Traces and Invariants for Non-Compact Manifolds

James L. Heitsch Department of Mathematics, Statistics and Computer Science University of Illinois at Chicago 851 S. Morgan Street (m/c 249) Chicago, Illinois 60607-7045 e-mail: heitsch@math.uic.edu

Abstract

We define a general trace for complete Riemannian manifolds of bounded geometry. This trace takes values in the Haefliger functions of the manifold and its existence allows the extension of the classical theory of invariants for compact manifolds to complete Riemannian manifolds of bounded geometry.

1 Introduction

The heat equation method for invariants of manifolds is now quite well developed and classical results (index theory, Lefschetz theory, betti numbers, Novikov-Shubin invariants, torsion, …) can be extended to new situations (foliations, non-compact manifolds) provided that an appropriate trace function can be defined, and that the asymptotics of the heat operator as t → ∞ (which is in general a global problem) can be handled. The asymptotics of the heat operator as t → 0 present no problem as these are local.

The L^{2} Index Theorem of Atiyah [A76] is a classical example of this sort of extension. The case considered there is a covering of a compact manifold and an operator lifted to the cover from the compact base. The trace is given by integration over a fundamental domain of the covering. In [R88I], John Roe provided an extension to non compact manifolds which admit a regular exhaustion. The exhaustion allows one to define a trace by averaging over the manifold, much as we do in the final section of this paper. Alain Connes extended the heat equation method to foliations which admit a transverse invariant measure. See [C90] and the references therein. Here the trace is given by integration over the placques of a cover by foliation charts with the resulting function being integrated against the transverse measure to obtain a real number. In [H95] and [HL99] we showed how to remove the requirement that the foliation admit an invariant transverse measure and obtained a general families index theorem. The trace in this case is given by integrating over the placques and taking the Haefliger cohomology class of the resulting differential form on the transverse space.

In this paper we define a general trace (the Haefliger trace) for any complete Riemannian manifold M of bounded geometry. This trace takes values in C_{h}(M) the space of Haefliger functions of M, which is a quotient space of the space of functions on a maximal uniformly separated discrete subset. The basic idea is to abstract the theory for foliations worked out in [HL90] to the case of a single leaf. The main innovation in this paper is the introduction of the Haefliger trace taking values in the space of Haefliger functions. As an example of the use of this trace, we outline a proof of a Lefschetz theorem for Dirac operators defined on any complete Riemannian manifold of bounded geometry.

We note here that C_{h}(M) agrees with the uniformly finite coarse homology group H_{0}^{uf}(M) defined by Block and Weinberger, [BW92], and that the application of uniformly finite coarse homology to index theory was anticipated in section 9 of [BW97]. The main idea of this paper is that there is a naturally defined geometric trace taking values in C_{h}(M), and given this, the geometric arguments of [HL90] and [HL99] are readily translated to this setting, giving index theorems, Lefschetz theorems, etc. This paper may be seen as alternate approach to invariants for non-compact manifolds which provides immediate access to results not readily obtainable from the Block-Weinberger view: e.g. Lefschetz Theory.

2 Haefliger functions and the trace

In this section, we define the space of Haefliger functions for a complete Riemannian manifold of bounded geometry and a trace which takes values in this space of functions.

2.1 Haefliger functions

Denote by M a complete Riemannian manifold of bounded geometry. This class of manifolds includes all compact manifolds and their covers as well as any leaf (and its covers) of any foliation of a compact manifold. There are also examples of such manifolds which cannot be a leaf of a foliation of any compact manifold, [AH98], [PS81].

Let r > 0. A lattice Γ for M is cover by open balls of radius r whose centers are at least a distance r apart. The bounded geometry of M then implies that there is an integer k so that any element of Γ intersects at most k other elements of Γ non-trivially.

To construct a lattice, let A ⊂ M be a maximal set of distinct points which are separated by a distance of at least r. Zorn’s lemma implies that such a set must exist. Then let Γ be the set of balls of radius r whose centers are the elements of A.

For any lattice Γ set

C^{0}(Γ) = {f : Γ → R | f is bounded}.

The space of Haefliger functions on M is a quotient space of this space. Denote by d(·, ·) the distance function on M. A permutation π of Γ is bounded if

sup_{U∈Γ} d(U, π(U)) < ∞.

The group G of all bounded permutations acts on C^{0}(Γ) by composition, that is

(π^{∗}f)(U) = f(π^{−1}(U)).

Set

H_{Γ} = linear span of {f − π^{∗}f | f ∈ C^{0}(Γ), π ∈ G}.

The space of Haefliger functions C_{h}(M) for M is the space

C_{h}(M) = C^{0}(Γ)/\overline{H}_{Γ},

where \overline{H}_{Γ} is the sup norm closure of H_{Γ}. Denote the class of f in C_{h}(M) by [f]. Then [f] = 0 provided that for all ϵ > 0 there is a representative f_{1} ∈ [f] so that sup_{U∈Γ} |f_{1}(U)| < ϵ.

Note that if f(U) = −f(π(U)) for all U ∈ Γ̂ ⊆ Γ, and Γ̂ ∩ π(Γ̂) = ∅, then we may set f(U) = 0 for all U ∈ Γ̂ ∪ π(Γ̂) without changing the Haefliger class of the function f. To see this, define g(U) = f(U) for U ∈ Γ̂ and g(U) = 0 otherwise. Then [g − π^{∗}g] = 0, and f_{1} = f − (g − π^{∗}g) satisfies f_{1}(U) = 0 for all U ∈ Γ̂ ∪ π(Γ̂), and f_{1}(U) = f(U) otherwise.

A priori, C_{h}(M) depends on the choice of lattice Γ. However, this is not the case.

Theorem 2.1 C_{h}(M) does not depend on Γ.

Proof: Let Γ_{1} and Γ_{2} be two lattices for M and set Γ = Γ_{1} ∪Γ_{2}. Now Γ may not be a lattice as the centers of distinct balls may be arbitrarily close, but there is still a k so that each element of Γ meets at most k other elements of Γ. It addition, C^{0}(Γ)/H_{Γ} still makes sense. With these observations, it is an easy exercise to show

C^{0}(Γ_{1})/H_{Γ_{1}} ≃ C^{0}(Γ)/H_{Γ} ≃ C^{0}(Γ_{2})/H_{Γ_{2}}.

Q.E.D.

Similarly, one can show that C_{h}(M) does not depend on r.

2.2 The Trace

Given a bounded measurable function g on M and any partition of unity Ψ = {ψ_{U} } subordinate to the cover of M given by the elements of the lattice Γ, define ∫ g ∈ C^{0}(Γ) to be the function

(∫ g)(U) = ∫_{U} ψ_{U} (x)g(x) dx

where dx is the volume form on M. It follows easily from the definitions that the Haefliger function [∫ g] does not depend on Ψ.

For any bundle E, denote by C(E) the bounded measurable sections of E, by C^{∞}(E) the bounded smooth sections, and by C_{0}^{∞}(E) the smooth sections with compact support. Denote by E ⊗ E^{∗} the bundle π_{1}^{∗}(E) ⊗ π_{2}^{∗}(E^{∗}) over M × M, where π_{1}, π_{2} : M × M → M are the two projections. For any k ∈ C(E ⊗ E^{∗}), k(x, x) is a linear map on E_{x} and so has a well defined trace tr(k(x, x)) which is a bounded measurable function on M. We define the Haefliger trace Tr(k) of k to be the Haefliger class of the function ∫ tr(k), that is

Tr(k) = [∫ tr(k)].

(If M is compact, any k ∈ C(E ⊗ E^{∗}) defines an operator on C(E) by

(ks)(x) = ∫_{M} k(x, y)s(y) dy,

and the Haefliger trace of k is the usual trace of k as in this case, C_{h}(M) = R.)

Denote by C_{s}^{∞}(E ⊗E^{∗}) the set of k ∈ C^{∞}(E ⊗E^{∗}) so that there is s > 0 with k(x, y) = 0 for all (x, y) with d(x, y) ≥ s. In addition we require that k and all its derivatives be bounded on M × M. Any k ∈ C_{s}^{∞}(E ⊗ E^{∗}) defines a bounded smoothing operator on L^{2}(E) with finite propagation speed and tr(k(x, x)) is a bounded smooth function on M.

An element k ∈ C^{∞}(E ⊗ E^{∗}) is basic if there are Γ̂ ⊂ Γ and π ∈ G so that for all U_{1} ≠ U_{2} ∈ Γ̂, U_{1} ∩ U_{2} = ∅ and π(U_{1}) ∩ π(U_{2}) = ∅ and

sup(k) ⊂ ∪_{Γ} U × π(U).

Note that if k is basic, then sup(k) ∩ (U × π(U)) is a compact subset of the open set U × π(U).

Proposition 2.2 Any k ∈ C_{s}^{∞}(E ⊗ E^{∗}) may be written as a finite sum k = k_{1} + ⋯ + k_{q} where each k_{i} is basic.

Proof: Choose s > 0 so that k(x, y) = 0 for all x and y with d(x, y) > s. Because of the bounded geometry of M and the fact that the centers of the elements of Γ are at least a distance r apart, Γ can be partitioned into a finite number of sets, Γ̂_{1}, … Γ̂_{n}, so that for all distinct U_{1}, U_{2} ∈ Γ̂_{ℓ}, ℓ = 1, … n, d(U_{1}, U_{2}) > 4r + 2s. This of course implies that U_{1} ∩ U_{2} = ∅. If (U_{1} × U_{3}) ∩ sup(k) ≠ ∅, then d(U_{1}, U_{3}) ≤ s so it further guarantees that if (U_{1} × U_{3}) ∩ sup(k) ≠ ∅ and (U_{2} × U_{4}) ∩ sup(k) ≠ ∅ then U_{3} ∩ U_{4} = ∅. This condition guarantees that the elements π_{i,j} ∈ G chosen below actually exist.

Again by the bounded geometry of M and the fact that the centers of the elements of Γ are at least a distance r apart, there is an integer p so that for each U ∈ Γ there are at most p elements of Γ whose product with U intersects sup(k) non-trivially. For a given U, denote these sets by Û_{1}, …, Û_{p}. (If there are fewer than p such sets, we complete the collection to a set of p elements by adding the empty set the required number of times). For i = 1, …, n, j = 1, …, p, choose π_{i,j} ∈ G so that π_{i,j}(U) = Û_{j} for all U ∈ Γ̂_{i}. Let {ψ_{U} } be a partition of unity subordinate to the cover given by Γ and set

k_{i,j}(x, y) = ∑_{U∈_{Γ_{i}}} ψ_{U} (x)ψ_{π_{i,j}(U)}(y)k(x, y).

Then each k_{i,j} is basic and k = ∑_{i,j} k_{i,j}. Q.E.D.

Let k_{1}(x, y) ∈ C(E ⊗ E^{∗}), and k_{2}(x, y) ∈ C_{s}^{∞}(E ⊗ E^{∗}), and define k_{1} ∘ k_{2} ∈ C(E ⊗ E^{∗}) by

k_{1} ∘ k_{2}(x, y) = ∫_{M} k_{1}(x, z)k_{2}(z, y) dz,

and similarly for k_{2} ∘ k_{1}. We have immediately

Proposition 2.3 k_{1} ∘ k_{2} and k_{2} ∘ k_{1} are elements of C(E ⊗ E^{∗}).

We now come to the central result of the paper, namely the fact that Tr satisfies the following trace property. It is this property which allows us to extend classical results from compact manifolds to complete manifolds of bounded geometry.

Theorem 2.4 Suppose k_{1}(x, y) ∈ C(E ⊗ E^{∗}), and k_{2}(x, y) ∈ C_{s}^{∞}(E ⊗ E^{∗}). Then

Tr (k_{1} ∘ k_{2}) = Tr (k_{2} ∘ k_{1}).

Proof: By Proposition 2.2 we may assume that k_{2} is basic with corresponding π, Γ̂. Choose a partition of unity {ψ_{U} } subordinate to Γ so that for all U ∈ Γ̂, ψ_{U} ×ψ_{π(U)} |_{sup(k_{2})∩(U×π(U))} = 1. Then the Haefliger function Tr(k_{1} ∘ k_{2}) is represented by the function f ∈ C^{0}(Γ) whose value at U ∈ Γ is

f(U) = ∫_{U} ψ_{U} (x)tr( ∫_{M} k_{1}(x, y)k_{2}(y, x)dy) dx =
∫_{U} ∫_{M} tr (k_{1}(x, y)k_{2}(y, x)) dy dx

which, because k_{2} is basic, equals

∫_{U} ∫_{π(U)} tr (k_{1}(x, y)k_{2}(y, x)) dy dx =
∫_{π(U)} ∫_{U} tr (k_{2}(y, x)k_{1}(x, y)) dx dy =
∫_{π(U)} ∫_{M} tr (k_{2}(y, x)k_{1}(x, y)) dx dy

again because k_{2} is basic, and this equals

∫_{π(U)} ψ_{π(U)}(y) ∫_{M} tr (k_{2}(y, x)k_{1}(x, y)) dx dy.

This last integral is the value at π(U) of the function g ∈ C^{0}(Γ) which represents Tr(k_{2} ∘ k_{1}), i.e. it is (π^{−1})^{∗}(g)(U). But g and (π^{−1})^{∗}(g) represent the same Haefliger function. Thus Tr(k_{1} ∘ k_{2}) = Tr(k_{2} ∘ k_{1}). Q.E.D.

3 Dirac Complexes and Lefschetz Functions

The Haefliger trace adapts extremely well to any situation where the heat equation method is used. As an example of this, we will outline a proof of a Lefschetz theorem for complete Riemannian manifolds of bounded geometry. This of course includes various index theorems as special cases.

Let M be a complete Riemannian manifold of bounded geometry. Fix a lattice Γ for M. We assume that each bundle over M comes equipped with a metric, and if the bundle is a complex bundle, that the metric is Hermitian. Unless otherwise stated, all objects considered here (e.g. metrics, connections, bundle maps, …) are assumed to be bounded in the sense that there are global bounds for the local expressions of the object and all its derivatives is any local orthonormal framing of the relevant bundle.

3.1 Dirac Complexes

A Dirac complex (E, d) on M consists of the following

1. E = (E_{0}, E_{1}, …, E_{ℓ}), a family of smooth finite dimensional complex vector bundles over M.

2. d = (d_{0}, d_{1}, …, d_{ℓ−1}), a family of differential operators where

d_{i} : C_{0}^{∞}(E_{i}) → C_{0}^{∞}(E_{i+1}),

and d_{i+1} d_{i} = 0.

3. Set E = ⊕_{i}E_{i} and denote the adjoint of d_{i} by d_{i}^{∗}. We assume that E is a Clifford bundle over the Clifford algebra of T^{∗}M, and that there is an Hermitian connection ∇ on E, compatible with Clifford multiplication, so that the operator

D = ⊕(d_{i} + d_{i−1}^{∗}) : C_{0}^{∞}(E) → C_{0}^{∞}(E)

is given by the composition

C_{0}^{∞}(E) →^{∇} C_{0}^{∞}(T^{∗}M ⊗ E) →^{m} C_{0}^{∞}(E)

where m is Clifford multiplication.

All the classical complexes (de Rham, Signature, Dolbeault, and Spin) give rise to Dirac complexes on M provided M supports the necessary geometric structures for these complexes to be defined.

Denote the space of L^{2} sections of E_{i} by L^{2}(E_{i}). The operators d_{i} and d_{i−1}^{∗} extend to densely defined unbounded operators

d_{i} : L^{2}(E_{i}) → L^{2}(E_{i+1}) and d_{i−1}^{∗} : L^{2}(E_{i}) → L^{2}(E_{i−1}).

As usual with possibly non-compact manifolds, we define

H^{i}(M, d) = ker(d_{i})/\overline{(im d_{i−1} ∩ dom d_{i}}).

It is necessary to mod out by \overline{im d_{i−1} ∩ dom d_{i}}, the closure of the image of d_{i−1} in the domain of d_{i}, as im d_{i−1} ∩ dom d_{i} is in general not a closed subspace.

Define

Δ_{i} = d_{i−1} d_{i−1}^{∗} + d_{i}^{∗} d_{i} : C_{0}^{∞}(E_{i}) → C_{0}^{∞}(E_{i}).

The operator D = ⊕(d_{i} ⊕ d_{i−1}^{∗}) is a Dirac operator and the fact that M is complete implies that D is essentially self adjoint [C73]. Thus D^{2} = ⊕Δ_{i} is also essentially self adjoint, and we can apply the spectral mapping theorem to each of the Δ_{i}.

Hodge Theory extends to complete manifolds and we have the following facts:

• L^{2}(E_{i}) = ker Δ_{i} ⊕ \overline{image d_{i−1}} ⊕ \overline{image d_{i}^{∗}}.

• The natural map ker Δ_{i} → H^{i}(E, d) is an isomorphism.

3.2 Geometric Endomorphisms

An endomorphism T = (T_{0}, …, T_{ℓ}) of a Dirac complex (E, d) is a collection of complex linear maps

T_{i} : C^{∞}(E_{i}) → C^{∞}(E_{i})

so that d_{i} T_{i} = T_{i+1} d_{i}. The most interesting of these are the so-called geometric endomorphisms [AB67] defined as follows. Let f : M → M be a smooth diffeomorphism of bounded distortion. For i = 0, …, ℓ, let A_{i} : f^{∗}(E_{i}) → E_{i} be a smooth bounded bundle map. For s ∈ C^{∞}(E_{i}) define

(T_{i}s)(x) = A_{i,x}(s(f(x))).

We assume that the A_{i} are chosen so that T = (T_{0}, …, T_{ℓ}) defines an endomorphism of (E, d).

Example. Let (E, d) be the de Rham complex of M and f : M → M a diffeomorphism of bounded distortion. For A_{i} take the ith exterior power of the dual df^{∗} of the differential df. Then for each i, T_{i} is just the map f^{∗} on the differential i forms on M.

As d_{i} T_{i} = T_{i+1} d_{i}, T_{i} induces a well defined map T_{i}^{#} : H^{i}(E, d) → H^{i}(E, d). We would like our Lefschetz theorem to relate ∑_{i} (−1^{i})tr T_{i}^{#} to indices defined on the fixed point sets of f. In general, H^{i}(E, d) is not finite dimensional, so the classical trace does not make sense. To get our theorem, we must recast T_{i}^{#} in terms of an operator on L^{2}(E_{i}) in order to apply the more general Haefliger trace.

Denote by P_{i} : L^{2}(E_{i}) → ker(Δ_{i}) the projection, and define

T_{i}^{∗} = P_{i}T_{i}P_{i} : L^{2}(E_{i}) → ker(Δ_{i}) ⊂ L^{2}(E_{i}).

We also denote by T_{i}^{∗} the map P_{i}T_{i}P_{i} ∘ j : ker(Δ_{i}) → ker(Δ_{i}) where j : ker(Δ_{i}) → L^{2}(E_{i}) is the injection. It follows from results of [HL90] that the Haefliger trace Tr (T_{i}^{∗}) exists and is finite.

Definition 3.1 The Lefschetz function L(T) of the geometric endomorphism T is the Haefliger function for M given by

L(T) = ∑_{i=0}^{ℓ}(−1)^{i}Tr (T_{i}^{∗}).

That this is a reasonable definition is a result of the fact that the diagram

[figure: ker Δ_{i}; ↓; H^{i}(E, d); T_{i}^{∗}; T_{i}^{#}; ker Δ_{i}; ↓; H^{i}(E, d)] commutes.

4 Fixed Point Indices and The Lefschetz Theorem

In this section, we describe the restrictions we place on the fixed point set N of f and state our Lefschetz theorem.

We assume that N is a disjoint union of closed submanifolds N = ∪N_{i} of M. As they are disjoint and closed, each N_{i} admits a positive function ϵ_{i}(x) so that

N_{ϵ}(N_{i}) = {x ∈ M | d(x, N_{i}) < ϵ_{i}(x)}

is an embedded normal disc bundle in M and we may assume that these disc bundles are pairwise disjoint.

We further assume that f is non-degenerate on N. This means that f_{∗} restricted to the normal bundle of N in M does not have 1 as an eigen value at any point in N, i.e. f has no transverse invariant directions. Note that the identity map I_{M} of M satisfies this condition.

Suppose that a is a smooth volume form on N, and denote its restriction to N_{i} by a_{i}. We define the Haefliger function ∫_{N} a as follows. Let {ψ_{U} } be a partition of unity subordinate to the cover by the elements of the lattice Γ. Then ∫_{N} a = [g] where g is the function on Γ

g(U) = ∑_{i} ∫_{N_{i}∩U} ψ_{U} a_{i}.

The fact that only a finite number of N_{i} ∩ U ≠ ∅ follows from the facts that \overline{U} is compact, f|_{N} = I_{N} and that the N_{i} are disjoint and closed.

Theorem 4.1 (Lefschetz Theorem) Let M, (E, d), f, A and T be as above. Associated to N is a smooth volume form a which depends only on f, A, the symbols of the Δ_{i}, the metrics and their derivatives to a finite order only on N, so that in C_{h}(M)

L(T) = ∫_{N} a.

For the classical complexes and T = f^{∗} acting on forms, a is given by the classical local integrand of the Atiyah-Singer G Index Theorem [ASe68]. So, if M is a compact manifold, special cases of this theorem recover the Atiyah-Singer Index Theorem, [AS68], the Atiyah-Singer G Index Theorem [ASe68], and the Atiyah-Bott Lefschetz Theorem for geometric endomorphisms, [AB67].

If (E, d) is an arbitrary Dirac complex and N_{i} = {x} is a single point, then

a_{i} = \frac{∑_{i=0}^{ℓ}(−1^{i}) tr A_{i,x}}{| det(I_{x} − df_{x})|} .

For the classical complexes with T = f^{∗} and N_{i} = {x}, the a_{i} may be further identified. For this see [AB67] and [AB68].

If f = I_{M}, then N = M and we define I(D), the index of D by I(D) = L(I_{M}). As an example of the theorems available in this case, we state only the following.

Theorem 4.2 Let M be an even dimensional complete Riemannian Spin manifold of bounded geometry. Let / D_{E}^{+} be the generalized Atiyah-Singer operator, [LM89], associated to the complex vector bundle E. Then in C_{h}(M)

I(/ D_{E}^{+}) = [ ∫ ch(E)Â(M)]

where ch(E) is the Chern class of E, and Â(M) is the Â genus of M.

There are similar theorems for the other classical operators.

5 An Example

In this section, we give an example of a non-compact complete Riemannian manifold M which is a covering of Σ_{4}, the surface of genus 4, and a diffeomorphism f : M → M of bounded distortion which has two fixed points in the interior of each fundamental domain of the covering M → Σ_{4}. This example is a single generic leaf of the foliation defined in Section 4 of [HL90], to which the reader is referred for the proofs of the claims made here.

Let Π ⊂ SL_{2}R be the subgroup generated by the elements α_{j} = θ^{−j}α θ^{j}, j = 0, …, 7 where θ is rotation by π/16 and

α = [ \begin{matrix}d & 0 \\ 0 & d^{−1} \end{matrix} ] ,

where d is chosen so that Π\SL_{2}R/SO_{2} = Σ_{4}. Let Π_{1} ⊂ Π be generated by {α_{i}^{±1}α_{j}^{±1} | i, j = 0, 3, 4, 7}∪{α_{i}^{±1}α_{j}^{±1} | i, j = 1, 2, 5, 6}. Then M = Π_{1}\SL_{2}R/SO_{2} and Π/Π_{1} ≃ Z_{2} ∗Z_{2} ≃ D_{∞} so M is non compact. Note that D_{∞} is solvable so also amenable.

Let f be the diffeomorphism of M determined by ρ, rotation of the Poincaré disc by π/2. Take for the fundamental domains of Σ_{4} a regular 16-gon centered at zero in the Poincaré disc and all its translates under Π. This determines a set of fundamental domains in M. A careful reading of Section 4 of [HL90] shows that f fixes each fundamental domain of M, (i.e. for any fundamental domain D in the Poincaré disc, there is an element γ ∈ Π_{1} so that ρ(D) = γ(D)). f acts on each fundamental domain of M by rotation by π/2. In each such fundamental domain, the only fixed points are the center and the 16 vertices. But the vertices are identified in groups of 8 by Π_{1} so in each of these fundamental domains there are exactly three fixed points and at each fixed point, df acts by rotation by π/2. By slightly altering the fundamental domains we have chosen (namely adding a small neighborhood of every other vertex and deleting a corresponding neighborhood of the other vertices), we obtain new fundamental domains so that each has exactly two fixed points contained in its interior.

M has all four classical complexes defined on it, and the local indices at the fixed points for T = f^{∗} are all non trivial. For the de Rham complex they are all −2, for the Signature complex −2i, for the Dolbeault complex with k = 0, they are 1/(1 − i), and for k = 1, they are i/(1 − i), and for the Spin complex they are ±i/\sqrt{2} depending on which lifting of df one uses.

6 Proof of Lefschetz Theorem

As noted above, each Δ_{i} is an essentially self adjoint operator on L^{2}(E_{i}). In addition, it is non-negative, so any bounded Borel function g on [0, ∞) applied to Δ_{i} yields a well defined bounded operator g(Δ_{i}) on L^{2}(E_{i}) whose Schwartz kernel we denote by k_{g}(x, y). Denote by S(R^{+}) the Schwartz class functions on R restricted to [0, ∞). If g is a Schwartz class function on R, we denote its Fourier transform by ĝ.

We now recall three theorems from [HL90]. The proofs for the case considered here are essentially the same as those given in [HL90] and so are omitted.

Theorem 6.1 If g ∈ S(R^{+}), then k_{g}(x, y) ∈ C^{∞}(E ⊗ E^{∗}), (so Tr (g(Δ_{i})) exists and is finite), and the map g → k_{g} is continuous. If ĝ ∈ C_{0}^{∞}(R), the compactly supported functions, then k_{g} ∈ C_{s}^{∞}(E ⊗ E^{∗}).

Theorem 6.2 Suppose that g ∈ S(R^{+}), B is a differential operator on E_{i} (whose coefficients, with respect to orthonormal bases, and all their derivatives are uniformly bounded), and T = (T_{0}, …, T_{ℓ}) is a geometric endomorphism of (E, d). Then Tr (BT_{i}g(Δ_{i})) and Tr (T_{i}Bg(Δ_{i})) exist and are finite.

Corollary 6.3 For i = 0, …, ℓ and t > 0, Tr (T_{i}e^{−tΔ_{i}}) exists and is finite.

Theorem 6.4 Tr (T_{i}^{∗}) = lim_{t→∞} Tr (T_{i}e^{−tΔ_{i}}).

Note that the convergence in Theorem 11 is pointwise and not uniform, so when applying a linear functional to C_{h}(M) as in Section 7 below, one needs to check that the functional commutes with taking limits. This does not happen in general. For an explicit counter example see [R88II], and for a systematic discussion, [GI00].

As the proof of the following theorem is where the trace property of Tr is most central, we will outline that proof here. For details see [HL90], Section 5.

Theorem 6.5 For all t > 0,

∑_{i=0}^{ℓ}(−1)^{i}Tr (T_{i}e^{−tΔ_{i}})

is independent of t.

Proof: We use Quillen’s formalism of the supertrace [Q85]. Let E = ⊕E_{i} and E^{+} = ⊕E_{2i} and E^{−} = ⊕E^{2i+1}. An operator A on E which preserves this splitting is called even and one that reverses it is called odd. The super trace applied to an even operator A is defined to be

Tr_{s}(A) = Tr(A |_{E^{+}}) − Tr(A |_{E^{−}}).

Note that if A and B are both even operators, then Tr_{s}(AB) = Tr_{s}(BA), while if both are odd, Tr_{s}(AB) = −Tr_{s}(BA).

Set

T = ⊕_{i=0}^{ℓ}T_{i}, d = ⊕_{i=0}^{ℓ−1}d_{i}, d^{∗} = ⊕_{i=0}^{ℓ−1}d_{i}^{∗}, Δ = ⊕_{i=0}^{ℓ}Δ_{i}.

T and Δ are even while d and d^{∗} are odd and these operators satisfy

Δ = dd^{∗} + d^{∗}d, dΔ = Δd, d^{∗}Δ = Δd^{∗}, and T d = dT.

It is not difficult to show that if φ(x) is an analytic function whose restriction to R^{+} is in S(R^{+}), then

dφ(Δ) = φ(Δ)d and d^{∗}φ(Δ) = φ(Δ)d^{∗}.

For s > t > 0 set

φ(x) = e^{−tx} − e^{−sx} and ψ(x) = φ(x)/x

and note that both restricted to R^{+} are in S(R^{+}). As

∑_{i=0}^{ℓ}(−1)^{i}Tr (T_{i}^{∗}e^{−tΔ_{i}}) = Tr_{s}(T e^{−tΔ}),

we must only show that Tr_{s}(T φ(Δ)) = 0.

By Theorem 6.2, we have that both Tr_{s}(T dd^{∗}ψ(Δ)) and Tr_{s}(T d^{∗}dψ(Δ)) exist and are finite, so we have

Tr_{s}(T dd^{∗}ψ(Δ)) + Tr_{s}(T d^{∗}dψ(Δ)) = Tr_{s}(T dd^{∗}ψ(Δ) + T d^{∗}dψ(Δ)).

But from the definitions, this last equals

Tr_{s}(T Δψ(Δ)) = Tr_{s}(T φ(Δ)),

which is what we want to show is zero. As Tr_{s}(T dd^{∗}ψ(Δ)) = Tr_{s}(dT d^{∗}ψ(Δ)), this will follow if we show that

Tr_{s}(dT d^{∗}ψ(Δ)) = −Tr_{s}(T d^{∗}dψ(Δ)).

Theorem 6.1 implies that if ψ_{n} ∈ S(R^{+}) converges to ψ, then

lim_{n→∞} Tr(dT d^{∗}ψ_{n}(Δ)) = Tr(dT d^{∗}ψ(Δ))

and

lim_{n→∞} Tr(T d^{∗}dψ_{n}(Δ)) = Tr(T d^{∗}dψ(Δ)).

Set ψ_{1}(x) = (1/2)ψ(x/2) and ψ_{2}(x) = e^{−tx/2} + e^{−sx/2}. Then ψ_{1}, ψ_{2} restricted to R^{+} are in S(R^{+}), and ψ = ψ_{1}ψ_{2}. Choose sequences ψ_{i,n} ∈ S(R^{+}) converging to ψ_{i} with ψ̂_{i,n} ∈ C_{0}^{∞}(R). Then

Tr_{s}(dT d^{∗}ψ(Δ)) = lim_{n→∞}T r_{s}(dT d^{∗}(ψ_{1,n}ψ_{2,n})(Δ))
= lim_{n→∞}T r_{s}(dT d^{∗}ψ_{1,n}(Δ)ψ_{2,n}(Δ))
= lim_{n→∞}T r_{s}(ψ_{2,n}(Δ)dT d^{∗}ψ_{1,n}(Δ))

by Theorem 2.4 (so it is here that we use the trace property of Tr). But

lim_{n→∞} T r_{s}(ψ_{2,n}(Δ)dT d^{∗}ψ_{1,n}(Δ)) = − lim_{n→∞} T r_{s}(T d^{∗}ψ_{1,n}(Δ)ψ_{2,n}(Δ)d),

since both ψ_{2,n}(Δ)d and T d^{∗}ψ_{1,n}(Δ) are odd operators. This last term equals

− lim_{n→∞}T r_{s}(T d^{∗}dψ_{1,n}(Δ)ψ_{2,n}(Δ)) = − lim_{n→∞}T r_{s}(T d^{∗}d(ψ_{1,n}ψ_{2,n})(Δ))
= −Tr_{s}(T d^{∗}dψ(Δ))

which completes the proof of Theorem 6.5. Q.E.D.

Corollary 6.6 L(T) = lim_{t→0} ∑_{i=0}^{ℓ}(−1)^{i}Tr (T_{i}^{∗}e^{−tΔ_{i}})

To finish the proof of our Lefschetz theorem, we have

Theorem 6.7 Associated to N is a smooth volume form a which depends only on f, A, the symbols of the Δ_{i}, the metrics and their derivatives to a finite order only on N so that in C_{h}(M)

lim_{t→0} ∑_{i=0}^{ℓ}(−1)^{i}Tr (T_{i}^{∗}e^{−tΔ_{i}}) = ∫_{N} a.

The proof of this theorem is a highly technical local computation and is identical to that given in [HL90] and so is omitted.

7 Functionals on C_{h}(M)

The space C_{h}(M) is somewhat difficult to understand. In the special case that M is compact, C_{h}(M) = R. One of the main results of [BW92], Theorem 3.1, is that C_{h}(M) ≠ 0 if and only if M satisfies Condition (15) below. However, even in the simple case of M = R, it is difficult to give a good description of this space. To obtain usable invariants, we need to exhibit interesting linear maps from C_{h}(M) to more tractable spaces, especially, maps to R or C. Given the result of Block and Weinberger quoted above, it should not be surprising that C_{h}(M) is particularly susceptible to averaging operations, one of which we give below. For more in this vein, see [R88I] and [A76].

Suppose that M is a complete Riemannian manifold of bounded geometry. Given x_{0} ∈ M and s ≥ 0, denote by B(x_{0}, s) the ball in M of radius s and center x_{0}. Assume that for some x_{0} ∈ M and any fixed t,

(7.1)  lim_{s→∞} \frac{vol(B(x_{0}, s))}{vol(B(x_{0}, s + t))} = 1.

Let Γ be a lattice in M and denote the center of U ∈ Γ by x_{U}. Set

Γ_{s} = {U ∈ Γ | d(x_{0}, x_{U}) < s}.

For any [f] ∈ C_{h}(M), define A : C_{h}(M) → R as follows.

A([f]) = lim_{s→∞} \frac{∑_{Γ_{s}} f(U)}{vol(B(x_{0}, s))}.

Proposition 7.2 A : C_{h}(M) → R is well defined.

Proof: We first show that

F (s) = \frac{∑_{Γ_{s}} f(U)}{vol(B(x_{0}, s))}

converges as s → ∞. Since the centers of the U s are uniformly separated and M has bounded geometry, there is a constant C so that for any difference of two concentric balls, B = B(x_{0}, s + t) − B(x_{0}, t), the number of elements of Γ with centers in B is bounded by C vol(B). If |f| is bounded by C_{f}, it follows immediately that

| ∑_{x_{U} ∈B} f(U)| ≤ C_{f} C vol(B).

In particular, this holds if t = 0, so F (s) is a bounded function and we need only show that for bounded t > 0,

lim_{s→∞} F (s + t) − F (s) = 0.

Note that t may vary with s, but it is a bounded function of s. Given any set B ⊂ M, denote by f(B) the sum

f(B) = ∑_{x_{U} ∈B} f(U)

and write B(s) for B(x_{0}, s) and vB(s) for vol(B(x_{0}, s)). Then

|F (s + t) − F (s)| = |\frac{f(B(s + t))}{vB(s + t)} − \frac{f(B(s))}{vB(s)}| ≤

|\frac{f(B(s))}{vB(s + t)} − \frac{f(B(s))}{vB(s)}| + |\frac{f(B(s + t) − B(s))}{vB(s + t)}| ≤

C_{f}C vB(s)|\frac{1}{vB(s + t)} − \frac{1}{vB(s)}| + C_{f}C|\frac{vB(s + t) − vB(s)}{vB(s + t)}| =

2C_{f}C \frac{vB(s + t) − vB(s)}{vB(s + t)} = 2C_{f}C(1 − \frac{vB(s)}{vB(s + t)})

which by our assumption goes to zero as s → ∞.

To show that A([f]) is independent of the choice of f, we need only show that for any f ∈ C^{0}(Γ) and π ∈ G,

lim_{s→∞} \frac{∑_{Γ_{s}}[f(U) − π^{∗}f(U)]}{vB(s)} = 0.

Let C_{f} be a bound for f (and so also for π^{∗}f), and let t be a bound on the distance that π moves elements of Γ, i.e. for all U ∈ Γ, d(x_{U}, x_{π(U)}) < t. Then

|\frac{∑_{Γ_{s}}[f(U) − π^{∗}f(U)]}{vB(s)}| = |\frac{f(B(s)) − π^{∗}f(B(s))}{vB(s)}|.

Since π moves elements of Γ at most a distance t, each element in f(B(s − t)) occurs in π^{∗}f(B(s)). Thus f(B(s)) and π^{∗}f(B(s)) differ on at most the number of points in B(s) − B(s − t). Thus

|\frac{f(B(s)) − π^{∗}f(B(s))}{vB(s)}| ≤ 2C_{f}C \frac{[vB(s) − vB(s − t)]}{vB(s)}.

As above, this last goes to zero as s → ∞. Q.E.D.

Recall that the manifold M of our example above supports the four classical complexes. It is not difficult to see that M also satisfies Condition 7.1. For the classical complexes, the operators Δ_{i} commute with the action of the covering group of M over Σ_{4}. The diffeomorphism f also commutes with this action, so we are in the situation considered by Atiyah in [A76]. It follows easily that for the operators in Theorem 11, the linear functional A is the Atiyah trace (given by integration over a single fundamental domain, which is a compact set) divided by the volume of the fundamental domain. As the Schwartz kernel of the T_{i}e^{−tΔ_{i}} is uniformly bounded and converges pointwise to the Schwartz kernel of T_{i}^{∗} and A is integration over a compact set, A does commute with the limit in Theorem 11. Thus we have the following.

Theorem 7.3 Let M and f be as in the example above and denote by V the volume of a fundamental domain for Σ_{4} in the Poincaré disc. Then • For the de Rham complex, A(L(f^{∗})) = \frac{−4}{V}

• for the Signature complex A(L(f^{∗})) = \frac{−4i}{V}

• for the Dolbeault complex with k = 0, A(L(f^{∗})) = \frac{2}{(1 − i)V} , and for k = 1, A(L(f^{∗})) =

\frac{2i}{(1 − i)V}

• for the Spin complex A(L(f^{∗})) = \frac{±i\sqrt{2}}{V} depending on which lifting of df one uses.

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