the eta invariant and parity conditions

http://www.elsevier.com/locate/aim

Advances in Mathematics 182 (2004) 173–203

The eta invariant and parity conditions$

Anton Savin� and Boris Sternin

Independent University of Moscow, Moscow, Russia

Received 7 November 2000; accepted 2 January 2003

Communicated by Daniel Freed

Abstract

We give a formula for the Z-invariant of odd-order operators on even-dimensionalmanifolds and even-order operators on odd-dimensional manifolds. Second-order operators

with nontrivial Z-invariants are found. This solves a problem posed by Gilkey.r 2003 Elsevier Science (USA). All rights reserved.

MSC: primary 58J28; 19K56; secondary 58J20; 19L64

Keywords: Eta invariant; Parity condition; K-theory; Linking index; Dirac operator; Spectral flow;

Elliptic operator

0. Introduction

1. Gilkey [Gil89] noticed that the Atiyah–Patodi–Singer Z-invariant is homotopyinvariant within the class of elliptic self-adjoint differential operators provided that

ordA þ dimM � 1 ðmod 2Þ: ð1Þ

More precisely, in this case the fractional part fZðAÞgAR=Z of the spectral Z-invariant is a homotopy invariant of the elliptic self-adjoint differential operator.This fractional part is determined by the principal symbol of the operator.Thus, we have the problem of computing the Z-invariant in topological terms and

the question of finding examples, showing the nontriviality of the invariant.

ARTICLE IN PRESS

$Supported by the Russian Foundation for Basic Research under Grants 02-01-00118 and 02-01-00928,

by Arbeitsgruppe Partielle Differentialgleichungen und Komplexe Analysis, Institut fur Mathematik,

Universitat Potsdam.�Corresponding author.

E-mail addresses: [email protected] (A. Savin), [email protected] (B. Sternin).

0001-8708/$ - see front matter r 2003 Elsevier Science (USA). All rights reserved.

doi:10.1016/S0001-8708(03)00077-X

Operators with nontrivial Z-invariant on even-dimensional manifolds wereconstructed in [Gil85]. It turned out that geometric first-order Dirac-type operatorscan have only dyadic Z-invariants

fZðAÞgAZ1

2

� ��Z:

Moreover, if the manifold is orientable, then the invariant is a half-integer. Aninteresting example is given by Dirac operators on (nonorientable!) pinc-manifolds

(see [Gil85]), e.g., on the even-dimensional real projective space RP2n: For a numberof manifolds, the Z-invariant of the Dirac operator was computed in [Gil85,Gil98],where applications to geometry were considered.In this paper, we give a formula for the Z-invariant of operators that satisfy (1).

Second-order operators with nonzero fractional part of the Z-invariant areconstructed on some odd-dimensional manifolds. This solves the problem posed in[Gil89].

2. Let us briefly describe the contents of the paper. The computational problem forthe fractional part of the Z-invariant is stated in the first section. To be definite, in thefollowing three section we deal with the case of even-order operators. Section 2 startsfrom the expression of the fractional part of the Z-invariant in terms of the spectralflow modulo n: Section 3 expresses the spectral flow in topological terms. Then weexpress the Z-invariant in terms of the linking index in K-theory. In Section 5 weindicate the changes needed for obtaining a formula for the fractional part of the Z-invariant for odd-order operators. The last section gives a second-order operator with

nonzero fractional part of the Z-invariant on the nonorientable product RP2n � S1:In the appendix, we give a formula for the action of antipodal involution on the K-

group of the Thom space for a real vector bundle.

3. The preliminary version [SS02] of this paper contained a formula for thefractional part of the Z-invariant. It was obtained by means of the reduction of thespectral invariant ZðAÞ to an (a priori) homotopy invariant of the spectral subspacegenerated by the eigenvectors of A corresponding to nonnegative eigenvalues (see[SS99,SS00]).The present paper contains a direct proof of the formula for the Z-invariant. This

proof is independent of the above-mentioned construction and the correspondingelliptic theory in subspaces defined by pseudodifferential projections.

1. Statement of the problem

Let M be a smooth closed n-dimensional manifold and

A : CNðM;EÞ-CNðM;EÞ

ARTICLE IN PRESSA. Savin, B. Sternin / Advances in Mathematics 182 (2004) 173–203174

an elliptic self-adjoint operator of positive order acting on smooth sections of avector bundle EAVectðMÞ: The spectral Z-function of A is defined by

Zðs;AÞ ¼ 12

Xsgn lijlij�s þ dim kerA

� �; ð2Þ

where the summation is taken over nonzero eigenvalues li of A with regard for theirmultiplicities. The series in (2) is absolutely convergent in the half-planeRe s4dimM=ordA: The spectral function Zðs;AÞ extends to the complex plane asa meromorphic function. Moreover, the residue at s ¼ 0 is zero [APS76a,Gil81]. Thevalue of the Z-function at the origin is called the Z-invariant of operator A:

ZðAÞ ¼ Zð0;AÞ:

The spectral Z-invariant is not a homotopy invariant of the operator: for a smoothfamily of operators At with parameter t; the function ZðAtÞ is piecewise smooth withintegral jumps at the values of t; where some eigenvalue of At changes its sign. At thesame time, the family

fZðAtÞgAR=Z

of fractional parts is smooth. It was noticed in [Gil89] that the fractional part fZðAÞgis a homotopy invariant of the operator for differential operators that satisfycondition (1). Condition (1) will be referred to as the parity condition.

Problem. Compute fZðAÞg under the parity condition in terms of the principalsymbol of A:

To apply the methods of algebraic topology to this problem, it is useful to enlargethe class of differential operators.

Definition 1. A pseudodifferential operator A is said to be admissible [Gil89] if forthe components akðx; xÞ homogeneous in x of degree k in the asymptotic expansionof the complete symbol

sðAÞðx; xÞBXjX0

ad�jðx; xÞ;

the following equality holds:

akðx;�xÞ ¼ ð�1Þkakðx; xÞ: ð3Þ

The basic properties of the Z-invariant, which are used in the computation of thefractional part fZðAÞg; are given in the following proposition (e.g., see [APS76a,Gil89]).

ARTICLE IN PRESSA. Savin, B. Sternin / Advances in Mathematics 182 (2004) 173–203 175

Proposition 1. Let A be an admissible elliptic self-adjoint operator satisfying the parity

condition. Then

1. fZðAÞgAR=Z is a homotopy invariant of the operator A; more precisely,

ZðA1Þ � ZðA0Þ ¼ sfðAtÞtA½0;1

for a smooth family At; where sfðAtÞtA½0;1 is the spectral flow (see [APS76a]) of the

family At;2. ZðWEÞ ¼ 0; where

WE : CNðM;EÞ-CNðM;EÞ

is a nonnegative Laplacian acting on section of a vector bundle E over an odd-

dimensional manifold; a similar statement is valid for powers of this operator:

ZðWlEÞ ¼ 0;

3. Zð�AÞ ¼ �ZðAÞ if A is invertible.

In subsequent sections, we compute the fractional part of the Z-invariant for even-order operators on odd-dimensional manifolds. Odd-order operators can be treatedsimilarly. The changes in the constructions and the proofs are given in Section 5.

2. An expression of the g-invariant in terms of the spectral flow

1. Let A be an admissible elliptic self-adjoint operator of even order d ¼ 2l satisfyingthe parity condition (i.e., the manifold is odd dimensional).We introduce the stable homotopy equivalence relation on this set of operators.

Definition 2. Operators with principal symbol equal to the direct sum of a positiveand a negative definite symbols are said to be trivial. Two operators A1 and A2 arestably homotopic if and only if, for some trivial operators A0 and A00; the direct sums

A1"A0 and A2"A00

are homotopic.

The set of equivalence classes of stably homotopic admissible self-adjoint even-order operators on M is denoted by SevðMÞ: This set is a group with respect to thedirect sum. The group SevðMÞ can be described in terms of K-theory.Indeed, the principal symbol sðAÞðx; xÞ is a Hermitian endomorphism of the

bundle the operator is acting on. By PþðsðAÞÞðx; xÞ denote the orthogonalprojection on the subspace generated by the eigenvectors of sðAÞðx; xÞ with positiveeigenvalues. For an even-order operator A; we obtain

sðAÞðx;�xÞ ¼ sðAÞðx; xÞ:


Consequently, the vector bundle ImPþðsðAÞÞ lifts to the projectivization

P�M ¼ S�M=fðx; xÞBðx;�xÞg:

Consider the mapping

SevðMÞ !wev

KðP�MÞ=p�KðMÞ;

½A / ½ImPþðsðAÞÞ ;

where p� : K Mð Þ-K P�Mð Þ is induced by the projection

p : P�M-M:

Remark 1. The bundle E can be assumed trivial, since A is equivalent to an operator

A"DlE0 in a trivial bundle, where E0 is the complementary bundle.

Lemma 1. wev is an isomorphism.

Proof. Let us construct the inverse

KðP�MÞ=p�KðMÞ �!w�1 SevðMÞ:

Consider an element ½L AKðP�MÞ; where LAVectðP�MÞ is a vector bundle. Let usembed it as a subbundle in the trivial bundle p�CN : The orthogonal projection on L

is denoted by P ¼ Pðx; xÞ:

P : p�CN-p�CN ; ImPCL:

Let us define the mapping w�1 by the formula

w�1½L ¼ ½A ;

where

A : CNðM;CNÞ-CNðM;CNÞ

is an arbitrary elliptic self-adjoint admissible even-order operator with principalsymbol

sðAÞðx; xÞ ¼ jxj2lð2Pðx; xÞ � 1Þ : p�CN-p�CN ;

and p : S�M-M is the projection. Let us show that this mapping is well defined.Indeed, consider two realizations of L as a subbundle

LCp�CN1 ; LCp�CN2 :


Then these subbundles are homotopic, i.e., there exists a homotopy

Pt : p�CN1þN2-p�CN1þN2 ;

of projections over P�M such that

ImP0 ¼ L"0Cp�CN1þN2 ; ImP1 ¼ 0"LCp�CN1þN2 :

The homotopy Pt can be lifted to a homotopy of admissible self-adjoint operatorsAt: Therefore, the equivalence class of A is independent of the realization of L as a

subbundle LCp�CN : This shows that w�1 is well defined. It is the inverse of wev byconstruction. The proof of Lemma 1 is complete. &

Remark 2. If we do not require the admissibility in the definition ofSevðMÞ; then thecorresponding groupSðMÞ (which is the group of stable homotopy classes of ellipticself-adjoint operators) was defined in [APS76a]. It was shown that a similar mapping

w : SðMÞ-KðS�MÞ=KðMÞ

is an isomorphism. It is also known that the latter group is isomorphic to K1c ðT�MÞ

(elements with compact supports) under the coboundary mapping

KðS�MÞ !d K1c ðT�MÞ:

In what follows, we denote by ½sðAÞ the corresponding element of the odd K-group:

½sðAÞ ¼ d½ImPþsðAÞ AK1c ðT�MÞ:

On an odd-dimensional M; the group KðP�MÞ=p�KðMÞ is a torsion group. Moreprecisely, the following result is valid.

Lemma 2 (Gilkey [Gil89]). Suppose that dimM is odd. Then K�ðP�MÞ=p�K�ðMÞ is a

2-torsion group, i.e. the orders of its elements are powers of two.

2. Using the last two lemmas, we see that for some NX1 there exists a homotopy

Bt : CNðM; 2NEÞ-CNðM; 2NEÞ; tA½0; 1 ;

B0 ¼ 2NA ¼ A"A"?"A|fflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflffl}2N times

; B1 ¼ WlE0"ð�Wl

E00 Þ ð4Þ

of admissible operators, where the operator B1 is a direct sum of powers of invertibleoperators

WE0 : CNðM;E0Þ-CNðM;E0Þ;

WE00 : CNðM;E00Þ-CNðM;E00Þ;


with positive definite principal symbols and the following vector bundle isomorph-ism is valid:

2NECE0"E00:

Applying Proposition 1 to the homotopy Bt; we obtain

ZðB1Þ � ZðB0Þ ¼ sfðBtÞtA½0;1 :

At the endpoints, we have

ZðB1Þ ¼ ZðWlE0 Þ � ZðWl

E00 Þ ¼ 0; ZðB0Þ ¼ 2NZðAÞ:

Thus, the fractional part of the Z-invariant of A is expressed as

fZðAÞg ¼ � 12NsfðBtÞtA½0;1

�:

Denote by

mod n-sfðBtÞtA½0;1 AZn

the spectral flow reduced modulo n: Then the last formula gives

fZðAÞg ¼ � 12Nmod 2N -sfðBtÞtA½0;1 : ð5Þ

Remark 3. Unfortunately, the formula for the spectral flow with values in a finitecyclic group (see [APS76a]) does not apply to the present case, since the operator Bt

does not have the form assumed in that paper. In the following section, we compute(5) in terms of the principal symbol sðBtÞ of the family.

3. Computation of the spectral flow modulo n

1. Let n be a given positive integer. Consider a smooth family At; tA 0; 1½ of ellipticself-adjoint operators such that

A0 ¼ nA; A1 ¼ WlE1"ð�Wl

E2Þ:

The residue modulo n of the spectral flow of the family ðAtÞ is denoted by

mod n-sfðAtÞtA½0;1 AZn ð6Þ

(cf. [APS76a]).The spectral flow for families of this form can be computed by means of the

Atiyah–Singer index formula for families. Namely, the spectral flow (6) can becomputed in terms of the difference construction

½sðAtÞ AKcðT�M;ZnÞ ð7Þ


in K-theory with coefficients Zn: The group on the right-hand side in (7) is defined as

KcðT�M;ZnÞ ¼ KcðT�M � Mn;T�M � ptÞ; ð8Þ

here Mn is the Moore space for the group Zn (see [APS76b, p. 428]). Let us represent

a generator of the reduced group K0ðMnÞ ¼ Zn by a line bundle g: Let us also fix atrivialization

ng Cb

Cn:

Eq. (8) shows that the elements of KcðT�M;ZnÞ can be represented as families ofelliptic symbols on M parametrized by the Moore space.Let us define the difference construction

½sðAtÞ ¼ ½ImPþðsðA1�tÞÞtA½0;1 #g; ImPþðsðA1�tÞÞtA½0;1

AKðB�M � Mn;B�M � pt,S�M � MnÞ

CKcðT�M � Mn;T�M � ptÞ; ð9Þ

where the fibers ImPþðsðA1�tÞÞ are located over the cospheres of M with radius t:The vector bundle isomorphism over S�M � Mn (i.e., for t ¼ 1) is defined by meansof the equivalences

ImPþðsðA0ÞÞ#g ¼ n ImPþðsðAÞÞ#g

C ImPþðsðAÞÞ#ng C1#b

ImPþðsðAÞÞ#Cn

C n ImPþðsðAÞÞ ¼ ImPþðsðA0ÞÞ:

We are now in a position to prove the formula for mod-n spectral flow.

Theorem 1. One has

mod n-sfðAtÞtA½0;1 ¼ �p!½sðAtÞ ; ð10Þ

where

p! : KcðT�M;ZnÞ-Kðpt;ZnÞ ¼ Zn ð11Þ

is the direct image mapping in K-theory with coefficients Zn: It is induced by the

mapping p : M-pt:

The proof is obtained in the third part of this section. The proof is based on theexpression of the spectral flow in terms of the index (see Lemma 3 below).


2. Consider a family ðBtÞtA½0;1

Bt : CNðM;EÞ-CNðM;EÞ

of elliptic self-adjoint operators. Suppose that the family starts and ends with directsums of positive- and negative definite operators:

B0 ¼ WE00"ð�WE00

0Þ; B1 ¼ WE0

1"ð�WE00

1Þ; E0

0;1"E000;1 ¼ E:

The spectral flow for families of this form is equal to the index of an elliptic operator.Consider the difference construction

½sðUÞ AKðB�M;S�MÞ

defined by the formula

½sðUÞ ¼ ½ImPþðsðBtÞÞtA½0;1 ; p�0E

01 ; p0 : B�M-M; ð12Þ

for the vector bundles

ImPþðsðBtÞÞtA½0;1 ; p�0E

01AVectðB�MÞ:

Here the base of ImPþðsðBtÞÞ is the cosphere space of radius t: The bundles p�0E01

and ImPþðsðBtÞÞtA½0;1 coincide over S�MCB�M:

It can be shown that ½sðUÞ corresponds to an elliptic operator

U : CNðM;E00Þ-CNðM;E0

1Þ ð13Þ

with principal symbol sðUÞ defined as the solution of the Cauchy problem

d

dtut ¼

d

dtPþðsðBtÞÞ;PþðsðBtÞÞ

� �ut; u0 ¼ 1 ð14Þ

at t ¼ 1:

sðUÞ ¼ u1:

Lemma 3. One has

sfðBtÞtA½0;1 ¼ �indU : ð15Þ

Proof. We shall use the following definition of the spectral flow, e.g., see [Mel93].There exists a partition

0 ¼ t0ot1o?otN�1otN ¼ 1 ð16Þ


of the unit segment and a set of real numbers flig0pipN�1 such that Bt has no

eigenvalue li on the segment tA½ti; tiþ1 : We set lN ¼ 0: Denote byPli

ðtÞ; tA½ti; tiþ1 the spectral projection for operator Bt on the sum of eigenspaces

corresponding to the eigenvalues greater than li: These families smoothly depend ont on the corresponding intervals. Then the spectral flow is equal to

sffBtgtA½0;1 ¼XN�1

i¼1indðPliþ1ðtiÞ;Pli

ðtiÞÞ; ð17Þ

where ind ðP;QÞ—is the relative index of a pair of projections with a compactdifference P � Q defined as the index of a Fredholm operator indðP;QÞ ¼def tðQ :

Im P-ImQÞ: For the spectral projections of a self-adjoint operator A one computesthis relative index explicitly

indðPlðAÞ;PmðAÞÞ ¼ sgnðm� lÞdimLl;m;

where Ll;m denotes the spectral subspace of A corresponding to the eigenvectors with

eigenvalues between l and m:On the other hand, on each of the segments ½ti; tiþ1 ; i ¼ 0;y;N � 1; consider the

solution Uit of the Cauchy problem (cf. (14))

d

dtUi

t ¼d

dtPli

ðtÞ;PliðtÞ

� �Ui

t ; Uit¼ti

¼ Id: ð18Þ

One can show the following properties of these operators.

1. The product UN�1tN

yU1t2

U0t1has the same principal symbol as the operator U

(since on the principal symbol level Eq. (18) is the same as (14));

2. Uitiþ1defines an isomorphism

Uitiþ1

: ImPliðtiÞ-ImPli

ðtiþ1Þ; ð19Þ

here ImPliðtÞ denotes the range of the corresponding projection.

Therefore, the composition

U ¼YN�1

i¼0Pliþ1ðtiþ1ÞUi

tiþ1: ImPl0ðt0Þ-ImPlN�1ðtNÞ

is a Fredholm operator with the principal symbol equal to sðUÞ: Hence,

ind U ¼ indU :


On the other hand, by the logarithmic property of the index we obtain

ind U ¼XN�1

i¼0indðPliþ1ðtiþ1ÞUi

tiþ1: ImPli

ðtiÞ-ImPliþ1ðtiþ1ÞÞ

¼XN�1

i¼0indðPliþ1ðtiþ1Þ : ImPli

ðtiþ1Þ-ImPliþ1ðtiþ1ÞÞ ¼ �sfðBtÞtA½0;1 :

In the second equality we use the invertibility of (19).This proves Lemma 3. &

Remark 4. There is a generalization of this result to the case of self-adjoint operatorfamilies over a parameter space ½0; 1 � X for a compact X : The notion of spectralflow in this case was introduced in [DZ98].

3. Proof of Theorem 1. Let us consider the family ðAtÞtA½0;1 as a family over ½0; 1 �Mn: Let At#1g be the corresponding family with coefficients in g: Consider thecomposition of these self-adjoint elliptic families:

Bt ¼At#1g; t40;

A�t; to0;

(tA½�1; 1 :

At t ¼ 0; the families are glued by means of the isomorphism

A0#1g ¼ nA#1gCA#1ng C1#b

A#1Cn ¼ nA ¼ A0:

We readily obtain

sfðBtÞtA½�1;1 ¼ sfðAtÞtA½0;1 ð½g � 1ÞAKðMnÞ: ð20Þ

On the other hand, the spectral flow for families of this form, i.e., for families thatstart and end with the direct sums of positive and negative definite operators, is equalto the index of an elliptic operator (see Lemma 3 and the subsequent remark). Thus,

sfðBtÞtA½�1;1 ¼ �indU :

The element ½sðAtÞ defined in (9) coincides with ½sðUÞ by construction (see (12)).Hence, Eq. (20) and the Atiyah–Singer index theorem for families give the desiredformula

mod n-sfðAtÞð½g � 1Þ ¼ �p!½sðAtÞ AKðMnÞ ¼ Zn:

The formula for spectral flow modulo n is proved. &


4. Let us apply the formula for the spectral flow to family (4). Denote by st anarbitrary homotopy of invertible self-adjoint even symbols

stðx;�xÞ ¼ stðx; xÞ; s0 ¼ 2NsðAÞ; s1 ¼ 1p�E1"ð�1p�E2Þ ð21Þ

on S�M: By Eq. (9) this homotopy defines an element

½st AKcðT�M;Z2N Þ:

Consider the natural inclusion

iN : Z2NCZ1

2

� ��Z:

Then the K-group with coefficients Z½12 =Z is defined as the direct limit

as N-N:

Kc T�M;Z1

2

� ��Z

� �¼def lim

-KcðT�M;Z2N Þ:

It is induced by the mappings

KcðT�M;Z2N Þ-KcðT�M;Z2Nþ1Þ-?; Z2NCZ2Nþ1y :

In this notation we obtain the following result for the fractional part of the Z-invariant.

Theorem 2. Let A be an admissible elliptic self-adjoint operator of a positive even order

on an odd-dimensional manifold.

1. One has

fZðAÞg ¼ 1

2Np!½st AZ

1

2

� ��Z; ð22Þ

where the direct image mapping

p! : KcðT�M;Z2N Þ-Kðpt;Z2N ÞCZ2N

is induced by p : M-pt:2. The element

iN� ½st AKc T�M;Z1

2

� ��Z

� �

is independent of the choice of the homotopy st:


Proof. The first assertion follows directly from (5) and Theorem 1.Let us prove the second assertion. Consider two homotopies st; s0t of the form

(21). We must prove the equality

iN 0

� ½st ¼ iN 0

� ½s0t AKcðT�M;Z2NþN0 Þ

for some N 0 and the natural inclusion

iN 0: Z2N-Z2NþN0 :

Consider the superposition of homotopies

st,s�t,s0t: ð23Þ

In other words, as the parameter tA½0; 3 increases, we start from the homotopy st;go along it in the opposite direction, and finally carry out the second homotopy s0t:Obviously, this superposition of homotopies is equivalent to s0t:Hence, we obtain thefollowing equality for the difference constructions:

½st,s�t,s0t ¼ ½s0t AKcðT�M;Z2N Þ:

By the definition of these elements (see (9)), we have

½st,s�t,s0t � ½st ¼ ½s�t,s0t #ð½g2N � 1Þ:

Here gn denotes the line bundle on the Moore space Mn; while

½s�t,s0t AKcðT�MÞ:

The family s�t,s0t is even with respect to the cotangent variables. We conclude thatthe element ½s�t,s0t lies in the range of the composition

K1ðP�MÞ=p�K1ðMÞ-K1ðS�MÞ=p�K1ðMÞ-KcðT�MÞ:

By virtue of Lemma 2, we have

2N 0 ½s�t,s0t ¼ 0

for some N 0: Consequently,

iN 0

� ½s0t � iN 0

� ½st ¼ 2N 0 ½s�t,s0t #ð½g2NþN0 � 1Þ ¼ 0:

This gives the desired formula

iN� ½st ¼ iN� ½s0t : &


4. The eta invariant and the linking index in K-theory

Formula (22) can be rewritten in a form more suitable for applications. In thissection, we show that the fractional part of twice the Z-invariant is determined by theelement of the group K1

c ðT�MÞ (see Remark 2). This element is defined by theprincipal symbol of the operator. The Z-invariant can be calculated as the linking

index in K-theory.

1. Let us define the linking pairing

/;S : TorKiþ1c ðT�MÞ � TorKiðMÞ-Q=Z ð24Þ

in the following way (cf. [MW00]).The short exact sequence

0-Z!iQ-Q=Z-0

induces the long exact sequence

?-KcðT�MÞ-KcðT�M;QÞ-KcðT�M;Q=ZÞ !@ K1c ðT�MÞ !i� K1

c ðT�M;QÞ

-? ð25Þ

in K-theory. Here the K-theories with coefficients Q and Q=Z are defined as thedirect limits (see [APS76a])

K�ðX ;QÞ ¼ limN-N

K�ðX ;ZÞ; Z �!�NZ;

K�ðX ;Q=ZÞ ¼ limN-N

K�ðX ;ZNÞ; ZnCZmn: ð26Þ

Note the natural isomorphism

K�ðX ;QÞCK�ðX Þ#Q:

Consider two torsion elements

xATorKiþ1c ðT�MÞ; yATorKiðMÞ:

Then i�x ¼ 0 and the exact sequence (25) gives the equality @x0 ¼ x for some

x0AKicðT�M;Q=ZÞ:

Definition 3. The number

/x; yS ¼ p!ðx0yÞ ð27Þ


is called the linking index of elements x and y: Here

p! : KcðT�M;Q=ZÞ-Kðpt;Q=ZÞ ¼ Q=Z:

Lemma 4. The product

x0yAKcðT�M;Q=ZÞ

is determined by x and y: It is independent of the choice of x0: Therefore, the linking

index /x; yS is well defined.

Proof. Sequence (25) shows that all x0 differ by the elements of the group

KicðT�MÞ#Q: However, the product

ðKicðT�MÞ#QÞ � TorKiðMÞ-K0

c ðT�MÞ#Q

is trivial. This proves the lemma. &

The linking pairing (24) is nondegenerate by Poincare duality for torsionsubgroups (e.g. see [SS02]).

Theorem 3. One has

f2ZðAÞg ¼ /½sðAÞ ; ½1� LnðMnÞ S; ð28Þ

where A is an admissible elliptic self-adjoint even order operator and LnðMnÞ is the

orientation bundle of the odd-dimensional manifold M:

Corollary 1. The invariant f2ZðAÞg is determined by the element ½sðAÞ AK1c ðT�MÞ:

Proof of Theorem 3. By virtue of (22), the fractional part of the Z-invariant isequal to

fZðAÞg ¼ 1

2Np!½st ; ½st AKcðT�M;Z2N Þ:

The element ½st satisfies the relation (see (25))

@½st ¼ ½sðAÞ AK1c ðT�MÞ:

Moreover, this element is invariant under the action of antipodal involution

a : T�M-T�M; aðx; xÞ ¼ ðx;�xÞ


on the K-group. Thus,

f2ZðAÞg ¼ 1

2Np!ð1þ a�Þ½st : ð29Þ

We show in the appendix that the involution a� acts on the K-group as the productwith the orientation bundle:

a�x ¼ ð�1Þn½LnðMnÞ x:

Substituting the last expression into (29), we obtain

f2ZðAÞg ¼ 1

2Np!ð½st ð1� LnðMnÞÞÞ:

The right-hand side of this expression coincides with the linking index (see (27), (26)).The theorem is proved. &

2. The formula for the fractional part of the Z-invariant can be rewritten in the formresembling the Atiyah–Singer index formula. In contrast with the index, which iscomputed under the collapsing map to the point, the Z-invariant is computed interms of the map to the projective space.The orientation bundle LnðMnÞ is a line bundle with the structure group Z2:

Consider the classifying mapping

f : Mn-BZ2 ¼ RPN;

i.e. a mapping f : Mn-RP2N such that

LnðMnÞCf �g;

here g is the tautological line bundle over the projective space RP2N : The mapping f

is uniquely defined up to homotopy. The reduced K-group of RP2N is cyclic:

KðRP2NÞ ¼ Z2N ;

moreover, the generator is 1� ½g (e.g. see [Ada62]). The Poincare duality for torsionsubgroups (24) implies that the value of the linking pairing with this element definesan isomorphism C

K1c ðT�RP2NÞ ¼ TorK1

c ðT�RP2NÞCZ2NCZ½12 =Z

x//x; ½1� g S: ð30Þ


Proposition 2. The fractional part of twice the Z-invariant is equal to

f2ZðAÞg ¼ f!½sðAÞ ;

f!½sðAÞ AK1c ðT�RP2NÞCZ2NCZ

1

2

� ��Z: ð31Þ

Proof. From the definitions of the linking index and the mapping f ; we obtain

f2ZðAÞg ¼ p!ð½sðAÞ � ½1� LnðMnÞ Þ ¼ p!ð½sðAÞ � ½1� f �g Þ:

Here � denotes the product

TorKiþ1c ðT�MÞ � TorKiðMÞ-KcðT�M;Q=ZÞ

from Lemma 4. By the functoriality of the direct image map p!; we have

f2ZðAÞg ¼ p0! f!ð½sðAÞ � f �½1� g Þ ¼ p0

!ð f!½sðAÞ � ½1� g Þ ¼ /f!½sðAÞ ; ½1� g S;

where p0 : RPN-pt: By virtue of (30), the last expression coincides with the desiredformula (31). The proposition is proved. &

Corollary 2. On an orientable manifold, the Z-invariant is at most a half-integer. In the

nonorientable case, the following estimate for its denominator is valid:

f2kþ1ZðAÞg ¼ 0; ð32Þ

on a 2k þ 1-dimensional manifold.

Proof. By the approximation theorem, we can choose a classifying mapping

f : Mn-RPn; n ¼ 2k þ 1:

The reduced K-groups for the projective spaces have the form

KðRP2kÞCKðRP2kþ1ÞCZ2k :

Consequently,

2k½1� LnðMnÞ ¼ 0:

Hence, we obtain the desired formula

f2kþ1ZðAÞg ¼ /½sðAÞ ; ½2kð1� LnðMnÞÞ S ¼ /½sðAÞ ; 0S ¼ 0: &


Remark 5. For Dirac type operators on even-dimensional manifolds, similarestimates were obtained in [Gil85].

5. Odd-order operators

1. Let

A : CNðM;EÞ-CNðM;EÞ

be an admissible elliptic self-adjoint operator of odd order on an even-dimensionalM:Let us introduce the stable homotopy equivalence relation on this set of operators.

Definition 4. The direct sums

A"ð�AÞ;

for an elliptic self-adjoint operator A are called trivial operators. Two operators A1and A2 are said to be stably homotopic if for some trivial A0;A00 the direct sums

A1"A0 and A2"A00

are homotopic.

The group of stable homotopy equivalence classes of odd order d ¼ 2l þ 1 ellipticoperators on M is denoted by SoddðMÞ:Principal symbols of odd order operators satisfy the equality

sðAÞðx;�xÞ ¼ �sðAÞðx; xÞ:

For the bundle defined by the positive spectral projection PþðsðAÞÞ; this gives

ImPþðsðAÞÞðx;�xÞ ¼ ½ImPþðsðAÞÞðx; xÞ >; ð33Þ

where > denotes the orthogonal complement.In contrast with the case of even order operators, the bundles on S�M that satisfy

(33) cannot be described in terms of some bundles over the projectivization P�M; aswas done in Section 2. However, the following result is valid. Its proof follows from[SS00].

Theorem 4. The group SoddðMÞ is a 2-torsion group.

Proof. For completeness of the presentation, let us recall the proof of this result. Itsuffices to show that for an arbitrary elliptic self-adjoint operator A of odd order


there exists a homotopy

2NAB2Nð�AÞ

for some N (this gives the desired triviality 2Nþ1AB2NðA"� AÞ).Let L be the bundle ImPþsðAÞCp�E on the cospheres. We obtain

a�L ¼ L>; ð34Þ

since the order of A is odd. Just as in the proof of the theorem for even-orderoperators, it suffices to construct a homotopy

2NLB2NL> ð35Þ

of subbundles that satisfy (34).We claim that the elements of KðS�MÞ are invariant under a� modulo 2-torsion.

Indeed, the projection S�M-P�M on an even-dimensional manifold M induces anisomorphism

KðP�MÞ#Z½12 -KðS�MÞ#Z½1

2

(this can be proved with the use of the Mayer–Vietoris principle).Hence, (34) implies the existence of an isomorphism

s : 2NL-2NL>:

Let us extend it to the entire bundle 2Np�E*2NL in accordance with thedecomposition

*s : 2Np�E ¼ 2NL"2Na�L-2Na�L"2NL ¼ 2Np�E

as

*sðxÞ ¼ sðxÞ"sð�xÞ:

Consider an even vector bundle isomorphism

*s" *s�1 : 2Nþ1p�E-2Nþ1p�E:

It maps 2Nþ1L to 2Nþ1a�L and is homotopic to the identity, since it is the sum ofmutually inverse isomorphisms. We take some homotopy

st; s0 ¼ 1; s1 ¼ *s" *s�1:

Then the desired homotopy of subbundles (35) satisfying (34) can be taken in theform

stð2Nþ1LÞ:

This proves the theorem. &


2. Just as in the case of even-order operators, Theorem 4 implies the existence of ahomotopy

Bt; tA½0; 1 ; B0 ¼ 2NA; B1 ¼ A0"ð�A0Þ; ð36Þ

of odd-order operators, where A0 is invertible. We again obtain ZðB1Þ ¼ 0; and theZ-invariant is expressed in terms of the spectral flow:

fZðAÞg ¼ � 12Nmod 2N -sfðBtÞtA½0;1 :

To compute the right-hand side of this equation, let us extend the family Bt to afamily satisfying the conditions of the theorem on the spectral flow modulo n: To thisend, it suffices to construct a homotopy

A0 0

0 �A0

!B

jA0j 0

0 �jA0j

!

of invertible operators. The desired homotopy can be defined, for example, by theformula (see [SSS02])

A0 0

0 �jA0j

!þ ðjA0j � A0Þ

sin2 j cos j sin j

cos j sin j cos2 j

!; jA 0;

p2

h i: ð37Þ

Now we can apply Theorem 1 to the superposition of homotopies (36) and (37). Onecan obtain an expression for the fractional part of the Z-invariant for odd-orderoperators along the lines of (22).Let us finally note that the formulas of Theorem 3 and Proposition 2, which

express the Z-invariant as a linking index, remain valid for odd-order operators oneven-dimensional manifolds.

3. Example. (pinc Dirac operator on RP2n; see [Gil85]). Consider the set of 2n � 2n

Clifford matrices e0; e1;y; e2n:

ekej þ ejek ¼ 2dkj :

For a vector v ¼ ðv0;y; v2nÞAR2nþ1; we define the linear operator

eðvÞ ¼X2ni¼0

viei : C2n

-C2n

:

Consider the self-adjoint symbol

sðDÞðx; xÞ ¼ ieðxÞeðxÞ : C2n

-C2n


on the unit sphere S2nCR2nþ1; where x is a tangent vector at a point xAS2n: Thesymbol sðDÞðx; xÞ is invariant under the involution

ðx; xÞ-ð�x;�xÞ:

Thus, it defines an elliptic symbol on the projective space RP2n: This symbol is equal

to the symbol of the pinc Dirac operator on RP2n; in particular, the spinor bundle ofthe projective space is trivial (see [Gil85]).Let us analyze the Z-invariant of this operator using the linking pairing.First of all, a straightforward computation shows that the orientation bundle of

RP2n is isomorphic to the line bundle g: Therefore, the orientation bundle is thegenerator of the reduced K-group

1� ½L2nðRP2nÞ AKðRP2nÞ:

On the other hand, the principal symbol of the self-adjoint pinc Dirac operator D onit defines a generator of the isomorphic group

½sðDÞ AK1c ðT�RP2nÞ ¼ TorK1

c ðT�RP2nÞCZ2N :

This was first proved in [Gil85]. However, the proof used the Z-invariant of thisoperator. To make our computation of the Z-invariant independent of the citedpaper, we sketch a direct topological proof.First of all, the K-group is computed geometrically

K1c ðT�RP2nÞCKcðT�RP2n � RÞ C

aKcðð2n þ 1ÞgÞ C

bKcðgÞ C

cZ2n :

Isomorphism ‘‘a’’ is defined by the classical vector bundle isomorphism

T�RP2n � R-ð2n þ 1Þg;

ðx; x; tÞ/ðx; tx þ xÞ;

‘‘b’’—is the Thom isomorphism:

KcðgÞ Cb

Kcðg"ngCÞ;

corresponding to the Whitney sum of n copies of the complexification gC of g:Finally, ‘‘c’’ is valid, since

KcðgÞ ¼ KðRP2nþ1ÞCZ2n :

Now the generators of these groups are constructed going in the opposite direction.A generator of KcðgÞ can be defined as the difference element of the bundle


morphism on the total space of g:

p�g-g� R; p : g-RP2n;

ðx; u; vÞ/ðx; uvÞ

(this is invertible outside the zero section). Then one applies the Thom isomorphismto this element (e.g. see [Kar78]). An explicit computation shows that the resulting

homomorphism of vector bundles over ð2n þ 1ÞgCT�RP2n � R coincides with the

homomorphism isðDÞðx; xÞ þ t; ðx; x; tÞAT�RP2n � R (up to a vector bundleisomorphism independent of x; t). The K-theory element corresponding toisðDÞðx; xÞ þ t by definition coincides with the element (see [APS76a])

½sðDÞ AKcðT�RP2n � RÞ:

This proves the desired statement.The nondegeneracy of the linking pairing (24) implies that

/2n�1½sðDÞ ; ½1� L2nðRP2nÞ S ¼ 12:

Hence, the Z-invariant of the pinc Dirac operator D has a ‘‘large’’ fractional part:

f2nZðDÞg ¼ /2n�1½sðDÞ ; ½1� L2nðRP2nÞ S ¼ 12:

This example shows that estimate (32) of the denominator of the Z-invariant is sharpon even-dimensional manifolds. Let us note that in [Gil85] the fractional part fZðDÞgwas computed exactly:

fZðDÞg ¼ 1

2nþ1: ð38Þ

6. Gilkey’s problem

In this section, we construct an even-order operator on an odd-dimensionalmanifold such that the Z-invariant has a nontrivial fractional part.

1. Consider the product RP2n � S1p: Here by S1p we denote the circle of length p: The

coordinates are denoted by x;j and the dual coordinates in the fibers of thecotangent bundle are x; t: Let us define a second-order elliptic differential operatorDon this manifold.

On the cylinder RP2n � ½0; p ; consider the expression

D0 ¼2 sin jð�i @

@jÞD � i cos jD Wxe�ij þ ð�i @@jÞeijð�i @

@jÞWxeij þ ð�i @

@jÞe�ijð�i @@jÞ 2 sin jði @

@jÞD þ i cos jD

!; ð39Þ


where D is the pinc Dirac operator on the projective space (see the previous section)and

Wx ¼ D2

is the Laplacian. The expression D0 is self-adjoint and elliptic. The self-adjointness isobvious, while the ellipticity is a consequence of the fact that the principal symbol

sðD0Þ ¼2t sin jsðDÞðx; xÞ x2e�ij þ eijt2

x2eij þ e�ijt2 �2t sin jsðDÞðx; xÞ

!

satisfies the equation

sðD0Þ2ðx; tÞ ¼ ðx2 þ t2Þ2

(i.e., D0 is a square root of the squared Laplacian).The principal symbol of D0 has the property

sðD0Þjj¼p ¼0 1

�1 0

!sðD0Þjj¼0

0 �11 0

!: ð40Þ

Let F be the vector bundle on RP2n � S1p obtained from the trivial bundle C2n

"C2n

on the cylinder RP2n � ½0; p by means of the following matrix transition function:

0 1

�1 0

!

defined on the base of the cylinder. It follows from (40) that the principal symbol

sðDÞ0 acts on the bundle F :

sðD0Þ : p�F-p�F ; p : S�ðRP2n � S1pÞ-RP2n � S1p:

Let

D : CNðRP2n � S1p;FÞ-CNðRP2n � S1p;FÞ

be a second-order elliptic self-adjoint differential operator obtained by smoothing

the coefficients of D0 on the product RP2n � S1p:

2. The following theorem answers a question posed in [Gil89].

Theorem 5. One has

f2ZðDÞg ¼ 1

2n�1:


Corollary 3. There exist even-order operators on odd-dimensional manifolds with

arbitrary dyadic Z-invariants.

The proof of the theorem is given at the end of the section. It is based on severalauxiliary statements, to which we now proceed.

3. The straightforward application of formula (28) or (31) to D would be rathercumbersome (it is equivalent to the straightforward computation of the direct imagein K-theory).Our computation of the Z-invariant is based on the following observation. The

principal symbol of D defines the element

½sðDÞ AK1c ðT�ðRP2n � S1pÞÞ;

which is representable as the symbol of the ‘‘cross product’’ [APS76a] of an ellipticself-adjoint operator on the projective space and an elliptic operator on the circlewith a nonzero index.Consider the first-order pseudodifferential elliptic operator D1

D1 ¼1

2½e�ijðQ þ jQjÞ þ eijðjQj � QÞ ; Q ¼ �i

d

dj; jA½0; 2p ; ð41Þ

on the circle S1 of length 2p: The formulas

D1eikj ¼

keiðk�1Þj; kX0;

�keiðkþ1Þj; ko0

(

show that indD1 ¼ 2:

Proposition 3. One has

½sðDÞ ¼ ½sðDÞ ½sðD1Þ ; ð42Þ

where

½sðDÞ AK1c ðT�RP2nÞ; ½sðD1Þ AKcðT�S1Þ:

Proof. Recall that the product ½sðDÞ ½sðD1Þ is defined by the so-called twisted

tensor product

sðDÞ#sðD1Þ ¼sðDÞ#1 1#sðD�

1Þ1#sðD1Þ �sðDÞ#1

!ð43Þ


of elliptic symbols. However, D1 is not a differential operator. Hence, the principalsymbol sðD1Þ is not a polynomial. Therefore, the symbol sðDÞ#sðD1Þ is not smooth(and the operator D#D1 is not pseudodifferential).A trick to overcome this difficulty was introduced in [Gil81]. Symbol (43) is

replaced by a homotopic smooth symbol. To this end, let us replace thenonpseudodifferential operator jQj in the expression (41) for D1 by the

pseudodifferential operatorffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiD2 þ Q2

pon the product of manifolds. The resulting

elliptic symbol

sðDÞðx; xÞ 12 ½eijðtþ 1Þ þ e�ijð1� tÞ

12½e�ijðtþ 1Þ þ eijð1� tÞ �sðDÞðx; xÞ

!; x2 þ t2 ¼ 1 ð44Þ

is linearly homotopic to (43).To compare the elements in (42), let us make one more homotopy. Note that the

off-diagonal entries of matrix (44) are constant in ðx; tÞ at j ¼ 0 and p: Therefore,symbol (44) is linearly homotopic to

s0ðx;j; x; tÞ ¼jsinjjsðDÞðx; xÞ 1

2½eijðtþ 1Þ þ e�ijð1� tÞ

12½e�ijðtþ 1Þ þ eijð1� tÞ �jsin jjsðDÞðx; xÞ

!:

It turns out that s0 can be transformed to sðDÞ by means of the quadraticcoordinate transformation

Fðx; tÞ ¼ ð2tx; t2 � x2Þ;

so that

s0ðx;j;Fðx; tÞÞ ¼ sðDÞðx;j; x; tÞ; jA½0; p :

Geometrically, this coordinate transformation is a two-to-one map of the spheresuch that the great circles passing through the north pole are covered twice. Let us

decompose the spherical bundle S�ðRP2n � S1pÞ into pieces so as to obtain a one-to-one mapping.

Consider the bundle S�ðRP2n � S1pÞ with circle of length p: The symbolsðDÞ is defined on this space. We decompose the cospherical bundle into twosubmanifolds

N ¼ S�ðRP2n � S1pÞ-ft40;jAð0; pÞg; ð45Þ

S ¼ S�ðRP2n � S1pÞ-fto0;jAð0; pÞg ð46Þ

in accordance with the sign of t:


The space S�ðRP2n � S12pÞ; where the symbol s0 is defined, corresponds to thecircle of length 2p: This time we take a different decomposition

W ¼ S�ðRP2n � S12pÞ-fta� 1;jAð0; pÞg;

E ¼ S�ðRP2n � S12pÞ-fta1;jAðp; 2pÞg:

The pieces N;W and S;E are pairwise homeomorphic (see Fig. 1):

F : N-W ; Fðx;j; x; tÞ ¼ ðx;j; 2tx; t2 � x2Þ;

C : S-E; Cðx;j; x; tÞ ¼ ðx;jþ p;�2tx; x2 � t2Þ:

In addition, the symbols satisfy the relations

s03F ¼ sðDÞ; s03C ¼0 1

�1 0

!sðDÞ

0 �11 0

!: ð47Þ

On the boundaries of N and S; the symbol sðDÞ depends only on the points of thebase RP2n � S1: Therefore, the following decomposition is valid:

½sðDÞ ¼ ½sðDÞjN þ ½sðDÞjS AK1c ðT�ðRP2n � S1ÞÞ: ð48Þ

Here we have used the isomorphism

KðS�ðRP2n � S1ÞÞ=KðRP2n � S1ÞCK1c ðT�ðRP2n � S1ÞÞ:

Similarly, the symbol s0 depends only on x;j on the boundary of W ;E:Consequently,

½s0 ¼ ½s0jN þ ½s0jS : ð49Þ

Substituting (47) into (48) and (49), we obtain the desired relation

½sðDÞ ¼ ½s0 ¼ ½sðDÞ ½sðD1Þ : &

ARTICLE IN PRESS

Fig. 1. Two decompositions of S�ðRP2n � S1Þ:

A. Savin, B. Sternin / Advances in Mathematics 182 (2004) 173–203198

4. Let M1;2 be two smooth closed manifolds. The product M1 � M2 is denoted by

M: Consider two elements

½s1 ATorK1c ðT�M1Þ; ½s2 AK0

c ðT�M2Þ:

Let us compute the linking index of the product

½s1 ½s2 ATorK1c ðT�MÞ

with the orientation bundle of M:

Proposition 4. Suppose that M2 is orientable. Then

/½s1 ½s2 ; ½1� LnðMnÞ S ¼ /½s1 ; ½1� LkðM1Þ Sp!½s2 ; k ¼ dimM1: ð50Þ

Proof. Consider the commutative diagram

where j is the multiplication (see Lemma 4) on the right by ½1� LkðM1Þ : Thehorizontal maps denote products in K-theory. By virtue of the orientabilityof the second factor, we have

1� ½LkðM1Þ ¼ 1� ½LnðMÞ :

Hence, (50) follows from this diagram if we use the direct image mapping

p! : K0c ðT�M;Q=ZÞ-Q=Z:

This proves Proposition 4. &

Remark 6. Formula (50) is an analog of the following well-known property of the Z-invariant (see [APS76a]): the Z-invariant of the cross product of operators is the

product of the Z-invariant of the first factor by the index of the second factor.

5. Proof of Theorem 5. By virtue of (28), we have

f2ZðDÞg ¼ /½sðDÞ ; ½1� L2nþ1ðRP2n � S1Þ S:

Formula (50) for the product and the decomposition (42) give

f2ZðDÞg ¼ /½sðDÞ ; ½1� L2nðRP2nÞ SindD1:


Consequently,

f2ZðDÞg ¼ f2ZðDÞgindD1 ¼1

2n2 ¼ 1

2n�1:

We have used (38) in the second equality. &

Acknowledgments

We are grateful to Prof. P. Gilkey for useful advice and a number of valuableremarks. We thank Prof. M. Karoubi for a fruitful discussion. The results wereannounced in [SS01] and reported at the Conferences ‘‘Geometric Analysis’’in Potsdam, Germany in 2000 and ‘‘Differential Geometry and Global analysis’’in Tianjin, China, 2002. We also thank the referees for their very usefulremarks.

Appendix. Antipodal involution and orientability

Let V be a real vector bundle over a compact space X : In the present appendix, weinvestigate the action of the antipodal involution

a : V-V ; aðvÞ ¼ �v

on the group K�c ðVÞ:

Theorem A.1. One has

a� ¼ ð�1Þn LnðVÞ : K�c ðVÞ-K�

c ðVÞ; n ¼ dimV :

Proof. 1. Let us first prove the desired formula for a vector bundle V of even rankn ¼ 2k: The elements of this group can be realized in terms of the differenceconstruction

½E;F ; s AK0c ðVÞ; E;FAVectðXÞ;

where

s : p�E-p�F ; p : SV-X

is an isomorphism over the spheres SV (we assume that V is equipped with an innerproduct).

Karoubi [Kar78,Kar68] showed that K0c ðVÞ is generated by elements with

quadratic transition functions, which we describe below.


Let ClðVÞ be the Clifford algebra bundle associated with V : Consider quadruplesof the form

ðE; c; f1; f2Þ; ðA:1Þ

where ðE; cÞ is a Clifford module over ClðVÞ; i.e. a complex vector bundle E and ahomomorphism of algebra bundles

c : ClðVÞ-EndðEÞ;

while the involutions

f1;2AEndðEÞ; f 21 ¼ f 22 ¼ 1

anticommute with the Clifford structure

f1;2cðvÞ þ cðvÞf1;2 ¼ 0; vAVCClðVÞ:

According to [Kar78], the group K0c ðVÞ is generated by difference constructions of

the form

½kerð f1 � 1Þ; kerð f2 � 1Þ; ð1� cðvÞf2Þð1þ cðvÞf1Þ : ðA:2Þ

It is clear that the antipodal involution a acts on quadruples (A.1) as

a�ðE; c; f1; f2Þ ¼ ðE;�c; f1; f2Þ:

Let us show that the quadruple

ðE;�c; f1; f2Þ

differs from

ðE; c; f1; f2Þ#L2kðVÞ

by a vector bundle isomorphism.Let us take a local frame e1; e2;y; e2k in V and consider the element

b ¼ ikcðe1Þycðe2kÞ:

Then b2 ¼ 1; and b anticommutes with the Clifford structure c and commutes withthe involutions f1;2:

bcðvÞ þ cðvÞb ¼ 0; fib ¼ bfi: ðA:3Þ

Under a change of the frame, the element b is multiplied by the sign of the transitionmatrix, i.e., globally it defines a vector bundle isomorphism

b : E-E#L2kðVÞ:


The commutation relations (A.3) acquire the form

b�1ðcðvÞ#1L2kðVÞÞb ¼ �cðvÞ; b�1ð fi#1L2kðVÞÞb ¼ fi:

Hence, the quadruples

a�ðE; c; f1; f2Þ and ðE; c; f1; f2Þ#L2kðVÞ

are isomorphic. This proves the theorem, since elements (A.2) generate the entire

group K0c ðVÞ:

2. We now prove that a� acts by the same formula on the odd K-group K1c ðVÞ if

the rank of V is even. To this end, we consider the pull-back p�V of V to the product

X � S1; where p : X � S1-X is the natural projection. The Kunneth formula gives

K0c ðp�VÞ ¼ K0

c ðVÞ"K1c ðVÞ: ðA:4Þ

For the antipodal involution of p�V over X � S1 the theorem holds. Thus, thetheorem holds for both K-groups of the original bundle V ; since the sum in (A.4) isinvariant under the involution and the induced involutions on the terms coincidewith the involution a� corresponding to the bundle V : This proves that the desiredformula holds for the graded K-groups.3. Finally, let V be of odd rank. We express the corresponding K-groups as

K�c ðVÞCK�þ1

c ðV"1Þ:

On the latter group of an even-dimensional bundle V"1 the involution acts asa� Id: This involution is a composition of the usual antipodal involution a� ð�IdÞand the antipodal involution Id� ð�IdÞ along the second factor. Therefore, weobtain the desired formula for a� : K�

c ðVÞ-K�c ðVÞ

a� ¼ ða� ð�IdÞÞ�ðId� ð�IdÞÞ� ¼ ½Ldim Vþ1ðV"1Þ ð�1Þ ¼ �½Ldim V ðVÞ :

Here we use the elementary fact: ðId� ð�IdÞÞ� ¼ �1: &

Corollary A.1. The same formula

a� ¼ ð�1ÞnLnðVÞ : K�c ðV ;ZnÞ-K�

c ðV ;ZnÞ

holds in K-theory with coefficients Zn:

Indeed, K-theory with coefficients in Zn is defined by means of the Moore spaceMn

K�c ðV ;ZnÞ ¼ K�

c ðV � Mn;V � ptÞ:

Applying the theorem to the product X � Mn and the pullback of V ; we obtain thedesired result in K-theory with coefficients.


Remark A.1. For a smooth manifold X with cotangent bundle V ¼ T�X ; (A.2)defines a class of elliptic symbols such that an arbitrary symbol is stably homotopicto a symbol of this class.

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the eta invariant and parity conditions

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