any case forms part of the definition of holomorphy in the polyhomogeneous calculus. The polyhomogeneity of the resolvent, discussed next, is used to show that the small eigenvalues and the corresponding eigenfunctions are, collectively, polyhomogeneous down to e = 0. We use the notation for polyhomogeneous functions from w above.

8.1 P o l y h o m o g e n e o u s s u r g e r y o p e r a t o r s . A space of polyhomogeneous surgery operators is determined by fixing an order, m, and an index family E for X 2. Then

~m,e <az. ( M ; E , F ) + e ~- 2 s ,phg, . . . ,E ,F) = ~ Aphs(X2;Hom=(E,F) | ~2~(Xs)) . (lS7)

As before, this means that an operator in this space is one with Schwartz kernel in the push-forward to M 2 x [0, e0] of the space of distributions on the right. In the present application only index sets with non-negative real parts arise, so we shall assume this below. We also assume throughout that the leading term of both Eds and Edb is (0, 0).

When all index sets have non-negative real parts and Re Els, Re Ers > 0 the polyhomogeneous space is contained in @m'r(M; E, F) where v is the smallest of inf Re Els, inf Re Ers, inf Re(Ed~ \ 0) and inf Re(Edb \ 0).


8.2 A c t i o n on p o l y h o m o g e n e o u s d i s t r i bu t i ons . Corresponding to polyhomogeneity of the kernels there is a refinement of (86). Let G = (Gss, Gbb) and 7"/ (H,s " " " = , Hbb) be index families for X~. The push-forward properties of polyhomogeneous conormal densities under b-fibrations, discussed in w applied to/~2,L imply that

fftm,E (Tt/f. ~ X E) n X "F) (188) A 6 =s ,phg U " " , E, F) ~ A : Aph s ( 8, --* -Aphg ( 8,

where Sss = ( E d s -{- Gss)U(Els -4- Gbb)

(189) Hbb = (Edb + Gbb)U (Ers + Cs~),

see also [gc].

8.3 C o m p o s i t i o n . The composition formula, (105), has a refinement closely related to (188) in that the composite of two polyhomogeneous surgery operators is also polyhomogeneous. More precisely

rn,g m' g' m+m' ,$" . q,,phs(M;E,P) . @8,p'hs(M;F,C) C @,,phs (M,E,G) (190)

where

E's = (E~ + E'~OO(E~ + E;~) (191)

E;~s = (Els-b E~b)U(Eds-bE[s)

E;~s = (Edb + E;s)O (Ers + E~s)-

Since the symbol homomorphism vanishes on the second summand in (187) composition is compatible with the symbol homomorphisms in the usual way. This composition formula is proved using the same result concerning the push-forward operation from w

8.4 H o l o m o r p h y and p o l y h o m o g e n e i t y . The holomorphy of a polyhomogeneous conormal function with respect to a parameter is somewhat subtle since, in general, the index sets, and hence the space, will depend on the parameter. An index set, E(A), depending on a parameter A in an open set W C C is holomorphic if for each )to 6 W and each 7 6 R \ Re E(A0) the polynomial

bE(s,7;~) = 1-I ( 8 - z) (192) (~,p)eE(A) Re z <7

is holomorphic for A near A0 in W. The region of holomorphy may depend on 7. The test operator in (23) is just bE(Vgf, 7; A) so it is natural to say that a holomorphic map

A, , u(A) 6 A L ( X ) , T < infReE(A) (193)


is holomorphic-polyhomogeneous up to Bgf if

~/ l r 0 X )t , ' bEs,( .k)(Vgf, ' ) ' ;) t)U E pgf(pgf) .,4 ( ) (194)

is holomorphic near A = )% for each 7 E R and each A0 E W. The holomorphy of an index family E is just the holomorphy of each

index set and we write u e Ah~_phg(X, W) (195)

to mean that W 9 A ~-~ u(A) is holomorphic as a map (193) and (194) holds for each boundary hypersurface of X. As before this definition is independent of the choices of radial vector fields and the spaces are local C ~ modules so the space AhC_phg(X, W; E) of holomorphic-polyhomogeneous conormal sections of a vector bundle, for a given holomorphic index family, is well- defined. Certainly u e .A~_phg(X,W;E ) implies that u()~) e fi[p~hg(X;E) for each A E W. This map is surjective if W is a small disk.

For 1-step polyhomogeneous conormal distributions with respect to an interior p-submanifold the notion of holomorphy is simpler, since the image space is independent of the parameter. The holomorphic surgery calculus, associated to a holomorphic index family s A E W, is then defined in the obvious way by replacing (187) by

~m,e (M W; E, F) = ,h-phg ~ , (196)

E 2 . 1 2 hol (W; V • (M; E, F)) + ,Ah_phg (Xs , W , Horns (F, E) | f~ (X~)) .

Composition of elements of the holomorphic surgery calculus satisfies the obvious extension of (190) and (191) with the resulting index families again holomorphic.

8.5 Polyhomogeneous b-calculus . The polyhomogeneous form of the b-pseudodifferential calculus on a manifold with boundary Y is discussed extensively in [Me2] and [Mel]. Here we are interested in the overblown b-calculus, allowing extra singularities at the off-diagonal corners, and even more in the holomorphic-polyhomogeneous overblown calculus.

Let Y be a compact manifold with boundary where the boundary has one or two components. In these two cases Y2 b has three or eight boundary hypersurfaces and these can be classified as front faces, off-diagonal front faces, left faces and right faces (in case OY is connected there are no off- diagonal front faces.) If E is a holomorphic index family for Y2 b set

,~,e . a c ty~ W;Homb(E) |189 . ~h_ob(Y, W , E ) = hol (W; ~I/~n(Y; E ) ) -~- r \ ob, (197)

V o l , 5 , 1995 A N A L Y T I C S U R G E R Y A N D T H E E T A I N V A R I A N T 6 9

Let pj , j = 0, 1 , . . . be the distinct eigenvalues of 8~ with multiplicities mj . Consider the parametrized index set defined by the holomorphic maps

C \ [#2, oc) 9 A J , ~j(A) = (#2 - A)�89 j = 0 , 1 , . . . , (198)

with the square-root taken to have positive real part , by taking (z, p) E E(A) if z -= qjTlj(A), qj E N0, for some non-empty set of indices j E J and p = ~ j my - 1. This is holomorphic and in [Me2] and [Mel] a somewhat more precise form of the following result is shown:

PROPOSITION 4. For the Dirac operator on an odd-dimensional exact b- spin manifold with boundary the resolvent family of the Dirac Laplacian satisfies

q(,~)(AE,o _/~)-1 e tIth2fb (Y, W ; S ) (199)

where W -- C \ [,2, ~ ) , the space on the left is defined by (197) with the index set C assigning E(A) to each left or right boundary face, E(A) U E(A) to each off-diagonal front face and 0UE(A)UE(A) to each true front face and q(A) is a polynomial with zeros precisely at the points of the discrete spectrum.

Evaluating q(A)(AE, 0 -- A) -1 at A = 0 gives the orthogonal projector, H0, onto the null space of A]E,0. This is a finite rank operator as in (111), except that now it can be seen that the null space is polyhomogeneous.

E(O) - - More precisely there is a basis of the null space of AE,0 in ,Aphg (M; S) where in case the boundary of M has two components E(0) is assigned to

AE(~) (-~-Y" S) near )~ = 0 both. Such sections can be extended to sections of ~'h-phg ~"=,

'T'-~ W; S) of the same so II0 can be extended to an element F(A) E =h-oh t*, rank. Sett ing G(A) = (AE,0 - A)- ' -- A-1F(A) gives a holomorphic family near A = 0 such that

(AE,0 -- A)G(A) = Id - F'(A) (200)

where F'(A) E @h-oh (]I, W; S) is of rank equal to the dimension of the null space of AE,0 and F ' (0) -- I-I0.

8.6 N o r m a l h o m o m o r p h i s m s . The normal homomorphisms N8 and Nb of (87), when restricted to the polyhomogeneous surgery calculus, with Eds and Edb having leading part 0, take values in the polyhomogeneous overblown b-calculus. As before we assume that Re El~, Re Ers > 0. Thus there are maps

~m' t (M" E, F) gb ,T,m,gt~7. E, F) ~ (Eds Els, Zr~) 8 k , X o b k " * ~ , ---~ , (201)

�9 ~ ' t ( g ; E, F) N, ~T,m,~/~. E, F) ~ = (Eab, El~, Er~)

7"0 R.R. MAZZEO AND R.B. MELROSE GAFA

Here we have used E g f t o denote the index set for Bgf f3 Bdb, gf = ds, Is, rs in the first case and BgfMBds, gf = db, ls, rs in the second. Notice that ~o*b(M) should require specification of index sets at the four faces of Bah M Bds, but in our case these index sets are always all equal, so we have shortened the notation accordingly. Compatibility between these normal operators and with the symbol homomorphism is as before. The normal homomorphism clearly maps the holomorphic-polyhomogeneous calculus into the overblown holomorphic-polyhomogeneous calculus.

8.7 P o l y h o m o g e n e i t y o f t h e r e so lven t . By a straightforward extension of the previous constructions we can now construct a polyhomogeneity parametrix uniformly down to e = 0.

PROPOSITION 5. Assuming (2), there is a holomorphic index family • for X 2 with index sets having leading part 0 at Bd~ and Bdb and otherwise with

- 2 , 3 r positive real parts for A E W = A E C\[#o 2, ee) and G, E ~ ,h_phg(M, W, S) such that

(AE,, -- A)Gr = Id - F near A = 0 (202)

where F has uniformly finite rank equaling No = dimnullAE,0.

Proof: The proof follows the lines of the proofs in w and w closely. First the parametrix for Z]E, 0 -- )~ in (200) can be extended to an element of

-2,.~" I ate E ffffs,h-phg (M, W; S) for some index family stated type. That is, Gr has b-normal operator equal'to Go and s-normal operator equal to the resolvent for N~(AE,c). Then (202) holds except that the remainder term

I d - ( A E , , - A)G~ = F ' + F "

where F ' is of rank No and F " has all normal operators vanishing. The composition formula (190) shows that the Neumann series for ( I - F " ) -1 can be asymptotically summed in the holomorphic-polyhomogeneous calculus. This yields an operator I - S with S in the polyhomogeneous calculus, but with rather complicated (but non-negative) index sets, such that

( I -F") ( I -S)=I -Q, QEC~(Xy;Hom~|

The Neumann series for ( I - Q ) - I converges and the semi-ideal property shows that the inverse is an operator, I - Q~, of the same type. Setting Gc = G~ �9 (Id - S) . (Id - Q') gives a parametrix of the desired type.

This procedure may introduce spurious elements into the index sets, the removal of which requires a separate argument.


It remains to examine the error term in (202). Since F is of uniformly finite rank the invertibility of Id - F reduces to the invertibility of the matrix obtained by projecting F onto the span of its range. The entries of this matr ix are obtained by taking inner products, on X~, of polyhomogeneous sections and so the determinant is itself both polyhomogeneous and holomorphic:

(Id - F ) - I exists -~ '.. q()~, e) ~ 0 (203)

where q(0, A) = A y~ The zeros, near A = 0, of this functions are necessarily real and smooth in e > 0, since they are the small eigenvalues of AE,e. In fact the eigenvalues are polyhomogeneous at e = 0 as can be seen by using the standard construction of Puiseux series to construct formal so- lutions. Furthermore this shows that the small eigenvalues can be divided into groups according to the leading term in their expansion at e = 0. In particular the eigenvalues which vanish rapidly at e = 0 are smooth.

It follows that the resolvent has a similar structure to the resolvent in (199), i.e.

PROPOSITION 6. Under the assumption (2) the resolvent family of the Dirac Laplacian satisfies

-2 ,g q(e, A)(AE,~ -- A) -a e r W; S) (204)

where W is a small neighbourhood of O, the holomorphic index set g has leading part 0 at Bds and Bdb and positive real part elsewhere and q(e, A) is a polynomial of degree dim nullAE,o vanishing precisely at the small eigenvalues. The projector H~ onto the eigenspaces corresponding to all small eigenvalues is an element oi . . . . ~gs,phg-~'e (M; S).

9. P r o o f o f t h e T h e o r e m

The results of the previous sections will now be applied to prove the main theorem as stated in the Introduction. Initially we shall strengthen the hypothesis (2) to the assumption that 0 is not in the spectrum of AE, 0.

Rewriting formula (3) for ~(e) gives

r/(e) = (Trt).Qr~).[tko t rF ] , t - l / 2

F - V ~ 8E,~exp(--tAE,~) (205)

Here the 'little trace,' tr, is the pointwise trace on the rescaled homomor- 2 phism bundle homa(S | E), cA, : X~ x [0, 1] r Xc,hs is the embedding of

72 R . R . M A Z Z E O A N D R.B. M E L R O S E G A F A

the lifted diagonal in the compactified double heat space (defined below), r8 is the map (38) and r t is the projection [0, e0] x [0, c~] -4 [0, e0].

In order to use the results from w regarding the push-forward and pull- back of polyhomogeneous conormal distributions by b-fibrations on compact manifolds with corners we compactify the t ime variable by introducing 1/t as a coordinate near infinity. More globally, consider the diffeomorphism

t [0, t , , t +--7 e [ 0 , 1 ) . (206)

Since X28 is a equal to X 2 x [0, oo) outside a compact set we can carry out the compactification (206) and define

Xr , 2 = [X~ x [0,1],;A8 x {0},S] . (207)

Then (205) is to be interpreted in terms of these compact spaces. The polyhomogenei ty of t~, s (tr F ) on Xs x [0, 1] follows immediate ly from Proposi- tion 3 since polyhomogenei ty is preserved when the heat kernel is restricted to A~, and also when the pointwise trace is taken. From this proposit ion the heat kernel is smooth up to each of the faces Bas(X~,) and Bdb(X~8 ). Thus t r F has index set 0 at the faces Bss(Xs) x R + and Bbb(X~) X R +. Since we are assuming for the t ime being tha t AE, 0 has trivial null space, we conclude tha t ~* F vanishes rapidly, hence has empty index set, at the A s

face t = oo. Finally, from the rescaling arguments of w the index set of L* F a t t = O i s O . As

We now consider the two push forwards in (205). The inner push forward by r e : X~2,hs ~ [0, E0] X [0, lit is s imply the trace of F for t fixed, and this is computed by Lemma 2. To apply this result we must compute each of the three terms in (121) when A is replaced by F(t).

PROPOSITION 7. F o r any t > 0, the pointwise traces

tr Ns(~E,e e -tAE'~ ) , tr RH(~JE,e e -~AE'~ ) and tr Ntt(~JE,~ e -~A~," )

each vanishes identically.

Proof: This vanishing occurs for purely algebraic reasons. First consider the second case,~given by restriction to H. As discussed in w the restriction operator to H is a homomorphism so

- - t A ~ RH(~JE,e e - t A E ' ' ) = ~ e H

is given in terms of the Dirac operator and heat kernel on H. Since ~r is even dimensional the spinor bundle is Z2-graded with the two summands

Vol .5 , 1995 A N A L Y T I C S U R G E R Y A N D T H E E T A I N V A R I A N T 73

identified by Clifford multiplication by a unit b-normal vector. The Dirac operator is odd and the heat kernel even so the trace vanishes identically.

The normal operators at H and H are the same, and are given in terms of the Dirac operator on H for an invariant metric. The spinor bundle is naturally the lift of the spinor bundle from H so again splits. The heat kernel is again even but the Dirac operator has an even part. However the two diagonal terms are of opposite signs under the identification by Clifford multiplication, so again the trace vanishes pointwise.

Thus all but the last of the three term in (121) vanish, so that the inner push forward of (205) is given as a sum ~1 (e, t) + log e ~2 (e, t) where ~2(0, t) = 0 and ~l(O,t) = b-Tr~(Nb(F)). Pushing forward at last by 7rt produces

~(e)=(~),(b-Tr~(Nb(F))) + rl(e) + loger2(E ) (208)

where rl r2 E A ~ _ for some T > 0. The first term on the right in (208) is by definition simply the b-e ta invariant ~?b(~JE,O) from [Me1].

The final step in the proof is to check that rl and r2 are both g ~ functions of e; they must then vanish at e = 0. For this we use the variation formula for the eta invariant discussed in w Since the derivative, ~)(~), is the sum fl(e) + logef2(e) where f~, f2 E C~([0, GoD, integrating from e = 0 shows that rl and r 2 are both smooth.

Only slight modifications of this argument are required when we relax the assumption that AE, 0 has no null space and prove the theorem assuming only the hypothesis (2). Using the projection, 1I, onto the small eigenvalues of AE,, in t > 0, the operator F splits as

F = (Id - [ i )F + [ IF = F ' + F " (209)

where F ' is in the surgery heat calculus with bounds, decaying exponentially as t --4 oc, and F" is uniformly of finite rank N. The proof carries through as before for F ~, showing that

(Trt).(Tr~). [t~,s t r F ' ] =yb(3E,o)+rx(e)+loger2(e) (210)

where rl , r2 are smooth and ~lb(3E,O) is the b-eta invariant for (I--[I0)3E,0 ---- ~iE,0, II0 being the orthogonal projector onto the null space. The remaining term involving F" may be computed algebraically for each e > 0, and is simply the signature of ~iE,~ on. the range of H~. For small e > 0, H~ splits further as H~ = [irap + [iphg corresponding to the rapidly vanishing and the polynomially vanishing eigenvalues. Since the polynomially vanishing eigenvalues are non-zero for small e > 0 they contribute a constant term to the signature. The rapidly vanishing eigenvalues on the other hand whilst smooth can, at least in principle, change sign infinitely often as e ~ 0. This gives (8) and concludes the proof of the theorem assuming only (2).


R e f e r e n c e s

[APSil] M.F. ATIYAH, V.K. PATODI, I.M. StaGER, Spectral asymmetry and Rieman- nian geometry I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69.

[APSi2] M.F. ATIVAH, V.K. PATOm, I.M. SINGER, Spectral asymmetry and Rieman- nian geometry II, Math. Proc. Camb. Phil. Soc. 78 (1975), 405-432.

[APSi3] M.F. ATXYAH, V.K. PATODI, LM. SXNGER, Spectral asymmetry and Pdeman- nian geometry III, Math. Proc. Camb. Phil. Soc. 79 (1976), 71-99.

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Rafe R. Mazzeo Richard B. Melrose Department of Mathematics Department of Mathematics Stanford University MIT Stanford, CA 94305 Cambridge, MA 02139 USA USA Submitted: November 1992 [email protected] rbm~math.mit.edu Revised Version: November 1994

analytic surgery and the eta invariant

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