Large scale geometry and index theoryLecture notes, summer term 2015, University of Regensburg.

Alexander Engel

July 14, 2015

Contents

1 Introduction 2

2 Uniform Roe algebra 32.1 Bounded geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 Algebraic smooth uniform Roe algebra . . . . . . . . . . . . . . . . . . . 5

3 K-theory and abstract indices 73.1 Algebraic K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.2 Topological K-theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83.3 Abstract indices of graded operators . . . . . . . . . . . . . . . . . . . . 9

4 Rough index classes of generalized Dirac operators 104.1 Generalized Dirac operators . . . . . . . . . . . . . . . . . . . . . . . . . 104.2 Rough index classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.3 Spin manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144.4 Ungraded operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.5 Weak version of the rough Novikov conjecture . . . . . . . . . . . . . . . 16

5 Cyclic homology and Chern–Connes character 17

6 Uniformly finite homology and rough characters 186.1 Uniformly finite homology . . . . . . . . . . . . . . . . . . . . . . . . . . 186.2 Rough character map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216.3 Proving the main theorem of this lecture . . . . . . . . . . . . . . . . . . 22

1

1 Introduction

Let M be a connected Riemannian manifold of bounded geometry and D a generalizedDirac operator over M . We will construct a rough index class ind(D) ∈ Ktop

∗ (C∗u(M)),where C∗u(M) denotes the uniform Roe algebra of M .

For applications it is important to know the cases where ind(D) 6= 0. There is thefollowing conjecture, which is a weak version of the so-called rough Novikov conjecture:

Conjecture 1.1. If M is uniformly contractible1, then ind( /D) 6= 0, where /D is the Diracoperator associated to a spin structure on M .

On the other hand, we will show the following lemma in the lecture:

Lemma 1.2. If M is spin and /D the associated Dirac operator, then ind( /D) = 0 if Mhas uniformly positive scalar curvature2.

It follows immediately that uniformly contractible manifolds do not have upsc-metrics.But even more, we may deduce the following corollary from the conjecture:

Corollary 1.3. Assume that the conjecture holds. Then no closed aspherical3 manifoldadmits a upsc-metric4.

The goal of this lecture is to introduce all the notions occuring in the above discussion(this will be the first half of the lecture) and then to show the following special case ofthe conjecture (this will be the second half of the lecture):

Main Theorem of this Lecture 1.4. If M has polynomial volume growth5 and ispolynomially contractible6, then Conjecture 1.1 holds for M , i.e., ind( /D) 6= 0.

Proving the above theorem will require us to introduce cyclic homology and uniformlyfinite homology, two important homology theories that one should know. Furthermore,we will have to use a generalization of the Atiyah–Singer index theorem, which we willdiscuss but of course not prove in this lecture.We will also deduce the following corollary from Theorem 1.4, which is Corollary 1.3

restricted to the virtually nilpotent case. For this we will have to dip into geometricgroup theory and discuss the very important connection between geometric group theoryand coarse geometry.

Corollary 1.5. Let M be an aspherical manifold with virtually nilpotent fundamentalgroup. Then M does not admit a upsc-metric.

1This means that every ball BR(x) ⊂M is contractible in the larger ball BS(x), where S depends onlyon R but not on x. It follows especially that M is spin.

2From now on we will abbreviate “uniformly positive scalar curvature” by upsc.3Being aspherical means πk(M) = 0 for k ≥ 2.4Since M is closed, there is no difference here between uniformly positive and positive scalar curvature.5This means that volBr(x) grows asymptotically as a polynomial function in r for r →∞.6This is a sharpening of being uniformly contractible: we want that S occuring in the definition ofuniform contractibility depends polynomially on R.

2

2 Uniform Roe algebra

2.1 Bounded geometry

Definition 2.1. A Riemannian manifold M has bounded geometry if

• the curvature tensor and all its derivatives are bounded, i.e., ‖∇k Rm(x)‖ < Ck forall x ∈M , k ∈ N0, and

• the injectivity radius is uniformly positive, i.e., inj-radM(x) > ε for all x ∈M andfixed ε > 0.

Lemma 2.2. If M has bounded geometry, then M is complete.If M has bounded geometry and is non-compact, then it has infinite volume.

Proof. The first statement follows directly from inj-radM(x) > ε since using this wemay extend every geodesic step-by-step to be defined for all time (at every step we mayincrease the time interval on which it is defined by ε).For the second statement we first use that since M is non-compact and complete we

may find for every x0 ∈ M a sequence (xi) ⊂ M with d(x0, xi)→∞ for i→∞ (sinceotherwise M would be bounded and therefore compact). Without loss of generality wemay assume that d(xi, xj) > 2ε for all i 6= j. Now we put a normal ball of radius εaround each point xi. They are all disjoint and since M has bounded geometry, theirvolumes are uniformly bounded from below, i.e., volBε(xi) > δ for some δ > 0.

Examples 2.3. There are plenty of examples of manifolds of bounded geometry:

• Compact manifolds.

• Coverings of compact manifolds equipped with the pullback metric.

• Homogeneous manifolds with an invariant metric.

• Leafs in a foliation of a compact manifold (this is non-trivial to show).

An important local characterization of bounded geometry is the following:

Lemma 2.4. Let the injectivity radius of M be uniformly positive.Then M has bounded geometry if and only iff for any 0 < r < inj-radM all derivatives

of transition functions between overlapping normal coordiante charts of radius r areuniformly bounded, i.e., for φ : Br(x)→ Rm and ψ : Br(y)→ Rm with Br(x)∩Br(y) 6= ∅we have |∂α(ψ φ−1)| < Cα for all multi-indices α and all such φ, ψ.

We also have nice partitions of unity on manifolds of bounded geometry:

Lemma 2.5. Let M have bounded geometry.For every 0 < ε < inj-radM /3 exists a countable covering of M by balls Bε(xi) with:

• the points xi ∈M for a uniformly discrete set, i.e., d(xi, xj) > δ, and

3

• the balls B2ε(xi) form a uniformly locally finite cover of M , i.e., there is a C ∈ Nsuch that every x ∈M lies in at most C of the balls.

Furthermore, there is a subordinate partition of unity 1 =∑ϕi, i.e., suppϕi ⊂ B2ε(xi),

such that in normal coordinates the derivatives of all the ϕi are uniformly bounded, i.e.,|∂αϕi| < Cα for all multi-indices α.

Let us turn now to function spaces on manifolds of bounded geometry. Unless otherwisestated we consider only complex-valued functions.

Definition 2.6. A function f ∈ C∞(M) is called Cr-bounded, if we have ‖∇kf‖∞ < Ckfor all k ≤ r.

By the above local characterization of bounded geometry, this is equivalent to demand-ing |∂αf | < Cα in all normal coordinate charts of a fixed radius and for all multi-indicesα with |α| ≤ r.

We set

Crb (M) := f ∈ C∞(M) : f is Cr-bounded,

C∞b (M) :=⋂r

Crb (M).

Definition 2.7. The global Hk-Sobolev norm of f ∈ C∞c (M) is defined by

‖f‖2Hk :=

k∑i=0

∫M

‖∇if(x)‖2dx.

The completion of C∞c (M) with respect to this norm is denoted by Hk(M). Note thatthis norm comes from an inner product, i.e., Hk(M) is actually even a Hilbert space.

If M has bounded geometry, the global Sobolev norm is equivalent to (but in generalnot equal to) the local ones given by

‖f‖2Hk =

∞∑j=1

‖ϕjf‖2Hk(B2ε(xj))

,

where ϕj is a partition of unity as in Lemma 2.5 and Hk(B2ε(xj)) denotes the usualSobolev norm on Rm, where we have B2ε(xj) ⊂ Rm via normal coordinates.Recall that on Rm we have ‖f‖Hk(Rm) = ‖(1 + |·|2)k/2 · f‖L2(Rm) which allows one to

define Sobolev norms for all k ∈ R. Using the above local Sobolev norms on a manifoldof bounded geometry M , we may therefore also define Sobolev norms on M for all k ∈ R.

Lemma 2.8. H−k(M) is the dual space to Hk(M).

Because of bounded geometry, the usual Sobolev embedding theorems are true:

Theorem 2.9. Let Mm have bounded geometry.Then we have for all k > r +m/2 continuous embeddings Hk(M) → Cr

b (M).

4

And last, we have to discuss vector bundles of bounded geometry. As for functions,unless otherwise stated we consider only complex vector bundles.

Definition 2.10. Let E →M be a vector bundle of bounded geometry equipped with abundle metric and a compatible connection.7 Then E is said to have bounded geometry,if the curvature tensor of E and all its derivatives are bounded.

Examples 2.11. As for manifolds of bounded geometry, there are also plenty of examplesof vector bundles of bounded geometry:

• flat bundles (by definition this are bundles where the curvature tensor vanishes),

• every vector bundle over a compact manifold,

• if M →M is a covering of a compact manifold, equipped with the pullback metric,then the pullback bundle E → M will have bounded geometry if equipped withthe pullback metric, and

• if M has bounded geometry, then its tangent bundle TM has bounded geometry.

Furthermore, the usual constructions like E∗, E ⊕ F , E ⊗ F and E F produce againvector bundles of bounded geometry out of such bundles.

We will also need an analogue of normal coordinates for vector bundles:

Definition 2.12. At a point x ∈M we choose an orthonormal basis for Ex and extendit to a framing of E in a normal coordinate ball around x by parallel translation alongradial geodesics. Such a framing is called synchronous.

Everything that we have discussed above for manifolds of bounded geometry appliesalso to vector bundles of bounded geometry. For local computations we always usenormal coordinate charts and synchronous framings.From now on, if not otherwise stated, we will only consider manifolds and vector

bundles of bounded geometry.

2.2 Algebraic smooth uniform Roe algebra

Definition 2.13. We setH∞(E) :=

⋂k∈N

Hk(E)

and equip it with the following topology: a linear map φ : V → H∞(E) is continuous ifand only if φ : V → Hk(E) is continuous for all k ∈ N.

Definition 2.14. Let H−∞(E) denote the dual space of H∞(E).

Lemma 2.15. We have H−∞(E) =⋃k∈NH

−k(E).

7Compatibility means ∇MX 〈e, f〉 = 〈∇EXe, f〉+ 〈e,∇EXf〉.

5

We put the following topology onH−∞(E): a linear map φ : H−∞(E)→ V is continuousif and only if the restrictions φ : H−k(E)→ V are continuous for all k ∈ N.

Definition 2.16. A continuous linear operator A : H−∞(E) → H∞(F ) is called asmoothing operator.

By the definition of the topologies on H−∞(E) and on H∞(F ) the following lemma isimmediate:

Lemma 2.17. A linear operator A : H−∞(E)→ H∞(F ) is continuous if and only if itsrestrictions are bounded operators H−k(E)→ H l(F ) for all k, l ∈ N.

Proposition 2.18. Let M be a manifold of bounded geometry and E,F → M vectorbundles of bounded geometry. Let A : H−∞(E)→ H∞(F ) be a smoothing operator.

Then A is an integral operator with kernel kA ∈ C∞b (F E∗). Furthermore, themap B(H−∞(E), H∞(F ))→ C∞b (F E∗) associating a smoothing operator its kernel iscontinuous.

Being an integral operator means that we have (A(s))(x) =∫MkA(x, y)s(y) dy. The

topology on B(H−∞(E), H∞(F )) is defined by: Ai converges to A if and only if therestrictions to H−k(E) converge to the corresponding restriction of A in the operator normfor operators H−k(E)→ H l(F ), for all k, l ∈ N. The topology on C∞b (F E∗) is definedby the family ‖∇i·‖∞ for i ∈ N, i.e., kAi converges to kA if and only if ‖∇i(kAi − kA)‖∞converges to 0 for all i ∈ N.The idea of the proof of the above proposition is the following (for simplicity, we

assume E = F = C): for y ∈M the Dirac distribution δy is defined on smooth functionsas δy(f) := f(y). Due to the Sobolev embedding theorem, we have δy ∈ H−k(M) forsome k ∈ N. So A(δy) is a smooth function, and in fact, it will be kA(·, y) if kA is theintegral kernel of A. If we do not know yet that A is an integral operator then we definekA by this (and then have to show that this is indeed the integral kernel of A).

Definition 2.19. An operator A has propagation ≤ S if supp(A(e)) ⊂ BS(supp(e)) forall sections e on which A is defined.

For a smoothing operator A it is clear that A has propagation ≤ S if and only if theintegral kernel kA of it satisfies kA(x, y) = 0 for all x, y ∈ M with d(x, y) > S. Theconverse also holds: a function k ∈ C∞b (F E∗) with k(x, y) = 0 for d(x, y) > S definesa smoothing operator with propagation ≤ S.

Definition 2.20. The algebraic smooth uniform Roe algebra is defined by

C∗−∞(E) := all smoothing operators on E with finite propagation ⊂ B(L2(E)).

Its closure in B(L2(E)) is the uniform Roe algebra C∗u(E).

Exercise 2.21. Let g, h ∈ C∞c (E). Show that the operator defined by s 7→ (s, g) · h isa finite propagation smoothing operator, i.e., an element of C∗−∞(E). Here we use thenotation (e, f) :=

∫M〈e(x), f(x)〉 dx for the inner product of L2(E).

6

3 K-theory and abstract indices

3.1 Algebraic K-theory

Let R be a ring. We denote by Idemn(R) the idempotent (n×n)-matrices with entries inthe ring R. We embed Idemn(R) into Idemn+k(R) by the map M 7→ diag(M, 0), wherediag(·, ·) denotes the diagonal matrix with the corresponding entries, and consider thedirect limit Idem∞(R) := lim−→ Idemn(R). Let e ∼ f for e, f ∈ Idem∞(R) if there existsan invertible z ∈ Mat∞(R)+ with zez−1 = f , where ·+ denotes the unitization of a ring.

Definition 3.1. If the ring R has a unit, we define Kalg0 (R) as the group completion of

the abelian monoid Idem∞(R)/∼, where addition is given by [e] + [f ] := diag(e, f).

The monoid Idem∞(R)/∼ is abelian since we have diag(e, f) ∼ diag(f, e) via the

invertible matrix(

0 11 0

). A model for the group completion is given by the Grothendieck

group construction, i.e., we may regard an element of Kalg0 (R) as a formal difference

[e] − [f ] with e, f ∈ Idem∞(R) and we have the relation [e] − [f ] = [e′] − [f ′] if thereexists an element g ∈ Idem∞(R) with diag(e, f ′, g) ∼ diag(e′, f, g).

Definition 3.2. If R has no unit, we define Kalg0 (R) to consist of all formal differences

[e]− [f ] with e, f ∈ Idem∞(R+) and e ≡ f mod Mat∞(R).

Let R be a ring with unit. We denote by GLn(R) the invertible (n × n)-matriceswith entries in R. We embed GLn(R) into GLn+k(R) by M 7→ diag(M, idk), and defineGL∞(R) := lim−→GLn(R).

Definition 3.3. We define Kalg1 (R) := GL∞(R)/[GL∞(R),GL∞(R)] if R has a unit. In

the case that R has no unit we use GLn(R) := M ∈ GLn(R+) : M ≡ idn mod Matn(R)and then do the same as for unital rings.

The group operation is given by multiplication of the matrices. Note that it is abelianby definition since we have taken the quotient by the commutator subgroup.

Remark 3.4. Note that our definition of Kalg1 (R) for non-unital rings R does not coincide

with the usual definition given in the literature. The reason is that we are more interestedin topological K-theory (to be defined further down) and our definition of the algebraicK1 for non-unital rings will make it easier for us to compare it to the topological K1.

Lemma 3.5. Let ϕ : R→ S be a homomorphism of rings. Then we have induced mapsϕ∗ : K

alg∗ (R)→ Kalg

∗ (S).

Let R be a ring and I ⊂ R a two-sided ideal. Denote by ι : I → R the inclusion andby π : R→ R/I the quotient map.

Theorem 3.6. There exists a map ∂ : Kalg1 (R/I) → Kalg

0 (I), such that the followingsequence is exact:

Kalg1 (R)

π∗−→ Kalg1 (R/I)

∂−→ Kalg0 (I)

ι∗−→ Kalg0 (R)

π∗−→ Kalg0 (R/I).

7

Remark 3.7. It is tempting to extend the above sequence to the left byKalg1 (I)

ι∗−→ Kalg1 (R).

But with our definition of Kalg1 for non-unital rings this will be in general not an exact

sequence anymore; see also Remark 3.4. If we would use the usual definition of Kalg1 for

non-unital rings, then we could do this.Though we will not prove the above theorem, we will at least define the boundary

map ∂ : Kalg1 (R/I) → Kalg

0 (I). We will assume that R is unital (the construction ofthis map for non-unital R is the same, but one has to make sure at every step that theconstruction still makes sense). Let u ∈ GLn(R/I). We choose a lift w ∈ GL2n(R) ofdiag(u, u−1) and define ∂([u]) := [wpnw

−1]− [pn] ∈ Kalg0 (I). Here pn denotes the diagonal

matrix diag(1, . . . , 1, 0, . . .) with n ones on the diagonal. Note that wpnw−1 ∈ Mat2n(I+),because diag(u, u−1) commutes with pn, i.e., wpnw−1 ≡ pn mod Mat2n(I).

Let us explain how to choose a lift w ∈ GL2n(R) of diag(u, u−1): first we note that we

have diag(u, u−1) =

(1 u0 1

)·(

1 0−u−1 1

)·(

1 u0 1

)·(

0 −11 0

). Now we choose lifts b and

c of u and u−1, where b and c do not need to be invertible. But nevertheless(

1 b0 1

)and(

1 0−c 1

)will be invertible. So we may set w :=

(1 b0 1

)·(

1 0−c 1

)·(

1 b0 1

)·(

0 −11 0

).

If R has no unit, this construction works analogously. Though we will have now ingeneral w ∈ GL2n(R+), i.e., the matrix w may have 1s at off-diagonal places, we stillhave wpnw−1 ≡ pn mod Mat2n(I).

3.2 Topological K-theory

Definition 3.8. Let H be a separable Hilbert space over C. A C∗-algebra will be for usa closed ∗-subalgebra of B(H).

We define topological K-theory Ktop∗ (A) for a C∗-algebra A analogously as algebraic

K-theory, but we use now a diffent equivalence relation: we use homotopy. So if A isunital we have Ktop

0 (A) given by the group completion of Idem∞(A)/∼, where e ∼ f if eand f can be joined by a continuous path inside Idem∞(A), and Ktop

1 (A) := GL∞(A)/∼with ∼ meaning now homotopies via a path of invertibles. If A is non-unital we do thesame adjustments as for algebraic K-theory.

Proposition 3.9. We do have Kalg0 (A) = Ktop

0 (A) and we have a natural surjectionKalg

1 (A) Ktop1 (A).

For the proofs we assume that A is unital. The non-unital case works analogously.

Proof for K1. Let x, y be invertible. Setting

wt :=

(x 00 1

)·(

cos t − sin tsin t cos t

)·(y 00 1

)·(

cos t sin t− sin t cos t

)we get a path of invertibles from diag(xy, 1) to diag(x, y). Setting

zt :=

(cos t − sin tsin t cos t

)·(x 00 y

)·(

cos t sin t− sin t cos t

)

8

we get a path of invertibles from diag(x, y) to diag(y, x). This means that we have

diag(xy, 1) ∼ diag(x, y) ∼ diag(y, x) ∼ diag(yx, 1),

i.e., multiplication is commutative in Ktop1 (A). But Kalg

1 (A) was defined as the quotientof GL∞(A) by its commutator subgroup, so by the universal property of the commutatorsubgroup we get a unique and natural surjection Kalg

1 (A) Ktop1 (A).

Proof for K0. We will show that the equivalence relations are actually the same. Firstlet e, f be idempotents with ‖e− f‖ < 1/‖2e− 1‖. We set v := (2e− 1)(2f − 1) + 1 andget 2− v = 2(2e− 1)(e− f) and so by the assumed inequality v will be invertible.8 Wealso have ev = vf , i.e., v−1ev = f .From the above it follows that if e and f are two homotopic idempotents, then they

are similar (i.e., conjugate by an invertible): subdivide the path from e to f into smallpieces so that for each piece the above is applicable to the endpoints of the piece.Now let e and f be similar, i.e., there exists an invertible z with zez−1 = f . Let wt

be a path of invertibles from 1 to diag(z, z−1); the existence of such a path was shownabove. Then et := wt · diag(e, 0) · w−1

t is a homotopy from e to f .

Theorem 3.10. Let J ⊂ A be a closed, two-sided ∗-ideal. Denote by ι : J → A theinclusion and by π : A→ A/J the quotient map. Then we have an exact sequence

Ktop1 (J)

ι∗−→ Ktop1 (A)

π∗−→ Ktop1 (A/J)

∂−→ Ktop0 (J)

ι∗−→ Ktop0 (A)

π∗−→ Ktop0 (A/J),

where the boundary map ∂ : Ktop1 (A/J)→ Ktop

0 (J) is defined as for algebraic K-theory.

3.3 Abstract indices of graded operators

Let A be a unital and graded algebra. The latter means that we have a decompositionA = A+ ⊕ A− as a vector space and additionally A± · A± ⊂ A+ and A± · A∓ ⊂ A− (sothis especially means that A+ is a subalgebra of A, but A− is not). Note that it followsthat 1 ∈ A+. Let B ⊂ A be an ideal with the induced grading B± = B ∩ A±.

Definition 3.11. We call F ∈ A an abstract Fredholm operator if F is odd, i.e., wehave F ∈ A−, and F 2 − 1 ∈ B. (Note that we then actually have F 2 − 1 ∈ B+.)

Definition 3.12. Suppose that there exists an invertible, odd element U ∈ A−. Thenwe define ind(F ) := ∂([UF ]) ∈ Kalg

0 (B+), where ∂ : Kalg1 (A+/B+) → Kalg

0 (B+) is theboundary map in the long exact sequence for algebraic K-theory.

Proof of well-definedness. Suppose W ∈ A− is another invertible. Then

[UF ] · [WF ]−1 = [UFF−1W−1] = [UW−1] ∈ Kalg1 (A+/B+),

because F is invertible in the quotient A+/B+ (this follows from F 2 − 1 ∈ B+). ButUW−1 is an invertible in A+ and not only in A+/B+, i.e, we have [UW−1] ∈ Kalg

1 (A+)and therefore ∂([UF ]·[WF ]−1) = [0] ∈ Kalg

0 (B+) by the long exact sequence (compositionof two consecutive maps in the sequence is zero). Therefore ∂([UF ]) = ∂([WF ]).

8Here we need the fact that if ‖a‖ < 1, then (1− a) is invertible.

9

Remark 3.13. Note that the existence of an invertible, odd element U ∈ A− is notguaranteed, i.e., there might be graded algebras without such an element (note that theidentity 1 ∈ A is always even, i.e., 1 ∈ A+). In this case we plainly can not define indexclasses for abstract Fredholm operators by the above method.From now on we assume that we always have such an element U in A.

Lemma 3.14. Let F and F ′ be two abstract Fredholm operators with F −F ′ ∈ B. Thenind(F ) = ind(F ′).If F is such that F 2 = 1 (in A and not only in A/B), then ind(F ) = 0.

Proof. For the first statement note that we have [UF ] · [UF ′]−1 = [1] ∈ Kalg1 (A+/B+),

because F = F ′ in the quotient A/B and so F (F ′)−1 = 1 ∈ A+/B+. But again, becauseactually [1] ∈ Kalg

1 (A+), the claim ∂([UF ]) = ∂([UF ′]) follows.For the second statement note that in this case we have [UF ] ∈ Kalg

1 (A+) and therefore∂([UF ]) = [0] ∈ Kalg

0 (B+).

4 Rough index classes of generalized Dirac operators

4.1 Generalized Dirac operators

Let M be a Riemannian manifold which we always assume to be connected and havebounded geometry and let S → M be a graded vector bundle which we also alwaysassume to have bounded geometry. Graded means here that we have an orthogonaldecomposition S = S+ ⊕ S−.

Definition 4.1. S is called a graded Dirac bundle if we have an R-linear bundle mor-phism TM → End−(S), where End−(S) denotes the odd endomorphisms of S, i.e.,endomorphisms mapping S± to S∓, with:

• 〈X · s1, s2〉+ 〈s1, X · s2〉 = 0,

• X · (X · s) = −‖X‖2 · s, and

• ∇SZ(X · s) = (∇M

Z X) · s+X · ∇SZs.

Here X · s means throwing X into the bundle morphism TM → End−(S) and thenapplying it to s. It is called the Clifford multiplication (hence the notation).

Note that from the first two points it follows that we have 〈X ·s1, X ·s2〉 = ‖X‖2 ·〈s1, s2〉.Especially, if ‖X‖ = 1 then Clifford multiplication with X is an isometry.

Definition 4.2. The Dirac operator D of a Dirac bundle S is defined as the composition

C∞(S)∇−→ C∞(S ⊗ T ∗M)→ C∞(S ⊗ TM)→ C∞(S),

where the second arrow is given by the Riemannian metric and the third by Cliffordmultiplication.

Since S is graded, D is an odd operator.

10

Operators that arise in the above fashion are called generalized Dirac operators. Inlocal coordinates they are given by D(s) =

∑i ei · ∇S

eis, where ei is an orthonormal

frame of TM .

Examples 4.3. We will give two examples of Dirac operators on compact manifolds (butnot of Dirac bundles since it is hard to define the Clifford multiplication on the followingexamples without discussing the Clifford algebra).

• The space of complex-valued forms Ω∗(M) =⊕

Ωk(M) can be made into aDirac bundle (i.e., we may define a Clifford multiplication on it). We grade itaccording to Ω∗(M) = Ωev(M) ⊕ Ωodd(M). The exterior derivative d becomesan odd operator and we let d∗ be its adjoint with respect to the inner product(α, β) :=

∫M〈α(x), β(x)〉dx. The associated Dirac operator is given by D = d+ d∗

and is called the de Rham operator.

• Let M be oriented and dim(M) = 4k. We define a grading operator ε on Ω∗(M)by ε ·α := (−1)ke1 · · · e4k ·α for (e1, . . . , e4k) an oriented orthonormal frame of TMand the dots in the definition of ε represent Clifford multiplication. This operatoris well-defined, i.e., independent of the choice of oriented orthonormal frame. Thatit is a grading operator means that we have ε2 = 1 and ε = ε∗. Therefore we maygrade Ω∗(M) by the ±1-eigenspaces of ε. This grading differs from the previousgrading via even / odd forms drastically. The Dirac operator in this case is stillgiven by the formula D = d+ d∗ and is called the signature operator.

Note that though in both examples the formula for both operators is the same, i.e.,d+d∗, they behave due to the different grading completely different with respect to indextheoretic considerations. This shows that the grading is an essential part of an operator.

We will discuss now important analytic properties of generalized Dirac operators. OnC∞c (S) we may define an inner product by (s1, s2) :=

∫M〈s1(x), s2(x)〉dx and get the

Hilbert space L2(S) by completing. Note that D is a densely defined, unbounded operatoron L2(S).

Lemma 4.4. D is symmetric, i.e., we have (Ds1, s2) = (s1, Ds1) for all s1, s2 ∈ C∞c (S).Even more, D is essentially self-adjoint, i.e., it has exactly one self-adjoint extension.

A crucial property of generalized Dirac operators is that they are elliptic. Withoutdefining this, we will mention two important consequences of it:

• There exists a finite propagation operator Q of order −1, i.e., Q defines an operatorHk(S)→ Hk+1(S) for all k ∈ R, with QP ≡ PQ ≡ id modulo finite propagationsmoothing operators.

• The norm ‖·‖H1 is equivalent to ‖·‖L2 + ‖D·‖L2 on H1(S).

From the second point we may conclude that for each k ∈ N there is a norm onHk(S) equivalent to the global one from Definition 2.7 such that D becomes essentiallyself-adjoint on Hk(S) equipped with it.

11

Proposition 4.5. The wave operator eitD has propagation |t|.

Lemma 4.6. Let f : R→ R be a function with compactly supported Fourier transformf(t) = (2π)−1/2

∫R f(x)e−itxdx.

Then f(D) is a finite propagation smoothing operator, i.e., f(D) ∈ C∗−∞(S).

Proof. We have f(D) = (2π)−1/2∫R f(t)eitDdt. Since eitD has propagation |t| and f is

compactly supported, we get that f(D) has finite propagation.We use integration by parts to get

∫f(t)eitDdt = −1/(iD) ·

∫f ′(t)eitDdt. Now this

does not make sense since D is not invertible. But we know that we have an operator Qof order −1 which is the inverse of D modulo finite propagation smoothing operators. Sowe may interpret −1/(iD) as iQ and then the above formula will hold true modulo finitepropagation smoothing operators. But the right hand side iQ ·

∫f ′(t)eitDdt is now an

operator of order −1 since eitD has order 0 and Q has order −1. Iterating this argumentwe get that f(D) is an operator of order −k for every k ∈ N, i.e., f(D) is a smoothingoperator. (Note that a function with compact Fourier transform is a Schwartz functionas is its Fourier transform, i.e., f is infinitely often differentiable.)

4.2 Rough index classes

Definition 4.7. A smooth function χ : R → R is called a normalizing function, ifχ(x)→ ±1 for x→ ±∞ and χ′ has compactly supported Fourier transform.

Lemma 4.8. Some basic facts about normalizing functions:

(i) Normalizing functions exist and in fact, every continuous function f : R→ R withf(x)→ ±1 for x→ ±∞ can be approximated uniformly by them.

If f is odd, i.e., f(x) = −f(−x), then the approximating normalizing functionsmay also be chosen to be odd.

(ii) If χ is a normalizing function then χ2−1 has compactly supported Fourier transform.

(iii) If χ and ψ are two normalizing functions then χ − ψ has compactly supportedFourier transform.

Now let D be a graded generalized Dirac operator and χ an odd normalizing function.Let A be the graded9 algebra of all finite propagation, order 0 operators10 on the gradedDirac bundle S on which D is defined. Then C∗−∞(S) ⊂ A is a two-sided ideal.The reason why we assume χ to be odd is the following lemma:

Lemma 4.9. χ(D) ∈ A−.

Note that the above lemma does not only contain the statement that χ(D) is odd,but also that χ(D) is a finite propagation operator of order 0, which is also a non-trivialstatement.

9The grading is by even / odd operators.10Being of order 0 means that the operator is an operator Hk(S)→ Hk(S) for all k ∈ R.

12

Definition 4.10. We define ind(D) ∈ Kalg0 (C∗−∞(S+)) as the abstract index class of the

operator χ(D).

Proof of well-definedness. First we have to show that χ(D) is an abstract Fredholmoperator; see Definition 3.11. That χ(D) is an element of the algebra A and that it isodd is exactly the statement of the above Lemma 4.9. That χ(D)2− 1 ∈ C∗−∞(S) followsfrom χ(D)2 − 1 = (χ2 − 1)(D) together with Lemma 4.8(ii) and Lemma 4.6.That ind(D) is independent of the choice of odd normalizing function follows from

Lemma 4.8(iii) together with Lemma 4.6 and the first part of Lemma 3.14.

Remark 4.11. Note that for the definition of ind(D) we need an invertible element U ∈ A−,i.e., we need an invertible, odd, finite propagation operator of order 0 on S.There are two possible solutions for this: firstly, if M is spin, even-dimensional and

has uniformly positive scalar curvature, then we may take U := χ( /D), where /D is thegraded Dirac operator associated to the spin structure of M (we will see later why this isan admissible choice in this case, and we will also discuss spin manifolds later). Withthis method we can only define index classes for abstract Fredholm operators acting onthe graded spinor bundle /S since only on this bundle U = χ( /D) is defined and gives anadmissible element. So this method here is quite restricted.Secondly, if M admits a nowhere vanishing normalized vector field with bounded

derivatives, then Clifford multiplication with this vector field will do the job. Note thatsuch a vector field exists on M if and only if the Euler class e(M) ∈ Huf

0 (M) vanishes.This method is much more general than the first one since here U is now defined forevery Dirac bundle over M .

It is not known to me if such an admissible element U exists in general. From now onwe will just assume that we have such an element U given somehow.

Lemma 4.12. Assume that 0 6∈ spec(D), where spec(D) denotes the spectrum of D.Then ind(D) = [0] ∈ Ktop

0 (C∗u(S+)).

Proof. Since the spectrum is always a closed subset of R, we have [−δ, δ] ∩ spec(D) = ∅for some δ > 0. Let f : R→ R be a continuous, odd function with f(x) = 1 for x ≥ δ. ByLemma 4.8(i) there are odd, normalizing functions χi approximating f uniformly. Sinceχi(D) ∈ A− for every i ∈ N, we get f(D) ∈ A− ⊂ B(L2(S)), and f(D)− χ(D) ∈ C∗u(S)for any normalizing function χ since χi(D)− χ(D) ∈ C∗−∞(S) for all i.

Forming ind(D) ∈ Kalg0 (C∗−∞(S+)) and mapping it then to Ktop

0 (C∗u(S+)) is the sameas directly forming ∂[Uχ(D)] ∈ Ktop

0 (C∗u(S+)) for [Uχ(D)] ∈ Ktop1 (A+/C∗u(S+)). But for

the latter we may equivalent use [Uf(D)] since f(D)− χ(D) ∈ C∗u(S). Because f 2 ≡ 1on the spectrum of D, we get f(D)2 = 1 and therefore ind(D) = [0] ∈ Ktop

0 (C∗u(S+)) bythe second statement in Lemma 3.14.

An important application of the above lemma is the following one to spin manifoldswith upsc-metrics. We will discuss spin manifolds directly afterwards.

Proposition 4.13. Let M be an even-dimensional spin manifold of bounded geometryand assume that M has uniformly positive scalar curvature.

13

Then ind( /D) = [0] ∈ Ktop0 (C∗u(/S

+)), where /D is the graded Dirac operator associated

to the spin structure of M (see the next section for a discussion of spin structures andtheir Dirac operators).

Proof. Let δ > 0 be a lower bound for the scalar curvature κ(x). Due to the Lichnerowiczformula /D

2= ∇∗∇+ κ/4, where κ : M → R is the scalar curvature of the Riemannian

manifold M , we get [−δ, δ] ∩ spec( /D2) = ∅ and therefore 0 6∈ spec( /D) by properties of

the functional calculus (which are needed to go from spec( /D2) to spec( /D)). The claim

now follows with the above Lemma 4.12.

4.3 Spin manifolds

The best way to define the spin groups is via Clifford algebras, but since we will notdiscuss Clifford algebras in this lecture, we have to use another definition: the spin groupsare double covers of the special orthogonal group.Since for n ≥ 3 we have π1(SO(n)) = Z/2Z, it follows that Spin(n) is the universal

cover of SO(n). In the case n = 2 we have π1(SO(2)) = Z and so Spin(2) is now notthe universal cover. What we have is Spin(2) ∼= U(1), where u ∈ Spin(2) acts on C byz 7→ u2z. Finally, we have Spin(1) = O(1).

Definition 4.14. Let us denote by π : Spin(n)→ SO(n) the covering homomorphism.Let M be an oriented, n-dimensional Riemannian manifold and let PSO →M denote theprincipal SO(n)-bundle of oriented orthonormal frames of TM .A spin structure on M is a choice of two-fold cover ξ : PSpin → PSO, where PSpin is a

Spin(n)-principal bundle, with ξ(pg) = ξ(p)π(g) for all p ∈ PSpin and g ∈ Spin(n).

Note that not every manifold admits a spin structure.

Facts 4.15. Some facts about spin manifolds:

• Only oriented manifolds may have a spin structure.

• An oriented manifold is spinable if and only if its second Stiefel-Whitney classw2(M) ∈ H2(M ;Z/2Z) vanishes.

• If π1(M) = 0 and dim(M) ≥ 5, then M is spinable if and only if the normal bundleof any embedded 2-sphere in M is trivial.

Examples 4.16. Some examples of spin manifolds:

• CP 2n+2 is spin,

• all oriented 3-manifolds are spinable and

• all oriented manifolds with H2(M ;Z/2Z) = 0 are spinable.

A non-example is CP 2n: it does not admit a spin structure.

14

For us the most important consequence of a choice of spin structure on an orientedRiemannian manifold is the complex spinor bundle /S that we may construct then. Thisis naturally a Dirac bundle and the associated Dirac operator is usually denoted by /D.

A crucial property of it is the Lichnerowicz formula: /D2= ∇∗∇+κ/4, where κ : M → R

is the scalar curvature of M . We have used this formula in the proof of Proposition 4.13to show that 0 6∈ spec( /D) if κ(x) > δ > 0 for all x ∈M .

If M is even-dimensional then /S is naturally graded, but if M is odd-dimensional then/S is ungraded. This is the reason why we have in Proposition 4.13 the restriction toeven-dimensional manifolds. Since we want to drop this restriction, we have to discussnow index theory of ungraded operators.

4.4 Ungraded operators

Let D be a generalized Dirac operator associated to a spinor bundle S, but now we donot demand them to be graded. Setting P := (χ(D) + 1)/2 for χ a normalizing function(here in the ungraded case we do not need it to be odd), we get P 2 = P in A/C∗−∞(S),where A is as before the algebra of all finite propagation, order 0 operators on S.

Let A ⊂ B(L2(S)) be the closure of A and remember that C∗u(S) = C∗−∞(S) is atwo-sided ideal in A. We set

ind(D) := [e2πiP ] ∈ Ktop1 (C∗u(S)).

Proof of well-definedness. The operator e2πiP is invertible, because of e2πiP e−2πiP = 1.Since P 2 ≡ P mod C∗−∞(S) we get

N∑n=0

1

n!(2πi)nP n ≡

N∑n=1

1

n!(2πi)n︸ ︷︷ ︸

→e2πi−1=0

·P + 1 mod C∗−∞(S). (4.1)

Taking N →∞ we get e2πiP ≡ 1 mod C∗u(S), where we have to pass from C∗−∞(S) to itsclosure C∗u(S) since we are taking a limit. So e2πiP defines an element of Ktop

1 (C∗u(S)).It remains to show the independence of the choice of χ: let ψ be another normalizing

function. Then χt(x) := t·χ(x)+(1−t)·ψ(x) is a continuous path of normalizing functions,so we get a path Pt of idempotents in A/C∗−∞(S) and therefore a path e2πiPt of invertibleoperators with e2πiPt ≡ 1 mod C∗u(S) for all t. So [e2πiP0 ] = [e2πiP1 ] ∈ Ktop

1 (C∗u(S)) sincethe relation in the definition of Ktop

1 (C∗u(S)) is exactly homotopy via invertibles.

Let us prove now analogues of Lemma 4.12 and Proposition 4.13 for ungraded operators.In contrast to Lemma 4.12 the spectral gap is now allowed to be anywhere in the spectrum.

Lemma 4.17. Let D have a spectral gap, i.e., there is some t ∈ R with t 6∈ spec(D).Then ind(D) = [1] ∈ Ktop

1 (C∗u(S)).

Proof. Since D has a spectral gap we may find a continuous function f with f(x) = ±1on spec(D) and f(x) → ±1 for x → ±∞. This f may be approximated uniformly by

15

normalizing functions χj by Lemma 4.8(i). Since χj(D) ∈ A for each j by the analogueof Lemma 4.9 for ungraded operators, we get f(D) ∈ A. For P := (f(D) + 1)/2 we getas above that e2πiP is invertible. Even more, due to the choice of our f we get P 2 = Pin A and not only in the quotient A/C∗u(S). So Argument (4.1) gives directly e2πiP = 1and not only that e2πiP is equivalent to 1 modulo C∗u(S). So [e2πiP ] = [1].

It remains to show that [e2πiP ] = ind(D) since f is usually not a normalizing function(the condition that the derivative f ′ must have compactly supported Fourier transformmight be violated). The operators Pj := (χj(D) + 1)/2 converge to P and all the Pjdefine the index class ind(D) = [e2πiPj ]. In unital Banach algebras the group of invertibleelements is open,11 and e2πiP = 1 is clearly invertible in the unitization of C∗u(S). Nowe2πiPj converge to e2πiP , and so there will be some J ∈ N such that for all j > J theoperator e2πiPj lies in a small open ball around e2πiP consisting only of invertibles. Thestraight line between e2πiPj and e2πiP will then be a homotopy showing [e2πiPj ] = [e2πiP ].Note that the straight line goes through elements which are all equivalent to 1 moduloC∗u(S), since this holds for both e2πiPj and e2πiP .

Proposition 4.18. Let M be an odd-dimensional spin manifold of bounded geometryand assume that M has uniformly positive scalar curvature.

Then ind( /D) = [1] ∈ Ktop1 (C∗u(/S)).

The proof of the proposition follows directly from Lichnerowicz’ formula which gives aspectral gap around 0 for /D due to the uniformly positive scalar curvature of M , andthen we use the above lemma.

4.5 Weak version of the rough Novikov conjecture

Combining Propositions 4.13 and 4.18 we see that if M is spin and its metric is upsc,then the rough index class of /D vanishes. But we have the following conjecture, which isa weak version of the so-called rough Novikov conjecture and states the non-vanishing ofthe index class in certain situations:

Conjecture 4.19. Let M be uniformly contractible12 and of bounded geometry.Then ind( /D) ∈ Ktop

∗ (C∗u(M)) does not vanish.13

So using Propositions 4.13 and 4.18 we immediately see that the conjecture impliesthat uniformly contractible manifolds of bounded geometry can not have uniformlypositive scalar curvature. But we can derive an even more interesting statement fromthe conjecture:

Lemma 4.20. Assume that the conjecture holds.Then no aspherical14 closed manifold M admits a upsc-metric.

11Exercise: show this. Hint: use that ‖a‖ < 1 implies that 1− a is invertible.12Being uniformly contractible means that every ball BR(x) ⊂M is contractible in the larger ball BS(x),

where S depends only on R. It follows that M is contractible in the usual sense and therefore spin.13Ktop

∗ (C∗u(M)) means either Ktop0 (C∗u(/S

+)) if dim(M) is even, or Ktop

1 (C∗u(/S)) if its odd.14Being aspherical means πk(M) = 0 for k ≥ 2.

16

Proof. Since M is aspherical its universal cover X is uniformly contractible if equippedwith the pullback metric. So by the conjecture the index class of the spin Dirac operator/DX on X does not vanish.But ifM would have a upsc-metric, then the pullback metric on X would also have upsc

and therefore the index class /DX should vanish—which it does not by the conjecture.

As stated in the introduction, the goal of this lecture is to give an idea of the proof ofthe following special case of the conjecture:

Main Theorem of this Lecture 4.21. Let M have polynomial volume growth15 andlet it be polynomially contractible16.

Then Conjecture 4.19 holds for M , i.e., ind( /D) 6= 0.

Examples 4.22. Les us give examples for polynomial volume growth and polynomialcontractibility.

• Euclidean space Rn has polynomial volume growth, but hyperbolic space Hn not.

• If X is the universal cover of a compact manifold M , equipped with the pull-backmetric, then the volume growth of X is the same as the one of π1(M). By a theoremof Gromov, the volume growth of a finitely generated group is polynomial if andonly if the group is virtually nilpotent.

• If the Ricci curvature of M is non-negative, then M also has polynomial volumegrowth.

For polynomial contractibility we have one major source of examples: if π1(M) is virtuallynilpotent and M is closed and aspherical (i.e., πi(M) = 0 for i ≥ 2), then the universalcover of M is polynomially contractible.

5 Cyclic homology and Chern–Connes character

Let A be an algebra over C, not necessarily unital.

Definition 5.1. The cyclic homology HC∗(A) is the homology of the complex

· · · b−→ Cλn(A)

b−→ Cλn−1(A)

b−→ · · · b−→ Cλ0 (A)→ 0,

where Cλn(A) := A⊗(n+1)/(1− λ)A⊗(n+1), λ is the operator

λ(a0 ⊗ a1 ⊗ · · · ⊗ an) := (−1)nan ⊗ a0 ⊗ · · · ⊗ an−1,

and b is the Hochschild operator

b(a0 ⊗ a1 ⊗ · · · ⊗ an) :=n−1∑j=0

(−1)ja0 ⊗ · · · ⊗ ajaj+1 ⊗ · · · ⊗ an + (−1)nana0 ⊗ · · · ⊗ an−1.

15This means that volBr(x) grows asymptotically as a polynomial function in r for r →∞.16This means that in the definition of uniform contractibility the S depends polynomially on R.

17

Example 5.2. Let us compute HC0(A), i.e., we have to compute ker(b : Cλ0 (A)→0)

im(b : Cλ1 (A)→Cλ0 (A)).

The numerator is just Cλ0 (A) = A/(1− λ)A = A, where the last equality is due to the

fact that λ acts as the identity on A.For the denominator we firstly note that we have Cλ

1 (A) = A2/〈a0 ⊗ a1 + a1 ⊗ a0〉and secondly b(a0 ⊗ a1) = a0a1 − a1a0. Since b(a0 ⊗ a1 + a1 ⊗ a0) = 0 we get that theoperator b descends to an operator on Cλ

1 (A), which is of course always the case but wehave not checked it above, and we conclude that the image of b : Cλ

1 (A)→ Cλ0 (A) is just

the commutator [A,A] ⊂ Cλ0 (A) = A.

So we get HC0(A) = A/[A,A].

Remark 5.3. If A is a complete locally convex17 topological algebra with jointly continuousmultiplication, we may define Cλ,cont

n (A) by using the completed projective tensor product⊗π instead of the algebraic one ⊗. This defines continuous cyclic homology HCcont

∗ (A).

Definition 5.4. The Chern–Connes characters ch∗,2n+∗ : Kalg∗ (A)→ HC2n+∗(A) for all

n ∈ N0 and ∗ = 0, 1 are defined by

ch0,2n([e]) :=(2n)!

n!(2πi)n tr e⊗(2n+1) for [e] ∈ Kalg

0 (A),

ch1,2n+1([u]) :=(2n+ 1)!

(n+ 1)!(2πi)n+1 tr

((u−1 − 1)⊗ (u− 1)

)⊗(n+1) for [u] ∈ Kalg1 (A).

If A is a topological algebra as above, we also get maps Ktop∗ (A)→ HCcont

∗ (A).

6 Uniformly finite homology and rough characters

In this section we will define a map HC∗(C∗−∞(E))→ Huf∗ (Y ) and by using a continuity

argument extend it to a map HCcont∗ (C∗pol(E))→ Hpol

∗ (Y ); for E →M a vector bundleof bounded geometry over the manifold M of bounded geometry and where Y ⊂M willbe a quasi-lattice. We will of course discuss in this section all the above occuring groups.

Combining it with the Chern–Connes characters from the last section we will get mapsKalg∗ (C∗−∞(E))→ Huf

∗ (Y ) and Ktop∗ (C∗u(E)) ∼= Ktop

∗ (C∗pol(E))→ Hpol∗ (Y ).

6.1 Uniformly finite homology

Definition 6.1. Let X be a metric space. It is said to have bounded geometry if itadmits a quasi-lattice, i.e., a subset Y ⊂ X with the properties

• there is a c > 0 such that Bc(Y ) = X and

• for all r > 0 there is a Kr > 0 such that #(Y ∩Br(x)) ≤ Kr for all x ∈ X.

17Locally convex means that there exists a family of semi-norms pα such that a subbase for neighborhoodsof y ∈ A is given by Uα,ε(y) with Uα,ε(y) := x ∈ A : pα(x− y) < ε.

18

Examples 6.2. Every Riemannian manifold M of bounded geometry is a metric space ofbounded geometry: any maximal set Y ⊂M of points which are at least a fixed distanceapart (i.e., there is an ε > 0 such that d(x, y) ≥ ε for all x 6= y ∈ Y ) will do the job. Wecan get such a maximal set by invoking Zorn’s lemma.

If (X, d) is an arbitrary metric space that is bounded, i.e., d(x, x′) < D for all x, x′ ∈ Xand some D, then any finite subset of X will constitute a quasi-lattice.

Let K be a simplicial complex of bounded geometry18. Equipping K with the metricderived from barycentric coordinates, the set of all vertices becomes a quasi-lattice in K.

Definition 6.3. Let X be a metric space. We equip the product X i+1 with the metric

d((x0, . . . , xi), (y0, . . . , yi)

)= max

0≤j≤id(xj, yj).

We write ∆ for the multidiagonal in X i+1.Then Cuf

i (X) denotes the vector space of all infinite formal sums

c =∑

axx

with x ∈ X i+1 and ax ∈ C satisfying the following three conditions:

1. There exists a K > 0 depending on c such that |ax| ≤ K.

2. For all r > 0 there is a Kr depending on c such that for all y ∈ X i+1 we have

#x ∈ Br(y) | ax 6= 0 ≤ Kr.

3. There exists an R > 0 depending on c such that ax = 0 if d(x,∆) > R.

The boundary map ∂ : Cufi (X)→ Cuf

i−1(X) is defined by

∂(x0, . . . , xi) =i∑

j=0

(−1)j(x0, . . . , xj, . . . , xi)

and extended by linearity to all of Cufi (X).

The resulting homology is the uniformly finite homology Huf∗ (X).

Definition 6.4 (Rough maps). Let f : X → Y be a (not necessarily continuous) map.We call f a rough map, if

• for all R > 0 there is an S > 0 such that we have

d(x1, x2) < R⇒ d(f(x1), f(x2)) < S

for all x1, x2 ∈ X, and

18That is, the number of simplices in the link of each vertex is uniformly bounded.

19

• for all R > 0 there is an S > 0 such that we have

d(f(x1), f(x2)) < R⇒ d(x1, x2) < S

for all x1, x2 ∈ X.

Two (not necessarily continuous) maps f, g : X → Y are called close, if there is anR > 0 such that d(f(x), g(x)) < R for all x ∈ X.Finally, two metric spaces X and Y are called roughly equivalent, if there are rough

maps f : X → Y and g : Y → X such that their composites are close to the correspondingidentity maps.

Lemma 6.5. If Y ⊂ X is a quasi-lattice, then Y and X are roughly equivalent.

Proof. For the map Y → X we may just take the inclusion and for X → Y we take thenthe map which maps x ∈ X to the point y ∈ Y with d(x, y) ≤ d(x, y′) for all y′ ∈ Y .This assignment is usually not well-defined since there might be different choices for ysatisfying the condition. In this case just pick randomly one of these y. Note that themap X → Y is usually not continuous.

Proposition 6.6. Uniformly finite homology is functorial under rough maps, i.e., anyrough map f : X → Y induces a map f∗ : Huf

∗ (X)→ Huf∗ (Y ) and we have (gf)∗ = g∗f∗

for rough maps f : X → Y and g : Y → Z.If two rough maps f, g : X → Y are close, then f∗ = g∗.

Proof. The map f∗ is defined by f∗(∑

axx)

:=∑axf(x). It is immediately checked that

the latter is an element of Cufi (y), that f∗ is a chain map (i.e., descends to homology),

and that we have (g f)∗ = g∗ f∗.Suppose now that f and g are close. This means that f, g : X × 0, 1 → Y is a

rough map. We will show now that ι0, ι1 : X → X ×0, 1 induce chain homotopic maps,from which it will follow that f∗ and g∗ are chain homotopic.We define h : Cuf

i (X)→ Cufi+1(X × 0, 1) by

h(x0, . . . , xi) :=i∑

j=0

(−1)j((x0, 0), . . . (xj, 0), (xj, 1), . . . , (xi, 1)

)and extended by linearity to Cuf

i (X). We get ∂h+h∂ = (ι1)∗− (ι0)∗, i.e., h is the neededchain homotopy.

Corollary 6.7. If X and Y are roughly equivalent spaces, then their uniformly finitehomology groups are isomorphic: Huf

∗ (X) ∼= Huf∗ (Y ).

Example 6.8. We have seen in Example 6.2 that for any bounded metric space any singlepoint in it constitutes a quasi-lattice. Combining Lemma 6.5 with Corollary 6.7 we getHuf∗ (X) ∼= Huf

∗ (pt) for every bounded metric space X.We have Huf

∗ (pt) = Huf0 (pt) ∼= C.

20

Let X be a simplicial complex of bounded geometry equipped with the metric derivedfrom barycentric coordinates, and let Y ⊂ X the set of vertices equipped with the inducedsubspace metric. We have an obvious inclusion C∞∗ (X) → Cuf

∗ (Y ), where C∞∗ (X) denotesthe simplicial L∞-chain complex with C-coefficients, which induces a canonical mapH∞∗ (X)→ Huf

∗ (Y ) from simplicial L∞-homology of X with C-coefficients to uniformlyfinite homology of the vertex set Y ⊂ X. Then we have the following proposition:

Proposition 6.9. If X is a uniformly contractible19 simplicial complex of boundedgeometry, then the above canonical map is an isomorphism

H∞∗ (X)∼=−→ Huf

∗ (Y ).

We won’t prove this proposition now, but we will see in the proof of Proposition 6.13the proof of the injectivity statement of the above proposition.

6.2 Rough character map

We are now going to define a map HC∗(C∗−∞(E))→ Huf∗ (Y ) for E →M a vector bundle

of bounded geometry over the manifold M of bounded geometry and Y ⊂M will be acertain quasi-lattice.

Definition 6.10. Let A0⊗· · ·⊗An ∈ C∗−∞(E)⊗(n+1). We define χ(A0⊗· · ·⊗An) ∈ Cufn (Y )

for Y ⊂M a quasi-lattice by

χ(A0 ⊗ · · · ⊗ An)(y0, . . . , yn) :=1

(n+ 1)!

∑σ∈Sn+1

(−1)σ tr(A0yσ(0) · · ·Anyσ(n)

),

where yi are the projection operators on L2(E) given by characteristic functions ofVyi ⊂M , where Vyy∈Y is as follows:M is a manifold of bounded geometry, so it admits a compatible20 triangulation X as

a simplicial complex of bounded geometry. If Y ⊂ X is the set of vertices, Y ⊂M is auniformly discrete quasi-lattice. Then we define

Vy := x ∈M : d(x, y) ≤ d(x, y′) for all y′ ∈ Y .

The operators Ai in the above definition are smoothing operators, i.e., have integralkernels ki ∈ C∞b (E∗E). Using that if we have a trace class operator T with continuousintegral kernel k(x, y) defined over a compact domain N , then trT =

∫Nk(x, x)dx, we

get the formula

tr(A0yσ(0) · · ·Anyσ(n)

)=

∫Vyσ(n)

dxn · · ·∫Vyσ(0)

dx0 tr(k0(xn, x0)k1(x0, x1) · · · kn(xn−1, xn)

).

19Recall that being uniformly contractible means that for every R > 0 there is an S > 0 such that everyball BR(x) ⊂ X is contractible inside the ball BS(x).

20Equipping X with the metric derived from barycentric coordinates, the identity X →M must be abi-Lipschitz map.

21

Note that in order for the map χ to be well-defined, we need Proposition 2.18 since weneed to know that operators from C∗−∞(E) are uniformly locally traceable21. This is thecrucial point where we need the bounded geometry assumption for M and E.

Lemma 6.11. The above map χ descends to Cλn(C∗−∞(E)) which then is a chain map,

i.e., induces a map HC∗(C∗−∞(E))→ Huf∗ (Y ) also denoted by χ.

Proof. The total anti-symmetrization in the definition of χ ensures that it is well-definedon Cλ

n(C∗−∞(E)), i.e., that it satisfies

χ(A0 ⊗ · · · ⊗ An) = χ((−1)nAn ⊗ A0 ⊗ · · · ⊗ An−1).

Furthermore, we have

∂(χ(A0 ⊗ · · · ⊗ An)

)(y0, . . . , yn−1) =

=∑

0≤j≤n

∑x∈Y

(−1)jχ(A0 ⊗ · · · ⊗ An)(y0, . . . , yj−1, x, yj, . . . , yn−1)︸ ︷︷ ︸=(−1)jχ(A0⊗···⊗AjAj+1⊗···⊗An)(y0,...,yj−1,yj ,...,yn−1)

= χ(b(A0 ⊗ · · · ⊗ An)

)(y0, . . . , yn−1)

which shows that χ is a chain map.

6.3 Proving the main theorem of this lecture

Recall that our main theorem was the following:

Main Theorem of this Lecture 6.12. Let M have polynomial volume growth22 andlet it be polynomially contractible23.

Then ind( /D) ∈ Ktop∗ (C∗u(M)) does not vanish.24

The proof follows from the discussion of the following diagram:

/D //_

ind( /D) ∈ Ktop∗ (C∗u(M))

Kalg∗ (C∗−∞(M))

χ

// Ktop∗ (C∗pol(M))

χ

PD[A] ∈ H∞∗ (X) // Huf∗ (Y ) // Hpol

∗ (Y )

21This means that Ay is a trace class operator for A ∈ C∗−∞(E) and y ∈ B(L2(E)) being a projectionoperator as in the definition of χ, and that ‖Ay‖tr is bounded from above independently of the choiceof y ∈ Y . (Sorry for using “y” for two different, but of course connected to each other, things here.)

22This means that volBr(x) grows asymptotically as a polynomial function in r for r →∞.23This means that in the definition of uniform contractibility the S depends polynomially on R.24Ktop

∗ (C∗u(M)) means either Ktop0 (C∗u(/S

+)) if dim(M) is even, or Ktop

1 (C∗u(/S)) if its odd.

22

• X is a compatible25 triangulation ofM as a simplicial complex of bounded geometryand Y ⊂ X is the set of vertices.

• Hpol∗ (Y ) is the following homology: for every n ∈ N0 we define for a uniformly finite

chain c =∑ayy ∈ Cuf

i (Y ) of a uniformly discrete space Y of bounded geometrythe norm

‖c‖∞,n := supy∈Y i+1

|ay| · length(y)n,

where length(y) = max0≤k,l≤i d(yk, yl) for y = (y0, . . . , yi).

Then we equip Cufi (Y ) with the family of norms (‖·‖∞,n + ‖∂·‖∞,n)n∈N0 , denote its

completion to a Fréchet space by Cpoli (Y ), and the resulting homology by Hpol

∗ (Y ).

We have maps Huf∗ (Y )→ Hpol

∗ (Y ) induces by the inclusion Cuf∗ (Y )→ Cpol

∗ (Y ), buta priori we can not say anything about them (like injectivity or surjectivity).

• Analogously as in the last point, we equip C∗−∞(M) with a certain family of norms,and denote its completion by C∗pol(M). The norms are chosen is such a way thatthe following two bullet point holds.

• The rough character map χ : C∗−∞(M)⊗(i+1) → Cufi (Y ) is continuous against the

topologies defined in the above two bullet points and has therefore an extension tothe completions, i.e., χ : C∗pol(M)⊗(i+1) → Cpol

i (Y ).

• The algebra C∗pol(M) is closed under holomorphic functional calculus26 and also allmatrix algebras over it. By a small lemma (which we do not have discussed) weget from it Ktop

∗ (C∗pol(M)) ∼= Ktop∗ (C∗u(M)).

• A is the topological index class of /D, i.e.,

A = 1− 124p1 + 1

5760

(− 4p2 + 7(p1 ∪ p1)

)+ · · · ∈ H0(M)⊕H4(M)⊕H8(M)⊕ · · · ,

where pi ∈ H4i(M) is the ith Pontryagin class of M .

Because by assumption both M and the spinor bundle /S have bounded geometry,the A-class is even a bounded class, i.e., defines an element of bounded de Rhamcohomology27 A ∈ H4∗

b,dR(M). Its Poincaré dual is therefore PD[A] ∈ H∞m−4∗(M),where m = dim(M).

• The commutativity of the diagram is an index theorem which generalizes the usualAtiyah–Singer index theorem (which is the case where M is compact).

25Equipping X with the metric derived from barycentric coordinates, the identity X →M must be abi-Lipschitz map.

26This means that if we are given an operator A ∈ C∗pol(M) and a holomorphic function f : D → C,where D ⊂ C is a neighbourhood of the spectrum of A, then f(A) ∈ C∗pol(M), where f(A) is definedby holomorphic functional calculus.

27Bounded de Rham cohomology is defined analogously as usual de Rham cohomology, but we use onlyforms bounded in the norm ‖α‖∞ := supx∈M (|α(x)|+ |dα(x)|).

23

The main theorem now follows from the diagram together with the following proposition:

Proposition 6.13. Let M be as in the main theorem demanded.Then the map H∞q (X)→ Huf

q (Y ) is bounded from below, where we use on H∞q (M) theL∞-seminorm ‖·‖∞ and on Huf

q (Y ) the ‖·‖∞,n-seminorm for a large enough n.

Proof of the main theorem. The top-dimensional component of PD[A] is the generator1 ∈ H∞m (X) due to our above discussion of the A-class. Since there are no (m + 1)-simplices in X, the seminorm on H∞m (X) is actually a norm (this does in general nothold for lower degrees than m), i.e., the (semi-)norm of [1] ∈ H∞m (X) is not zero.By the above proposition the seminorm of the image of [1] ∈ H∞m (X) in Huf

m (Y ) istherefore also not zero. Since Hpol

∗ (Y ) was defined as the homology of the completedchain complex used to define Huf

∗ (Y ), the map Huf∗ (Y )→ Hpol

∗ (Y ) is an isometry, andtherefore the image of [1] ∈ H∞m (X) in Hpol

∗ (Y ) has also non-zero seminorm. Especiallyit follows that it is not the zero element.

By commutativity of the diagram it follows that ind( /D) ∈ Ktop∗ (C∗u(M)) is mapped to

something non-trivial in Hpol∗ (Y ) and so the rough index class is not zero.

Proof of Proposition 6.13. First recall how we can prove that for uniformly contractiblespaces the natural map H∞q (X)→ Huf

q (Y ) is injective (this is part of the statement ofProposition 6.9): given vertices y ∈ Y i we can construct a simplicial (i − 1)-ball ∆y

by inductively exploiting the uniform contractibility of X and such that ∆y has thefollowing properties: its vertices are exactly y if we regard ∆y as the triangulation of aq-simplex, ∆y is contained in an S ′-ball around y, and the boundary of ∆y is the chain∑

(−1)j∆(y0,...,yj ,...,yq). Here S ′(R) = S(S(S(. . . (R)))), where we apply S (i − 1)-timesand R is the length of the simplex y. This defines a chain map Cuf

q (Y )→ C∞q (X) andthe composition C∞q (X)→ Cuf

q (Y )→ C∞q (X) is the identity. Passing to homology weget a left inverse to the map H∞q (X)→ Huf

q (Y ).In order to prove our theorem we now have to quantify the above argument. So let

c =∑ayy ∈ Cuf

q (Y ) and denote by ∆c ∈ C∞q (X) the simplicial chain that we get by theabove construction. Let the growth of Y be estimated by volBR(y) ≤ D · RM for ally ∈ Y (we will usually just write volBR if the point y is understood) and recall that thedependence of S on R was by assumption polynomial, i.e., S ′(R) can be estimated byS ′(R) ≤ C ·RN . Let us introduce the value

‖c‖R−[1] := supy∈BR−[1](∆)

|ay|,

where BR−[1](∆) = BR(∆)−BR−1(∆) for ∆ ⊂ Y q+1 the diagonal, and then we have theestimate

|∆y0 | ≤∑R∈N

‖c‖R−[1] · volBS′(R) · volBR−[1] · (volBR)q−1

on the coefficient of the simplex ∆y0 of ∆c. We arrive at this estimate in the followingway: we have to estimate which y may contribute to ∆y0 and our construction of ∆c issuch that ∆y0 must lie in the S ′(R)-ball of y for this to happen. Now we go through

24

all y with maximal edge length between R− 1 and R which may constribute to ∆y0 bypicking a point in the S ′(R)-ball of y0 (which will be one of the vertices of a potentiallycontributing y), picking a second point in the (R− [1])-ball of the first point (which willbe our second vertex of y and together with the first one they will produce the maximaledge length of y), and then picking the missing q − 1 vertices of y (which must lie in theR-ball around the first picked point since y must have edge lengths smaller than R).

Now we plug in our estimates for S ′(R) and volBR to get

‖∆c‖∞ ≤∑R∈N

‖c‖R−[1] ·DCMRNM ·DRM ·Dq−1RM(q−1)

= Dq+1CM∑R∈N

‖c‖R−[1] ·RM(N+q).

Furthermore, we have for all n ∈ N0

‖c‖∞,n = supy∈Y q+1

|ay| · length(y)n

≥ supR∈N

supy∈BR−[1](∆)

|ay| · (R− 1)n

= supR∈N‖c‖R−[1] · (R− 1)n

which gives the estimate

‖c‖R−[1] ≤ 2n‖c‖∞,nRn

for all R ≥ 2 (this restriction is because we used R− 1 ≥ R/2 for R ≥ 2). We set nown := M(N + q) + 2 and combine our estimates to get

‖∆c‖∞ ≤ Dq+1CM2n‖c‖∞,n∑R∈N≥2

R−2 +Dq+1CM‖c‖1−[1]

≤ Dq+1CM(2nπ2/6 + 1)‖c‖∞,n.

Equipping Cufq (Y ) with the norm ‖·‖∞,n + ‖∂·‖∞,n for n := M(N + q) + 2 and C∞q (X)

with the norm ‖·‖∞ + ‖∂·‖∞, we get from the above estimate and the fact that our mapCufq (Y )→ C∞q (X) is a chain map, that it is bounded. Therefore it induces a bounded

map Hufq (Y ) → H∞q (X) and since this is the left inverse to H∞q (X) → Huf

q (Y ) we getthat the latter must be bounded from below.

25