Riemann-Roch-Grothendieck and Torsion for Foliations

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 and Connor Lazarov Department of Mathematics Herbert Lehman College, CUNY Bronx, New York 10468 e-mail: lazarov@lcvax.lehman.cuny.edu

Abstract

We prove a Riemann-Roch-Grothendieck theorem for the characteristic classes of a flat vector bundle over a foliation whose graph is Hausdorff. We assume that the strong foliation Novikov-Shubin invariants of the flat bundle are greater than three times the codimension of the foliation. Using transgression, we define a torsion form which in the odd acyclic case determines a Haefliger cohomology class which only depends on the foliation and the flat bundle. We construct examples where this torsion class is highly non-trivial. Key Words: Riemann-Roch-Grothendieck Theorem, torsion, foliation. AMS Subject Classifications: 19L10, 57R20, 58J35

∗Partially supported by NSF Grant DMS 9400676

1 Introduction

In this paper we extend the results of Bismut and Lott [BL95] for a flat complex vector bundle E over a fibration to such bundles over a foliation F of a compact manifold M. First we prove a Riemann-Roch-Grothendieck Theorem for the characteristic classes of E. That is, we relate the flat classes of E to those of its “direct image” on the “space of leaves of F.” As most manifolds are not the total space of a fibration, but do admit positive codimension foliations, this result represents a significant extension of the Bismut-Lott result. We assume that the graph of F is Hausdorff, that the projection onto the kernel of the Laplacian of E over F is smooth and that the strong foliation Novikov-Shubin invariants of E over F are greater than three times the codimension of the foliation. Next, using transgression, we define a torsion form in the space of Haefliger forms of F which in the odd acyclic case is closed. The class it defines in the Haefliger cohomology of F depends only on the foliation and the flat bundle. We show how the results here encompass many previous results on torsion. Finally, we show that the foliation torsion is highly non-trivial for Seifert fibrations of generalized lens spaces.

In the 1930’s Reidemeister and Franz, [Rei35], [Fra35], defined an invariant for simplicial complexes, the so called Reidemeister torsion. It assigns a real number to a flat unitary bundle E over a finite simplicial complex K whose twisted cohomology H^{∗}(K; E) is zero. Reidemeister torsion distinguishes lens spaces which are homotopy equivalent but not homeomorphic. In [RS71] Ray and Singer defined analytic torsion for acyclic flat bundles over a compact manifold M and conjectured that their torsion is the same as the Reidemeister torsion. This conjecture was proven independently by Cheeger [Che79] and Müller [Mül78]. The equality of these two invariants can be extended to the case of a non unitary E, [Mül93], [BZ92]. Based on the well known relationship between Reidemeister torsion and the first algebraic K-group of fields, Wagoner [Wag78] suggested that these torsions should extend to invariants for families of compact spaces, i.e. fiber bundles, and should be related to the higher algebraic K-theory groups. On the topological side, this extension was achieved by Igusa [Igu91] and Klein [Kle] using Waldhausen’s K-theory of spaces and parametrized Morse theory. On the analytic side, deep and important results on torsion for holomorphic families have been obtained by Gillet and Soulé [GS91a], and Bismut, Gillet and Soulé [BGS88]. In the smooth category, Bismut and Lott [BL95] defined a higher torsion for fiber bundles. Their torsion form is a differential form on the base space of the bundle and under suitable acyclicity conditions it gives a smooth topological invariant of the bundle. They show that it extends the Ray-Singer torsion and that when the base space is S^{2}, their torsion agrees with the higher torsion of Igusa-Klein. A natural problem is then to extend the Cheeger-Müller theorem to torsion for families.

The heat equation method has proven to be a powerful tool in the theory of invariants of manifolds. The fact that the asymptotics of the trace of the heat operator as t → 0 are completely local suggested that many results for manifolds could be extended to foliations of compact manifolds. See [Con86], [HL90], where, respectively, the Index Theorem, and the Lefschetz Theorem were proven for foliations using the heat equation method. In [Qui85] Quillen conjectured that there should be a heat equation proof of the Families Index Theorem using superconnections. This program was carried out by Bismut in [Bis86]. See also [BV87]. In [HL99], we apply the results here to extend the results in [Hei95] and obtain a Families Index Theorem for generalized Dirac operators defined along the leaves of a foliation whose graph is Hausdorff and whose strong Novikov-Shubin invariants are sufficiently positive. In [BL95], Bismut and Lott used superconnections and the heat equation method to prove a Riemann-Roch-Grothendieck Theorem for flat vector bundles over a fiber bundle. When the primary invariants occurring in this theorem are zero, they constructed a families torsion invariant for such bundles.

There are three major problems to overcome in order to extend such results to foliations. First, one needs to extend Duhamel’s formula for the derivative of a family of heat operators to such operators defined along the leaves of a foliation. This was done in [Hei95] provided that the holonomy (or homotopy) graph of the foliation is Hausdorff. Second, one needs a trace on the appropriate von Neumann algebra. Originally, it was assumed that the foliation admitted an invariant transverse measure in order to obtain a real valued trace. In [Hei95] we showed that the existence of such a measure was not necessary, and one can define a trace which takes values in the Haefliger cohomology of the foliation. Third, one needs good control of the asymptotics as t → ∞. In the families case, this control follows from the fact that the positive spectrum of the fiberwise Laplacian under consideration is uniformly bounded away from zero. For foliations, the spectrum is not so well behaved and can in fact be all of the non-negative reals. To obtain our results, we replace the strong condition of having a uniform gap about zero by the weaker assumptions that the projection onto the kernel of the Laplacian of E over F is smooth and that the strong foliation Novikov-Shubin invariants of E over F are sufficiently positive.

The main technical results of this paper are in Section 2. Proposition 2.4 relates the Haefliger exterior derivative to the superconnection C_{t}. Theorem 2.8 gives the asymptotics of the modified heat operator under consideration. The RRG Theorem for E over F is an immediate corollary. In Section 3 we define the foliation torsion, T (F; E) for E over F. It is a Haefliger form on a complete transversal of the foliation. We define the foliation Ray-Singer torsion T (F; E) and show that it is the order zero part of T (F; E). We prove that in the odd acyclic case, T (F; E) is a closed form whose Haefliger cohomology class depends only on F and E. In Section 4, we relate the foliation torsion to other torsion invariants and compute examples.

Readers who are familiar with [BL95] will note that the results presented here are extensions to foliations of the results of that paper. Our contribution is showing how to overcome the considerable technical difficulties involved in making that extension.

It is a pleasure to thank Jean-Michel Bismut, Donggeng Gong, Leon Karp, John Lott, and Melvin Rothenberg for helpful conversations. In particular, we would like to thank Gong and Rothenberg for giving us a preliminary copy of their paper [GR96]. This allowed us to significantly weaken the hypotheses of Theorem 2.8. Finally, part of this work was done while the first author was visiting the Institut des Hautes Etudes Scientifique and the University of Paris VII. He would like to thank them both for their warm hospitality.

2 Riemann-Roch-Grothendieck for foliations

In this section we combine the results of [BL95] and [Hei95] to prove a Riemann-Roch-Grothendieck theorem for flat vector bundles over a compact foliated manifold. Our guiding principle is that the appearance of bundles in [BL95] is a red herring in the sense that the objects one is really interested in are the spaces of sections of the bundles. A paradigm for this idea is that the ‘bundle’ W we define below is not really a bundle, but we may still consider its space of sections, which is the space of smooth sections of the bundle ∧^{∗}T^{∗}F_{s} ⊗Ẽ over the manifold G.

The proof of our RRG Theorem is long and complicated, so we have divided this section into subsections of more manageable proportions.

2.1 Some definitions

We assume that all objects are smooth unless specified otherwise. If A is a manifold and B is a bundle, we denote by C^{∞}(A; B), C_{c}^{∞}(A; B), Ω(A; B), and Ω_{c}(A; B) the smooth sections, the smooth sections with compact support, the smooth B valued forms, and the smooth B valued forms with compact support, respectively.

Let F be a oriented codimension q foliation of a compact oriented n = p + q dimensional Riemannian manifold M. The metric g on M gives rise to a metric on each leaf of F and this metric is well defined up to quasi-isometry. The holonomy groupoid G of F consists of equivalence classes of paths γ : [0, 1] → M such that the image of γ is contained in a leaf of F. Two such paths are equivalent if γ_{1}(j) = γ_{2}(j), j = 0, 1 and the holonomy along them is the same. There are two natural maps s, r : G → M given by s([γ]) = γ(0), r([γ]) = γ(1). G has two natural foliations F_{s} and F_{r} whose leaves are respectively s^{−1}(x), r^{−1}(x) for x ∈ M. We will work primarily with the foliation F_{s}. For each x ∈ M, denote by L_{x} the leaf of F through x and set L̃_{x} = s^{−1}(x). Note that r : L̃_{x} → L_{x} is the holonomy cover of L_{x} and is a local isometry for the metric on G defined below.

There is a canonical lift of the normal bundle ν of F to a bundle ν_{G} ⊂ T G so that T G = T F_{s} ⊕ T F_{r} ⊕ ν_{G} and r_{∗}ν_{G} = ν, s_{∗}ν_{G} = ν. It is given as follows. Let [γ] ∈ G with s([γ]) = x. Denote by exp : ν → M the exponential map. Given X ∈ ν_{x} and t ∈ R sufficiently small, there is a unique curve γ_{t} : [0, 1] → M so that

i) γ_{t}(0) = exp tX  ii) γ_{t} ⊂ L_{exp tX}  iii) γ_{t}(s) ∈ exp(ν_{γ(s)}).

In particular γ_{0} = γ. Thus the family [γ_{t}] in G defines a tangent vector X̂ ∈ T G_{[γ]}. It is easy to check that s_{∗}(X̂) = X and r_{∗}(X̂) is the parallel translate of X along γ to ν_{r[γ]}.

The metric g on M induces a canonical metric g on G by declaring that the subbundles T F_{s}, T F_{r} and ν_{G} are mutually orthogonal and on T F_{r} ⊕ ν_{G} defining g to be s^{∗}g and on T F_{s} defining it to be r^{∗}(g|T F).

The homotopy groupoid H of F is the covering of G which consists of equivalence classes of leafwise paths where two such paths are equivalent if they are homotopy equivalent in their leaf, relative to their end points. The homotopy groupoid has essentially the same properties as the holonomy groupoid.

In order to use the results of [Hei95] we will assume that G and H are Hausdorff. Riemannian foliations and foliations constructed from the suspension of analytic group actions always have Hausdorff holonomy graphs. Note that if G is Hausdorff, then H is also, but the converse is not true. For example the suspension of a non identity diffeomorphism of S^{1} which is the identity on an open set has non-Hausdorff holonomy graph, but Hausdorff homotopy graph. Note also that the results of [Hei95] are still true if G is replaced by H. We will work here with a Hausdorff G with the understanding that all results are also true for any Hausdorff H.

Let E be an complex vector bundle over M and set Ẽ = r^{∗}(E). One of our basic objects of study will be the ‘fiber bundle’ W over M with ‘fiber’ over x the space W_{x} = C_{c}^{∞}(L̃_{x}; ∧^{∗}T^{∗}F_{s} ⊗Ẽ). As noted above, W is not usually a fiber bundle as it fails to be locally a product, but we may still speak of its smooth sections, C^{∞}(M; W) = C_{c}^{∞}(G; ∧^{∗}T^{∗}F_{s} ⊗ Ẽ). Note also that W is Z and Z_{2} graded using the gradings on forms.

Now suppose that E is a flat Hermitian vector bundle over M with metric h^{E} and flat connection ∇^{E}. Using the pullback ˜∇^{E} of ∇^{E} to Ẽ, the ordinary de Rham operator on G can be extended to a de Rham operator d^{E} on sections of ∧^{∗}T^{∗}G⊗Ẽ. In particular, any connection on Ẽ may be viewed as a map from C^{∞}(Ẽ) to C^{∞}(∧^{1}T^{∗}G ⊗ Ẽ). If ω ∈ C^{∞}(∧^{k}T^{∗}G) and ϕ ∈ C^{∞}(Ẽ), then d^{E}(ω ⊗ ϕ) = dω ⊗ ϕ + (−1)^{k}ω ∧ ˜∇^{E}(ϕ). Acting on Ω(M; W) ≃ Ω_{c}(G; Ẽ) d^{E} satisfies (d^{E})^{2} = 0. In addition, for all ω ∈ Ω^{k}(M) and ϕ ∈ Ω_{c}(G; Ẽ), we have

d^{E}(s^{∗}ω ∧ ϕ) = s^{∗}(d^{M}ω) ∧ ϕ + (−1)^{k}s^{∗}ω ∧ d^{E}ϕ.

Here d^{M} is the ordinary de Rham operator on M. Thus d^{E} is a flat super connection on W of degree 1.

Denote the adjoint of d^{E} by (d^{E})^{∗}. Let N be the number operator on W, that is N multiplies an element of Ω(M; C_{c}^{∞}(∧^{k}T^{∗}F_{s} ⊗ Ẽ)) = C_{c}^{∞}(G; ∧^{k}T^{∗}F_{s} ⊕ ∧^{∗}(T^{∗}F_{r} ⊕ ν_{G}^{∗}) ⊗ Ẽ) by k. Denote by (d^{E})^{∗_{t}} the adjoint of d^{E} with respect to the metric obtained by scaling the leafwise metric on M by 1/t. Then

(d^{E})^{∗_{t}} = t^{−N}(d^{E})^{∗}t^{N}.

Define

C_{t} = \frac{1}{2}t^{N/2}[(d^{E})^{∗_{t}} + d^{E}]t^{−N/2} = \frac{1}{2}[t^{−N/2}(d^{E})^{∗}t^{N/2} + t^{N/2}d^{E}t^{−N/2}]

and

D_{t} = \frac{1}{2}t^{N/2}[(d^{E})^{∗_{t}} − d^{E}]t^{−N/2} = \frac{1}{2}[t^{−N/2}(d^{E})^{∗}t^{N/2} − t^{N/2}d^{E}t^{−N/2}].

These are the operators we will use to prove our results. They are operators on Ω(M; W) which we also think of as operators on Ω_{c}(G; Ẽ). Note that C_{t}^{∗} = C_{t} and D_{t}^{∗} = −D_{t}.

A careful reading of sections III a-e of [BL95] shows that all computations and constructions there are local on their total space, so that these computations and constructions are still valid for d^{E}, C_{t} and D_{t} on our total space G. In particular, both C_{t} and D_{t} depend only on the leafwise metric on F (not on the global metric on M) and the choice of normal bundle ν. In addition, the operator D_{t} is a pure leafwise operator (that is, it differentiates only F_{s} variables), so it commutes with the natural action of ∧^{∗}T^{∗}M on ∧^{∗}T^{∗}G ⊗ Ẽ. Thus it is an element of Ω(M; End(W)). The same is true of D_{t}^{2} = −C_{t}^{2} and so also of the associated heat operator e^{D_{t}^{2}} = e^{−C_{t}^{2}}. Let P_{t}(y, z) be the Schwartz kernel of the leafwise operator e^{D_{t}^{2}}, where y, z ∈ G with s(y) = s(z). P_{t} satisfies the crucial estimate (3.3) of [Hei95], so it is smooth in all its variables. Because e^{D_{t}^{2}} ∈ Ω(M; End(W)) and (D_{t}^{2})^{∗} = D_{t}^{2}, its pointwise trace satisfies tr(P_{t}(y, y)) ∈ r^{∗}(∧^{∗}T^{∗}M)_{y} = ∧^{∗}T^{∗}M_{x}, where x = s(y). That is, this trace is real valued. In fact more is true.

Lemma 2.1 P_{t}(y, y) takes values in ∧^{∗}ν_{G}^{∗} ⊗ End(W), so tr(P_{t}(y, y)) ∈ (∧^{∗}ν_{G}^{∗})_{y} = ∧^{∗}ν_{x}^{∗}.

Proof: Let U ≃ R^{q} × R^{p} be a foliation chart in M with transversal T ≃ R^{q} × {0} and let π : U → T be the projection. For any b ∈ U, L̃_{b} is isometric to L̃_{x}, where x = π(b) ∈ T, where the isometry is given by composition with any path in the placque of b from b to x. This isometry also induces an isometry between Ẽ|L̃_{b} and Ẽ|L̃_{x}. The construction of D_{t} depends only on the metric on L̃ and the structures on Ẽ. On s^{−1}(U), the structure is isometric to the obvious structure on s^{−1}(T) × R^{p} over T × R^{p}, with leaves L̃_{x} × {a}, x ∈ T, a ∈ R^{p}. Because the metrics used on s^{−1}(T) × R^{p} are independent of a ∈ R^{p}, the ∗ operator is also independent of a and it is easy to see that D_{t} does not involve any differentiation in the a coordinate. More precisely, for (x, a) ∈ T × R^{p}, D_{t,(x,a)} = D_{t,x} where D_{t,x} is the operator defined on s^{−1}(T). Thus e^{D_{t,x}^{2}} takes values in ∧^{∗}T^{∗}T_{x} ⊗ End(W |_{s^{−1}(T)}) and ∧^{∗}T^{∗}T_{x} = ∧^{∗}ν_{x}^{∗}, so we are done. □

Note that Lemma 2.1 also holds for the objects we construct below out of D_{t}, e.g. A_{ϵ,t}, D̄_{ϵ,t}, ηP_{ϵ}D_{t}P_{ϵ}e^{(P_{ϵ}D_{t}P_{ϵ})^{2}}, possibly with ∧^{∗}ν_{x}^{∗} replaced by ∧^{∗}ν_{x}^{∗} ⊗ C. It is also remains true if we replace tr by the pointwise supertrace tr_{s} associated to the Z_{2} grading on W.

2.2 Haefliger traces

In order to define the trace function we will use, we first recall the notion of the Haefliger forms and Haefliger cohomology of F, [Hae80]. Let U = {U_{1}, …, U_{r}} be a good cover of M by foliation charts, [HL90]. Let T_{i} ⊂ U_{i} be the transversal and set T = ∪ T_{i}. We may assume that the T_{i} are disjoint. Let P be the holonomy pseudogroup induced by F on T. Let Ω_{c}^{k}(T) be the space of smooth k forms on T with compact support and denote by Ω_{c}^{k}(T/P) the quotient of Ω_{c}^{k}(T) by the vector subspace generated by elements of the form α − h^{∗}α where h ∈ P and α ∈ Ω_{c}^{k}(T) has support contained in the range of h. Give Ω_{c}^{k}(T/P) the quotient topology of the usual C^{∞} topology on Ω_{c}^{k}(T). The exterior derivative d^{T} : Ω_{c}^{k}(T) → Ω_{c}^{k+1}(T) induces a continuous differential d^{H} : Ω_{c}^{k}(T/P) → Ω_{c}^{k+1}(T/P).

Note that Ω_{c}^{k}(T/P) and d^{H} are independent of the choice of cover U. Set Ω_{c}^{∗}(F) = Ω_{c}^{∗}(T/P) ⊗ C and consider the complex {Ω_{c}^{∗}(F), d^{H}} and its cohomology by H_{c}^{∗}(F). They are, respectively, the Haefliger forms and Haefliger cohomology of F.

If F is the foliation given by a fiber bundle M → B, Ω_{c}^{∗}(F) is the usual de Rham complex of B tensored with C, and H_{c}^{∗}(F) is the usual de Rham cohomology of B tensored with C.

As the bundle T F is oriented, there is a continuous open surjective linear map, called integration over the fiber,

∫_{F} : Ω^{p+k}(M) → Ω_{c}^{k}(F)

which commutes with the de Rham operators d^{M} and d^{H}. It is given as follows. Choose a partition of unity {ϕ_{i}} subordinate to U. Given ω ∈ Ω^{p+k}(M), ϕ_{i} ω ∈ Ω_{c}^{p+k}(U_{i}). Integrate ϕ_{i} ω along the fibers of the submersion π_{i} : U_{i} → T_{i} to obtain ∫_{U_{i}} ϕ_{i} ω ∈ Ω_{c}^{k}(T_{i}). Define ∫_{F} ω

to be the class of ∑_{i} ∫_{U_{i}} ϕ_{i} ω in Ω_{c}^{k}(F). It is independent of the choice of the ϕ_{i} and of the

cover U. The construction obviously extends to a map ∫_{F} : Ω^{p+k}(M) ⊗ C → Ω_{c}^{k}(F).

Let K be a section of ∧^{∗}T^{∗}M ⊗End(W), and write its leafwise Schwartz kernel as K^{x}(y, z) where x ∈ M and y, z ∈ L̃_{x}. We say that K is equivariant if it takes values in ∧^{∗}ν^{∗} ⊗End(W) and for all u ∈ r^{−1}(x), K^{s(u)}(uy, uz) = (u^{∗} ⊗ 1)K^{x}(y, z), where u^{∗} : ∧^{∗}ν_{x}^{∗} → ∧^{∗}ν_{s(u)}^{∗} is the action induced by the holonomy along u. We say that K is super exponentially decaying if it is smooth in x, y, and z and given multi indices α, β, γ and a real number S > 0, there are constants C_{1}, C_{2} > 0 such that for all x, y, and z,

|\frac{∂^{|α|+|β|+|γ|}}{∂x^{α}∂y^{β}∂z^{γ}} K^{x}(y, z)| ≤ C_{1} exp[\frac{−d_{x}(y, z)^{2}}{C_{2}} ]. (2.2)

Here ∂/∂y and ∂/∂z come from the coordinates on L̃_{x} obtained from the good cover U, ∂/∂x comes from the coordinates of U, and d_{x}( , ) is the distance on L̃_{x}. We say that K is measurable and super exponentially decaying if it is measurable in x, smooth in y, and z and Equation 2.2 holds for all β and γ, with α = 0.

Examples of such smooth equivariant super exponentially decaying sections are given by p(D_{t})e^{D_{t}^{2}} where p(z) is any polynomial. The equivariance follows from Lemma 2.1 above and the fact that D_{t} depends only on the leafwise metric (not on the global metric on M) and on the choice of normal bundle ν. The super exponential decay is classical and follows because −D_{t}^{2} = C_{t}^{2}, a leafwise elliptic operator whose leading order symbol is given by t times the metric. See Theorem 3.11 of [BL95] and the remarks on p. 520 of [Hei95].

Let dx be the volume form along the leaves of F. For x ∈ M, we denote also by x ∈ L̃_{x} the class of the constant path at x. Then tr_{s}(K^{x}(x, x)) is a smooth section of the complexified normal bundle ∧^{∗}ν^{∗} ⊗ C of F, so tr_{s}(K^{x}(x, x))dx is a smooth C valued differential form on M. The Haefliger trace and supertrace of K are defined to be

Tr(K) = ∫_{F} tr(K^{x}(x, x))dx and Tr_{s}(K) = ∫_{F} tr_{s}(K^{x}(x, x))dx,

where tr_{s} is the pointwise supertrace associated to the Z_{2} grading on W. If K^{x}(y, z) is bounded and measurable, we may also define Tr(K) and Tr_{s}(K) and we obtain a bounded measurable differential form on the transversal T.

As an application of these ideas, we extend a result from [BL95] to our situation. Recall that P_{t}(y, z) is the Schwartz kernel of e^{D_{t}^{2}} = e^{−C_{t}^{2}}. By Theorem 3.1 of [Hei95], Tr_{s}(P_{t}) is a closed Haefliger form, and its cohomology class [Tr_{s}(P_{t})] in H_{c}^{∗}(F) is independent of the metric on M. Since the parameter t affects only the metric, [Tr_{s}(P_{t})] is independent of t. In [Hei95] we interpreted [Tr_{s}(P_{t})] as the non-normalized Chern character of the ‘index bundle’ H^{∗}(F; E) over M (defined below). We now compute the normalized form of this Chern character. Denote by e(T F) the Euler form of the Riemannian bundle T F over M and by e(F) its cohomology class. Define the Euler class χ(F) of F to be the Haefliger cohomology class of the form ∫_{F} e(T F). Let φ : Ω_{c}^{∗}(F) ↦ Ω_{c}^{∗}(F) be the linear map which for

all α ∈ Ω_{c}^{k}(F) is φ(α) = (2πi)^{(−k/2)}α. Finally, let rk(E) be the rank of the vector bundle E.

Theorem 2.3 (Theorem 3.15, [BL95]) For all t > 0, φ(Tr_{s}(P_{t})) = rk(E)χ(F).

Proof: Tr_{s}(P_{t}) = Tr_{s}(e^{D_{t}^{2}}) and by Proposition 3.5 of [Hei95], we have

\frac{∂}{∂t}Tr_{s}(e^{D_{t}^{2}}) = Tr_{s}([D_{t}, \frac{∂D_{t}}{∂t}]e^{D_{t}^{2}}).

This is equal to

Tr_{s}(D_{t} \frac{∂D_{t}}{∂t} e^{D_{t}^{2}} + \frac{∂D_{t}}{∂t} D_{t}e^{D_{t}^{2}}) = Tr_{s}(D_{t} \frac{∂D_{t}}{∂t} e^{D_{t}^{2}} + \frac{∂D_{t}}{∂t} e^{D_{t}^{2}}D_{t}) = Tr_{s}([D_{t}, \frac{∂D_{t}}{∂t} e^{D_{t}^{2}}]),

which is zero by Lemma 2.6 below. A standard rescaling argument and local index theory then give the result. See [BL95], pp. 333-4. □

We remark that this construction of a trace and supertrace with values in the Haefliger forms of F allows the extension to arbitrary foliations of the results in [HL90] as well as the Connes foliation index theorem [Con86]. In essence, one erases all mention of the invariant transverse measure and rereads the proofs replacing the traces used there by the above Haefliger traces.

2.3 A result on traces

Proposition 2.4 If K be a smooth equivariant section of ∧^{∗}T^{∗}M ⊗ End(W) which is super exponentially decaying, then d^{H}(Tr_{s}(K)) = Tr_{s}([C_{t}, K]).

Note that when we take commutators of operators, we always take graded commutators. As [C_{t}, D_{t}] = 0, we have immediately

Corollary 2.5 Let p(z) be any polynomial, then d^{H}(Tr_{s}(p(D_{t})e^{D_{t}^{2}})) = 0.

We begin the proof of Proposition 2.4 with

Lemma 2.6 Suppose that H is a continuous equivariant section of ∧^{∗}T^{∗}M ⊗ End(W) with uniformly bounded coefficients. Then Tr_{s}([H, K]) = 0.

Proof: Denote by k(x, y, z) the pointwise supertrace tr_{s}([H, K]), where x ∈ M and y, z ∈ L̃_{x}, and the operators are composed at the point z. That is, k(x, y, z) = tr_{s}(H^{x}(y, z)K^{x}(z, y) ± K^{x}(y, z)H^{x}(z, y)), where the ambiguity of the signs occurs because we are using graded commutators. Let λ > 0 and denote by G_{λ} the Green’s operator for the positive equivariant operator D_{λ} = λ − D_{t}^{2}. For sufficiently large N, HG_{λ}^{N} and G_{λ}^{N}H have continuous bounded Schwartz kernels and are equivariant. In addition, D_{λ}^{N}K and KD_{λ}^{N} are smooth, equivariant and super exponentially decaying. By noting that HK ± KH = HG_{λ}^{N}D_{λ}^{N}K ± KD_{λ}^{N}G_{λ}^{N}H, we may assume that the Schwartz kernel H^{x}(y, z) of H is continuous and bounded. Now k(x, y, z) also satisfies Equation 2.2 and it is equivariant. Recalling that we also denote by x ∈ L̃_{x} the class of the constant path at x, we need to show

∫_{F} ∫_{L_{x}} k(x, x, z)dz dx = 0,

where the first x in k(x, x, z) is in M and the second is in L̃_{x}.

Let U_{G} = {(U_{i}, U_{j}, γ_{ijk})} be the cover of G corresponding to the good cover U of M. Here, U_{i}, U_{j} ∈ U and γ_{ijk} is a leafwise path from U_{i} to U_{j}. The open set (U_{i}, U_{j}, γ_{ijk}) consists of the classes of all leafwise paths which start in U_{i}, end in U_{j} and are homotopic to γ_{ijk} through leafwise paths which also start in U_{i} and end in U_{j}. It is easy to see that U determines the cover U_{G}.

Let {ϕ_{i}} be a partition of unity subordinate to the cover U. For each (U_{i}, U_{j}, γ_{ijk}), define ϕ_{ijk} : G → R by ϕ_{ijk}(z) = ϕ_{j}(r(z)) for z ∈ (U_{i}, U_{j}, γ_{ijk}), and ϕ_{ijk}(z) = 0 otherwise. Then for each x ∈ M, {ϕ_{ijk}} restricted to L̃_{x} is a partition of unity subordinate to the cover of L̃_{x} by its placques in the various (U_{i}, U_{j}, γ_{ijk}).

On the transversal T_{i} ⊂ U_{i}, ∫_{F} ∫_{L_{x}} k(x, x, z)dz dx is represented by the Haefliger form

∫_{P_{i}} ϕ_{i}(x) ∫_{L_{x}} k(x, x, z)dz dx, where P_{i} is the placque of x ∈ U_{i}. Because of the equivariance

of k, this is equal to ∫_{P_{i}} ϕ_{i}(y) ∫_{L_{x}} k(x, y, z)dz dy, where now x = P_{i} ∩ T_{i} and y ∈ P_{i} is thought

of as the class of the path in P_{i} from x to y. Now we have

∫_{P_{i}} ∫_{L_{x}} ϕ_{i}(y)k(x, y, z)dz dy = ∑_{j,k} ∫_{P_{i}} ∫_{L_{x}} ϕ_{i}(y)ϕ_{i,j,k}(z)k(x, y, z)dz dy =

∑_{j,k} ∫_{P_{i}} ∫_{P_{j}} ϕ_{i}(y)ϕ_{j}(z)k(x, y, z)dz dy,

where z ∈ P_{j} ⊂ U_{j} is thought of as the element of (U_{i}, U_{j}, γ_{ijk}) starting at x and ending at z. Note that with this interpretation, ϕ_{i,j,k}(z) becomes ϕ_{j}(z). For the chart (U_{i}, U_{i}, γ), γ a constant path, ∫_{P_{i}} ∫_{P_{i}} ϕ_{i}(y)ϕ_{i}(z)k(x, y, z)dz dy = 0 by symmetry. For the chart (U_{i}, U_{j}, γ_{ijk}),

consider the form I_{ijk} = ∫_{P_{i}} ∫_{P_{j}} ϕ_{i}(y)ϕ_{j}(z)k(x, y, z)dz dy on T_{i}. On T_{j} ⊂ U_{j} there is the

corresponding form I_{jik} = ∫_{P_{j}} ∫_{P_{i}} ϕ_{j}(z)ϕ_{i}(y)k(x, z, y)dy dz for the chart (U_{j}, U_{i}, γ_{ijk}^{−1}). The crucial point is that because we are using graded commutators, I_{ijk} = h_{γ_{ijk}}^{∗}(−I_{jik}). Thus if we move the term I_{jik} to T_{i} using the holonomy map h_{γ_{ijk}} associated to γ_{ijk}, it will cancel

I_{ijk} and we obtain a new representative of ∫_{F} ∫_{L} k, which at x ∈ T_{i} is given by

∫_{P_{i}} ∫_{L_{x}} ϕ_{i}(y)(1 − ϕ_{ijk}(z))k(x, y, z)dz dy,

and for x ∈ T_{j} is given by

∫_{P_{j}} ∫_{L_{x}} ϕ_{j}(z)(1 − ϕ_{jik}(y))k(x, z, y)dy dz.

Doing this procedure a finite number of times, say over all γ_{ijk} of less than a fixed leafwise length, we obtain a representative for ∫_{F} ∫_{L} k so that on each T_{i} it is dominated by

∫_{P_{i}} ∫_{_{L_{x}}−B_{R}(x)} |k(x, y, z)|dz dy,

where R is as large as we like. Here B_{R}(x) is the ball of radius R in L̃_{x} centered at x. Let A and B be upper bounds for the volume and diameter of the placques in the cover U. Then Equation 2.2 implies

\begin{matrix}∫_{P_{i}} ∫_{_{L_{x}}−B_{R}(x)} |k(x, y, z)|dz dy ≤ A ∫_{_{L_{x}}−B_{R}(x)} C_{1}e^{(−d_{x}(y,z)^{2}/C_{2})}dz ≤ \\ AC_{1}e^{(−(R−B)^{2}/2C_{2})} ∫_{_{L_{x}}−B_{R}(x)} e^{(−d_{x}(y,z)^{2}/2C_{2})}dz \end{matrix}

as x and y are in the same placque P_{i}, and d_{x}(y, z) ≥ R − B since d_{x}(x, z) > R. As L̃_{x} grows at most exponentially, the integral is bounded, and this expression can be made arbitrarily small by taking R large. □

Proof of Proposition 2.4: Let x ∈ U, where U ∈ U, the good cover of M. In general, the set s^{−1}(U) is not a product neighborhood of L̃_{x}. However, it does contain a neighborhood of L̃_{x} which is diffeomorphic to an open subset of L̃_{x} × U containing L̃_{x} × {x}. To see this, recall that any compact set K ⊂ L̃_{x} has an open neighborhood Ũ_{K} ⊂ s^{−1}(U) of K of the form K × U_{K}, where U_{K} ⊂ U. See [Hur], [Win83]. The subsets of Ũ_{K} of the form {y} × U_{K} are the image under the exponential map of a neighborhood of zero in (T F_{r} ⊕ ν_{G})_{y}. Thus these product neighborhoods are coherent, and taking a union of them, we obtain an open neighborhood Ũ ⊂ s^{−1}(U) of L̃_{x} which we may identify with an open subset of L̃_{x} × U containing L̃_{x} × {x}. We call Ũ a coordinate neighborhood adapted to x. Now define the differential operator d̃^{M} on G by declaring that on L̃_{x} it is given by d^{M} in the coordinates on Ũ, i.e. d^{M} differentiates in the U directions only. Then d̃^{M} is globally well defined and equivariant, and on Ũ it is given by

d̃^{M} = d^{M} + J,

where J is a local leafwise differential operator defined on L̃_{a} ∩ Ũ, depending smoothly on a ∈ U which is zero at x.

We may write C_{t} as C_{t} = d̃^{M} + H, where H is a smooth equivariant section of ∧^{∗}T^{∗}M ⊗ End(W) with uniformly bounded coefficients, i.e. H satisfies the hypothesis of Lemma 2.6. Thus we have

Tr_{s}([C_{t}, K]) = Tr_{s}([d̃^{M}, K]) = ∫_{F} tr_{s}([d̃^{M}, K])dx.

If x ∈ T_{i}, then ∫_{F} tr_{s}([d̃^{M}, K])dx is represented by the form whose value at x is

∫_{P_{i}} tr_{s}([d̃^{M}, K])dy

where as above, we consider the placque P_{i} of x as being contained in L̃_{x}. Let Ũ be a coordinate neighborhood adapted to x. Then on L̃_{x} ⊂ Ũ, we have d̃^{M} = d^{M} and we may write d^{M} = d^{F_{r}} + d^{T} where d^{F_{r}} is the exterior derivative along the leaves of F_{r} and d^{T} is the exterior derivative on the transversal T_{i}. The equivariance of K implies that tr_{s}(K) is constant on the leaves of F_{r}, so on L̃_{x} we have

tr_{s}([d̃^{M}, K]) = tr_{s}([d^{M}, K]) = tr_{s}(d^{M}(K)) = d^{M}tr_{s}(K) = d^{T} tr_{s}(K).

As d^{T} induces d^{H}, we are done. □

2.4 Some more definitions

The Hodge duality operator ∗ on C_{c}^{∞}(G; ∧^{∗}T^{∗}F_{s}) extends to an operator on the space C_{c}^{∞}(G; ∧^{∗}T^{∗}F_{s} ⊗ Ẽ) = C^{∞}(M; W). This gives a C^{∞}(M) ⊗ C valued inner product h^{W} on C^{∞}(M; W), where for s_{1}, s_{2} ∈ C^{∞}(M; W), and x ∈ M,

(< s_{1}, s_{2} >_{h^{W}})(x) = ∫_{L_{x}} s_{1}(x) ∧ ∗s_{2}(x).

Denote by d^{F} the de Rham operator along the leaves of F_{s} acting on C^{∞}(M; W). Let (d^{F})^{∗} be the leafwise adjoint of d^{F} with respect to h^{W}, i.e. (d^{F})^{∗} on W_{x} is the adjoint of d^{F} on W_{x}. The operator

V = (d^{F})^{∗} − d^{F}

is then an element of C^{∞}(M; End(W)). For each x ∈ M, V_{x} = V |W_{x} extends to a densely defined skew-adjoint operator acting on the L^{2} completion L^{2}(L̃_{x}; ∧^{∗}T^{∗}F_{s}⊗Ẽ) of W_{x}. Define H^{∗}(F; E) to be Ker(V), the direct integral of the Ker(V_{x}).

Denote by P the orthogonal projection onto Ker(V). It is a standard result that its Schwartz kernel K_{P}^{x}(y, z) is leafwise smooth, that is restricted to L̃_{x} it is smooth in y, z ∈ L̃_{x}. We assume that it is also transversely smooth, so that K_{P} is a smooth section of ∧^{∗}T^{∗}M ⊗ End(W). In the fiber bundle case, i.e. the case considered in [BL95], this is automatic as there is a fiberwise uniform gap about 0 in the spectrum of V. It is also true in the fiber bundle case if we consider the homotopy graph H of F, see [GR96]. Finally, it is true in the acyclic case, P = 0, which is the one of interest for torsion invariants. In that case, all of the holonomy covers are L^{2} acyclic. Examples of odd dimensional manifolds which are L^{2} acyclic are K(π, 1) manifolds with π amenable [CG86], non-positively curved locally symmetric spaces, and manifolds with sectional curvature pinched sufficiently closely to −1, [DX84].

Define the connection ∇^{W} on the bundle W as follows. Let X be a vector field on M. Denote by X̃ the vector field on G which is the unique section of T F_{r} ⊕ν_{G} satisfying s_{∗}(X̃) = X. Locally in G, the flow generated by X̃ sends leaves of F_{s} to leaves of F_{s}. This implies that the Lie derivative L_{X}, which is a local operator, acts on C^{∞}(M; W) = C_{c}^{∞}(G; ∧^{∗}T^{∗}F_{s} ⊗ Ẽ). Given φ ∈ C^{∞}(M; W) and a point y ∈ G, define (∇_{X}^{W} φ)(y) = (L_{X}φ)(y). ∇_{X}^{W} φ is then a well defined section of W. Note further that for f ∈ C^{∞}(M; R), ∇_{fX}^{W}φ = s^{∗}(f)∇_{X}^{W} φ, and ∇_{X}^{W} s^{∗}(f)φ = s^{∗}(Xf)φ + s^{∗}(f)∇_{X}^{W} φ, so ∇^{W} is indeed a connection.

Denote by (∇^{W})^{∗} the adjoint of ∇^{W} with respect to h^{W} and set

ω = (∇^{W})^{∗} − ∇^{W} .

We regard this as an element of Ω(M; End(W)), as well as a pointwise operator on sections of ∧^{∗}T^{∗}F_{s} ⊗ Ẽ over G.

Denote by (∇^{E})^{∗} the adjoint with respect to the metric h^{E} of the flat connection ∇^{E} on E. Regard E as a superbundle with E_{+} = E and E_{−} = 0. Recall that φ : Ω_{c}^{∗}(F) ↦ Ω_{c}^{∗}(F) is the linear map which for all α ∈ Ω_{c}^{k}(F) is φ(α) = (2πi)^{(−k/2)}α.

Set

f(z) = ze^{z^{2}}
f(∇^{E}, h^{E}) = (2πi)^{1/2}φtr_{s}(f(\frac{1}{2}[(∇^{E})^{∗} − ∇^{E}]) ∈ Ω^{∗}(M) ⊗ C,
f(∇^{H^{∗}(F;E)}, h^{H^{∗}(F;E)}) = (2πi)^{1/2}φTr_{s}(f(P \frac{1}{2}ωP)) ∈ Ω_{c}^{∗}(F)
and  f(C′_{t}, h^{W}) = (2πi)^{1/2}φTr_{s}(f(D_{t})) ∈ Ω_{c}^{∗}(F).

Note that f(∇^{H^{∗}(F;E)}, h^{H^{∗}(F;E)}) ∈ Ω_{c}^{∗}(F) since both P and ω are smooth, and f(C′_{t}, h^{W}) ∈ Ω_{c}^{∗}(F) since f(D_{t}) is a smooth section of ∧^{∗}T^{∗}M ⊗ End_{S}(W), where End_{S}(W) is the space of smoothing operators.

In order to extend Theorem 3.16 of [BL95], we need to assume that the strong foliation Novikov-Shubin invariants of E over F, [HL94], are at least three times the codimension of F. More specifically, denote by P_{ϵ} the spectral projection associated to the interval (0, ϵ) for the non-negative self adjoint operator −V^{2}/4. We assume that for each choice of metric on M there is β > 3q so that for all sufficiently small ϵ the Haefliger function Tr(P_{ϵ}) satisfies

Tr(P_{ϵ}) ∼ O(ϵ^{β}). (2.7)

When we say that a family of Haefliger forms is O(t^{β}) as t → 0 or t → ∞ we mean that it has a representative ψ_{t} defined on some transversal T, which is measurable in t, so that the function ∥ψ_{t}∥_{T} on T satisfies ∥ψ_{t}∥_{T} = O(t^{β}) as t → 0 or t → ∞. Here ∥ ∥_{T} is the pointwise norm on forms on T induced from the metric on M.

The fact that for any two good covers of M by foliation charts there is a integer m so that any placque of the first cover intersects at most m placques of the second cover implies easily that this condition does not depend on the choice of good cover. By the results of [HL94], this condition is independent of the metric on M.

Having positive strong foliation Novikov-Shubin invariants is guaranteed under various geometric conditions and all examples known to us satisfy this condition. See [GS91b],[Lot92], [NS86], and [Roe88]. Condition 2.7 always holds in the fiber bundle case as P_{ϵ} = 0 for some positive ϵ. For the same reason, it also holds in the foliation case where the holonomy groupoid is a fiber bundle with compact fibers (e.g. Example 4.4). It is not difficult to construct examples satisfying 2.7 where P_{ϵ} ≠ 0 for all positive ϵ. In particular, any foliation of T^{8} with all holonomy covers equivalent to R^{7} will do. Many generally, examples can be constructed by considering the natural foliations of flat bundles over manifolds whose L^{2} Novikov-Shubin invariants are sufficiently positive. The real obstruction to constructing more such examples is the paucity of information about the Novikov-Shubin invariants of non-compact manifolds.

2.5 The main technical theorem

The following is our main technical theorem.

Theorem 2.8 (Theorem 3.16, [BL95]) For all t > 0, the form f(C′_{t}, h^{W}) is real, odd and closed. Its Haefliger cohomology class is independent of t, and the metrics g on M and h^{E} on E. As t → 0,

(2.9) f(C′_{t}, h^{W}) = \{ \begin{matrix}_{∫F} e(T F)f(∇^{E}, h^{E}) + O(t) & if dim F is even, \\ O(\sqrt{t}) & if dim F is odd. \end{matrix}

There is γ > 0 so that as t → ∞

f(C′_{t}, h^{W}) = f(∇^{H^{∗}(F;E)}, h^{H^{∗}(F;E)}) + O(t^{−γ}). (2.10)

Proof: D_{t} is an odd element of of Ω(M; End(W)) and f(z) is an odd holomorphic function, so f(D_{t}) is also an odd element of Ω(M; End(W)). Thus for any 2k vector X ∈ ∧^{2k}T M, f(D_{t})(X) is an odd element of End(W), so it interchanges the two subpaces W^{±} ⊂ W which define the Z_{2} grading. This implies immediately that f(D_{t})(X) must have trace and supertrace zero, so f(C′_{t}, h^{W}) is an odd element of Ω_{c}^{∗}(F). By Corollary 2.5 it is closed. To see that it is real, note that D_{t}^{∗} = −D_{t} implies that f(D_{t})^{∗} = −f(D_{t}). This coupled with the fact that Tr_{s}(f(D_{t})) is an odd Haefliger form implies that f(C′_{t}, h^{W}) is real valued. (See sections I (c), (d) and (e) of [BL95].) Theorem 3.1 below implies that the Haefliger class of f(C′_{t}, h^{W}) is independent of t. The de Rham cohomology class of e(T F) is independent of g and that of f(∇^{E}, h^{E}) is independent of h^{E} by Theorem 1.11 of [BL95]. Thus Equation 2.9 implies that the Haefliger cohomology class of f(C′_{t}, h^{W}) is independent of g and h^{E}.

To prove Equation 2.9 we use the proof in [BL95], with minor additions. Let z be an odd Grassmann variable. Given α ∈ Ω_{c}^{∗}(F) ⊗ C[z], we can write it in the form α = α_{0} + zα_{1}, where α_{0}, α_{1} ∈ Ω_{c}^{∗}(F). Put α^{z} = α_{1}. As D_{t}^{2} = −C_{t}^{2}, we have

Tr_{s}(f(D_{t}^{2})) = Tr_{s}[exp(−C_{t}^{2} + zD_{t})]^{z}.

For t > 0, let ψ_{t} ∈ End(Ω_{c}^{∗}(F) ⊗ C[z]) be such that if α ∈ Ω_{c}^{∗}(F) ⊗ C[z] has total degree k, then ψ_{t}(α) = t^{−k/2}α. Then

Tr_{s}[exp(−C_{t}^{2} + zD_{t})] = ψ_{t}Tr_{s}[exp(t(−C_{1}^{2} + zD_{1}))].

By standard results on heat kernels, Tr_{s}[exp(t(−C_{1}^{2} + zD_{1}))] has an asymtotic expansion in t as t → 0. This expansion contains only integral powers of t if dim F is even, and only half-integral powers of t if dim F is odd. It then follows that f(C′_{t}, h^{W}) has an asymtotic expansion in t of the same type.

To calculate the t → 0 limit of Tr_{s}[exp(−C_{t}^{2} + zD_{t})]^{z}, we note that C_{t} and D_{t} satisfy the Lichnerowicz-type identity of Theorem 3.11 of [BL95] (which is purely local). Standard rescaling and local index arguments then show that the pointwise supertrace satisfies

lim_{t→0} (2πi)^{1/2}φtr_{s}[exp(−C_{t}^{2} + zD_{t})]^{z} = e(T F)(2πi)^{1/2}φtr_{s}[exp(\frac{ω^{2}}{4} + z \frac{ω}{2} )]^{z}.

All the objects we use come from objects on the compact manifold M and so are of bounded geometry. Thus the convergence above is uniform and we may interchange taking the limit and integrating over the fiber of F. This has the effect of replacing the pointwise supertrace tr_{s} by the Haefliger supertrace Tr_{s}. Doing so gives Equation 2.9 once we note that when dim F is odd, the Euler form e(T F) = 0.

All that remains is to prove Equation 2.10. It is quite complicated for two reasons: the cohomology of V may be non-zero; and the Green’s operator of −V^{2} is in general not bounded. This second obstruction occurs when P_{ϵ} ≠ 0 for all ϵ > 0. In the remainder of this section, we will prove Equation 2.10 in the case that P_{ϵ} = 0 for some ϵ > 0. We will still consider the operator P_{ϵ} in what we do, since we will need those more general results to prove the equation in all its generality. The reader may find it easier to read the rest of this section for the first time with the following assumptions in mind: ϵ is fixed, i.e. in Equation 2.15 ϵ does not depend on t, and for this fixed ϵ, P_{ϵ} = 0. Then G_{ϵ} is bounded independently of t and in the resulting analysis, 1/a may be replaced by 0.

To begin, we first filter the space M of all sections of ∧^{∗}T^{∗}M ⊗End(W) by the subspaces M_{i} of sections of ∑_{j≥i} ∧^{j}T^{∗}M ⊗ End(W). Filter the space N of all sections of ∧^{∗}T^{∗}M ⊗ End_{S}(W) similarly, where End_{S}(W) is the space of smoothing operators. An element A ∈ M assigns to each k vector X ∈ ∧^{k}T M_{x} an operator A(X) on the space W_{x} = C_{c}^{∞}(L̃_{x}; ∧^{∗}T^{∗}F_{s}⊗ Ẽ), or more properly its L^{2} completion. We say A is bounded if

∥A∥ ≡ sup{∥A(X)∥ | X ∈ ∧^{∗}T M, ∥X∥ = 1} < ∞.

We say A is smooth (resp. measurable) if it is a smooth (resp. measurable) section of ∧^{∗}T^{∗}M⊗ End(W).

Denote by ∥ ∥_{r,s} the Sobolev r, s norm on End(W_{x}). We say A ∈ N is bounded if for each r and s,

∥A∥_{r,s} ≡ sup{∥A(X)∥_{r,s} | X ∈ ∧^{∗}T M, ∥X∥ = 1} < ∞.

Because of the uniformly bounded geometry of the L̃_{x}, it follows from standard estimates that any bounded A ∈ N has a Schwartz kernel K_{A} which is leafwise smooth and pointwise uniformly bounded, as are all its leafwise derivatives. In particular, for sufficiently large s, and for any x ∈ M and y, z ∈ L̃_{x}, the pointwise norm of K_{A}^{x}(y, z), ∥K_{A}^{x}(y, z)∥ ≤ C∥A∥_{−s,s} where C is a constant depending only on s. Thus estimates on ∥A∥_{−s,s} translate directly into estimates on Tr(A) and Tr_{s}(A). In particular, if A_{t} ∈ N satisfies ∥A_{t}∥_{−s,s} = O(t^{α}) as t → ∞, this implies Tr(A_{t}) = O(t^{α}) also.

Given ϵ > 0, denote by Q_{ϵ} the spectral projection associated to the interval [ϵ, ∞) for the non negative self adjoint operator −V^{2}/4. Note that P_{ϵ} is a bounded measurable element of N and Q_{ϵ} is a bounded measurable element of M. Recall (see Equation (3.95) of [BL95]) that there is an element ĉ ∈ M_{2}, so that

D_{t} = \frac{\sqrt{t}}{2}V + \frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ, (2.11)

and note that V ∈ M_{0} and ω ∈ M_{1}. Set

D̄_{ϵ,t} = (P + Q_{ϵ})D_{t}(P + Q_{ϵ}) + P_{ϵ}D_{t}P_{ϵ},

and

A_{ϵ,t} = (P + Q_{ϵ})(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ)P_{ϵ} + P_{ϵ}(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ)(P + Q_{ϵ}).

As V commutes with the spectral projections of −V^{2}/4, and (P + Q_{ϵ})P_{ϵ} = 0 = P_{ϵ}(P + Q_{ϵ}), we have that D_{t} = D̄_{ϵ,t} + A_{ϵ,t}.

We now use an extension, due to Gong and Rothenberg [GR96] and involving ideas which go back to Müller, of the material in Chapter 9 of [BGV92]. In particular, we show that Tr_{s}(f(D̄_{ϵ,t})) satisfies Equation 2.10. When P_{ϵ} = 0 for some ϵ > 0 this completes the proof of Theorem 2.8, since in that case, D̄_{ϵ,t} = D_{t}. In the next section, we will finish the proof in general by showing that Tr_{s}(f(D_{t})) = Tr_{s}(f(D̄_{ϵ,t})) + O(t^{−γ}) as t → ∞ for some γ > 0.

Set D = D_{1} and D̂_{ϵ} = (P + Q_{ϵ})D(P + Q_{ϵ}). Then D̄_{ϵ} ≡ D̄_{ϵ,1} = D̂_{ϵ} + P_{ϵ}DP_{ϵ}. Set T_{ϵ,t} = Q_{ϵ}D_{t}Q_{ϵ}, and T_{ϵ} = T_{ϵ,1}.

Proposition 2.12 There is a bounded measurable section g_{ϵ} in M with g_{ϵ} −I and g_{ϵ}^{−1} −I ∈ N_{1} so that under the decomposition W = P W ⊕ Q_{ϵ}W ⊕ P_{ϵ}W,

g_{ϵ}D̄_{ϵ}^{2}g_{ϵ}^{−1} ≡ | \begin{matrix}(P \frac{1}{2}ωP)^{2} & 0 & 0 \\ 0 & T_{ϵ}^{2} & 0 \\ 0 & 0 & (P_{ϵ}DP_{ϵ})^{2} \end{matrix} | mod | \begin{matrix}N_{3} & 0 & 0 \\ 0 & N_{2} & 0 \\ 0 & 0 & 0 \end{matrix} | ,

and

g_{ϵ}D̄_{ϵ}g_{ϵ}^{−1} ≡ | \begin{matrix}P \frac{1}{2}ωP & 0 & 0 \\ 0 & T_{ϵ} & 0 \\ 0 & 0 & P_{ϵ}DP_{ϵ} \end{matrix} | mod | \begin{matrix}N_{2} & N_{2} & 0 \\ N_{2} & N_{2} & 0 \\ 0 & 0 & 0 \end{matrix} | .

Proof: Denote by G_{ϵ} the Green’s operator of −(Q_{ϵ} \frac{1}{2}V Q_{ϵ})^{2}, i.e. G_{ϵ} is zero on P W ⊕ P_{ϵ}W and is the inverse of −(Q_{ϵ} \frac{1}{2}V Q_{ϵ})^{2} on Q_{ϵ}W . It is a bounded operator with norm ||G_{ϵ}|| ≤ ϵ^{−1}.

The proof proceeds by induction on i and we need only prove the analogous result for D̂_{ϵ} restricted to P W ⊕ Q_{ϵ}W. For i = 1, we set g_{1} = I and we have

D̂_{ϵ}^{2} = | \begin{matrix}P (\frac{1}{2}ω + \frac{1}{2} ĉ)P & P (\frac{1}{2}ω + \frac{1}{2} ĉ)Q_{ϵ} \\ Q_{ϵ}(\frac{1}{2}ω + \frac{1}{2} ĉ)P & T_{ϵ} \end{matrix} |^{2} = | \begin{matrix}X_{1} & Y_{1} \\ Z_{1} & T_{1} \end{matrix} | ∈ | \begin{matrix}N_{2} & N_{1} \\ N_{1} & M_{0} \end{matrix} |

where T_{1} ≡ T_{ϵ}^{2} mod N_{2}. Note that X_{1} − Y_{1}G_{ϵ}Z_{1} ≡ (P \frac{1}{2}ωP)^{2} mod N_{3}.

Now we inductively define g_{i} ∈ M with g_{i} − I ∈ N_{1} and

g_{i}D̂_{ϵ}^{2}g_{i}^{−1} = | \begin{matrix}X_{i} & Y_{i} \\ Z_{i} & T_{i} \end{matrix} | ∈ | \begin{matrix}N_{2} & N_{i} \\ N_{i} & M_{0} \end{matrix} |

with T_{i} ≡ T_{ϵ}^{2} mod N_{2}. Note that this last condition implies

| \begin{matrix}0 & 0 \\ 0 & 1 − G_{ϵ}T_{i} \end{matrix} | and | \begin{matrix}0 & 0 \\ 0 & 1 − T_{i}G_{ϵ} \end{matrix} | ∈ M_{1}. (2.13)

Set

h_{i} = | \begin{matrix}1 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 1 \end{matrix} | , and | \begin{matrix}X_{i+1} & Y_{i+1} \\ Z_{i+1} & T_{i+1} \end{matrix} | = h_{i} | \begin{matrix}X_{i} & Y_{i} \\ Z_{i} & T_{i} \end{matrix} | h_{i}^{−1}.

Since | \begin{matrix}0 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 0 \end{matrix} | is a nilpotent operator, h_{i}^{−1} = ∑_{k=0}^{∞}(−1)^{k} | \begin{matrix}0 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 0 \end{matrix} |^{k} is a finite

sum, and

h_{i}^{−1} ≡ | \begin{matrix}1 & Y_{i}G_{ϵ} \\ −G_{ϵ}Z_{i} & 1 \end{matrix} | mod N_{2i}.

Thus

| \begin{matrix}X_{i+1} & Y_{i+1} \\ Z_{i+1} & T_{i+1} \end{matrix} | ≡ | \begin{matrix}1 & −Y_{i}G_{ϵ} \\ G_{ϵ}Z_{i} & 1 \end{matrix} | | \begin{matrix}X_{i} & Y_{i} \\ Z_{i} & T_{i} \end{matrix} | | \begin{matrix}1 & Y_{i}G_{ϵ} \\ −G_{ϵ}Z_{i} & 1 \end{matrix} | mod N_{2i}.

Using Equation 2.13, a direct computation, see [BGV92] p. 274, shows that Y_{i+1} and Z_{i+1} ∈ N_{i+1}, and T_{i+1} ≡ T_{ϵ}^{2} mod N_{2}. Setting g_{i+1} = h_{i}g_{i}, we may continue the induction to obtain g_{ϵ} = g_{n+1} ⊕ 1, as N_{n+1} = 0. As in [BGV92], we have that X_{n+1} ≡ X_{1} − Y_{1}G_{ϵ}Z_{1} mod N_{3}, so g_{ϵ}D̄_{ϵ}^{2}g_{ϵ}^{−1} has the required form.

To prove the formula given for D̄_{ϵ}, note that

g_{2} = | \begin{matrix}1 & −P \frac{1}{2}ωQ_{ϵ} \frac{1}{2}V Q_{ϵ}G_{ϵ} \\ G_{ϵ}Q_{ϵ} \frac{1}{2}V Q_{ϵ} \frac{1}{2}ωP & 1 \end{matrix} | mod N_{2}

and

g_{2}^{−1} = | \begin{matrix}1 & P \frac{1}{2}ωQ_{ϵ} \frac{1}{2}V Q_{ϵ}G_{ϵ} \\ −G_{ϵ}Q_{ϵ} \frac{1}{2}V Q_{ϵ} \frac{1}{2}ωP & 1 \end{matrix} | mod N_{2}.

A direct computation then shows that

g_{2}D̂_{ϵ}g_{2}^{−1} = | \begin{matrix}P \frac{1}{2}ωP & 0 \\ 0 & T_{ϵ} \end{matrix} | mod | \begin{matrix}N_{2} & N_{2} \\ N_{2} & N_{2} \end{matrix} |

For i > 2, g_{i} ≡ I mod N_{2}, so we are done, except for showing that g_{ϵ} and g_{ϵ}^{−1} are bounded. To see this, note that the uniformly bounded geometry of the leaves implies that all of the operators used in the construction of g_{ϵ} and g_{ϵ}^{−1}, which are elements of M, are differential operators whose coefficients are uniformly bounded. Thus they yield a bounded element of N when composed with a bounded element of N. This fact, combined with the fact that G_{ϵ} is bounded, implies that g_{ϵ} and g_{ϵ}^{−1} are bounded (by bounds which depend on ϵ). □

An important fact to be gleaned from a careful analysis of the construction of g_{ϵ} is the following.

The number of G_{ϵ} s appearing in any monomial ocurring in g_{ϵ} is less than or equal to the number of ω s and ĉs ocurring in the same monomial.

(2.14)

To see this, use the fact that (Q_{ϵ} \frac{1}{2}V Q_{ϵ})^{2}G_{ϵ} = (Q_{ϵ} \frac{1}{2}V Q_{ϵ})G_{ϵ}(Q_{ϵ} \frac{1}{2}V Q_{ϵ}) = −Q_{ϵ} to reduce the number of G_{ϵ} s. Also, note that the only terms in g_{ϵ}D̄_{ϵ}g_{ϵ}^{−1} and g_{ϵ}D̄_{ϵ}^{2}g_{ϵ}^{−1} which are not bounded elements of N are T_{ϵ}, and T_{ϵ}^{2}, respectively.

Now for all t > 0, let ψ_{t} be the map of M which multiplies a section of ∧^{k}T^{∗}M ⊗End(W) by t^{−k/2}. Then

D_{t} = \sqrt{t}ψ_{t}Dψ_{t}^{−1}, D̄_{ϵ,t} = \sqrt{t}ψ_{t}D̄_{ϵ}ψ_{t}^{−1}, D̂_{ϵ,t} = \sqrt{t}ψ_{t}D̂_{ϵ}ψ_{t}^{−1} and T_{ϵ,t} = \sqrt{t}ψ_{t}T_{ϵ}ψ_{t}^{−1}.

We now couple the variables ϵ and t. (If P_{ϵ} = 0 for some ϵ > 0, this step is not necessary). The β of Condition 2.7 satisfies β > 3q. Choose a real number a ∈ (6, 2β/q) and set

ϵ = t^{− \frac{1}{a}} .

Then

∥G_{ϵ}∥_{s,s} ≤ t^{\frac{1}{a}} for all s. (2.15)

Lemma 2.16 Any element A ∈ N_{k} which is in g_{ϵ} − I, g_{ϵ}^{−1} − I or is modded out in Proposition 2.12, satisfies, for all r and s,

∥ψ_{t}Aψ_{t}^{−1}∥_{r,s} is at least O(t^{− \frac{k}{2} + \frac{k}{a}}) as t → ∞.

Proof: All such A are sums of words in the leafwise operators P, Q_{ϵ}, G_{ϵ}, V, ω and ĉ. Each word contains at least one P. Any word containing no G_{ϵ} s is at worst a differential operator of finite order, say s_{0}, with uniformly bounded coefficients. As such, its s, s − s_{0} Sobolev norm is finite and independent of t. Now take any word W in A and assume that it is in N_{ℓ}, i.e. that it is purely of base degree ℓ ≥ k. Then ψ_{t}W ψ_{t}^{−1} = t^{− \frac{ℓ}{2}} W. Suppose for definiteness that

W = W_{1}G_{ϵ}W_{2}P W_{3}G_{ϵ}W_{4}

where the W_{i} are words which contain no G_{ϵ} s. Then at worst, W_{i} has finite order s_{i}, so that ∥W_{i}∥_{s,s−s_{i}} is finite for all s. Then we have

∥W ∥_{r,s} ≤ t^{− \frac{ℓ}{2}} ∥W_{1}∥_{k_{1},s}∥G_{ϵ}∥_{k_{1},k_{1}}∥W_{2}∥_{k_{2},k_{1}}∥P ∥_{k_{3},k_{2}}∥W_{3}∥_{k_{4},k_{3}}∥G_{ϵ}∥_{k_{4},k_{4}}∥W_{4}∥_{r,k_{4}}

where k_{1} = s + s_{1}, k_{2} = s + s_{1} + s_{2}, k_{3} = r − (s_{3} + s_{4}) and k_{4} = r − s_{4}. Now ∥P ∥_{k_{3},k_{2}} ≤ 1 for any k_{3}, k_{2} and ∥G_{ϵ}∥_{s,s} ≤ t^{\frac{1}{a}} for all s, so

∥W ∥_{r,s} ≤ t^{− \frac{ℓ}{2} + \frac{2}{a}} ∥W_{1}∥_{k_{1},s}∥W_{2}∥_{k_{2},k_{1}}∥W_{3}∥_{k_{4},k_{3}}∥W_{4}∥_{r,k_{4}} = Ct^{− \frac{ℓ}{2} + \frac{2}{a}} ,

where C is independent of t. It follows from Remark 2.14 that an arbitrary such W can contain at most ℓ G_{ϵ} s, so a trivial extension of the above argument gives the Lemma. □

Note that any A as in Lemma 2.16 is nilpotent since it is at least in N_{2}. Since a > 6, if A ∈ N_{3}, then ∥ψ_{t}Aψ_{t}^{−1}∥_{r,s} is at least O(t^{−(1+γ)}) for some γ > 0. Thus we have

D̄_{ϵ,t} = \sqrt{t}ψ_{t}D̄_{ϵ}ψ_{t}^{−1} = ψ_{t}g_{ϵ}^{−1}ψ_{t}^{−1} | \begin{matrix}P \frac{1}{2}ωP + O(t^{− \frac{1}{2} + \frac{2}{a}}) & O(t^{− \frac{1}{2} + \frac{2}{a}}) & 0 \\ O(t^{− \frac{1}{2} + \frac{2}{a}}) & T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}) & 0 \\ 0 & 0 & P_{ϵ}D_{t}P_{ϵ} \end{matrix} | ψ_{t}g_{ϵ}ψ_{t}^{−1}

and

D̄_{ϵ,t}^{2} = tψ_{t}D̄_{ϵ}^{2}ψ_{t}^{−1} =

ψ_{t}g_{ϵ}^{−1}ψ_{t}^{−1} | \begin{matrix}(P \frac{1}{2}ωP)^{2} + O(tt^{−(1+γ)}) & 0 & 0 \\ 0 & T_{ϵ,t}^{2} + O(tt^{−1+ \frac{2}{a}}) & 0 \\ 0 & 0 & (P_{ϵ}D_{t}P_{ϵ})^{2} \end{matrix} | ψ_{t}g_{ϵ}ψ_{t}^{−1}.

Proposition 2.17 We may assume that ψ_{t}g_{ϵ}ψ_{t}^{−1} = I = ψ_{t}g_{ϵ}^{−1}ψ_{t}^{−1}.

Proof: To see this, recall that ψ_{t}g_{ϵ}ψ_{t}^{−1} = ψ_{t}g_{n+1}ψ_{t}^{−1} ⊕ 1 and

ψ_{t}g_{n+1}ψ_{t}^{−1} = I + | \begin{matrix}O(t^{− \frac{1}{2} + \frac{1}{a}}) & O(t^{− \frac{1}{2} + \frac{1}{a}}) \\ O(t^{− \frac{1}{2} + \frac{1}{a}}) & O(t^{− \frac{1}{2} + \frac{1}{a}}) \end{matrix} |

and similarly for ψ_{t}g_{n+1}^{−1}ψ_{t}^{−1}. Let A and B be the upper left hand 2 by 2 submatrices of the 3 by 3 matrices for D̄_{ϵ,t} and D̄_{ϵ,t}^{2}, respectively, given above. Then D̂_{ϵ,t} = ψ_{t}g_{n+1}^{−1}ψ_{t}^{−1}Aψ_{t}g_{n+1}ψ_{t}^{−1} and D̂_{ϵ,t}^{2} = ψ_{t}g_{n+1}^{−1}ψ_{t}^{−1}Bψ_{t}g_{n+1}ψ_{t}^{−1}. Let η be the grading operator used in defining the supertrace and note that ∥η∥_{s,s} = 1 for all s. Then we have

Tr_{s}(f(D̂_{ϵ,t})) = Tr(ηψ_{t}g_{n+1}^{−1}ψ_{t}^{−1}Ae^{B}ψ_{t}g_{n+1}ψ_{t}^{−1})
= Tr(η(I + O(t^{− \frac{1}{2} + \frac{1}{a}}))Ae^{B}(I + O(t^{− \frac{1}{2} + \frac{1}{a}})))
= Tr(ηAe^{B}) + Tr(ηO(t^{− \frac{1}{2} + \frac{1}{a}})Ae^{B}) + Tr(ηAe^{B}O(t^{− \frac{1}{2} + \frac{1}{a}})) +
Tr(ηO(t^{− \frac{1}{2} + \frac{1}{a}})Ae^{B}O(t^{− \frac{1}{2} + \frac{1}{a}}))
= Tr(ηAe^{B}) + O(t^{− \frac{1}{2} + \frac{1}{a}}),

since

Lemma 2.18 ∥Ae^{B}∥_{s,s} is bounded for all s.

Proof:

Ae^{B} = | \begin{matrix}(P \frac{1}{2}ωP + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{(P \frac{1}{2} ωP)^{2}+O(t^{−γ})} & O(t^{− \frac{1}{2} + \frac{2}{a}})e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})} \\ O(t^{− \frac{1}{2} + \frac{2}{a}})e^{(P \frac{1}{2} ωP)^{2}+O(t^{−γ})} & (T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})} \end{matrix} | .

The proof of Proposition 2.19 below implies that the elements in the right column have bounded s, s norm for all s. To deal with the other terms, note that P, O(t^{− \frac{1}{2} + \frac{2}{a}}) and O(t^{−γ}) are bounded smoothing operators and ω is an order zero differential operator with uniformly bounded coefficients, so is a bounded operator also. In addition, (P \frac{1}{2}ωP)^{2}, O(t^{−γ}) and their sum are nilpotent, so this term is a finite sum of the form

(P \frac{1}{2}ωP + O(t^{− \frac{1}{2} + \frac{2}{a}})) ∑(P \frac{1}{2}ωP)^{2j_{1}}O(t^{−γ})(P \frac{1}{2}ωP)^{2j_{2}}O(t^{−γ}) ⋯ O(t^{−γ})(P \frac{1}{2}ωP)^{2j_{k}},

where all the operators are bounded elements of N . This completes the proof of the Lemma and Proposition 2.17. □

Note the abuse of notation here. We are really working on subspaces of W, and so should use P (P \frac{1}{2}ωP + O(t^{− \frac{1}{2} + \frac{2}{a}}))P e^{(P \frac{1}{2} ωP)^{2}+O(t^{−γ})}P in place of (P \frac{1}{2}ωP + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{(P \frac{1}{2} ωP)^{2}+O(t^{−γ})}, and similarly for the other terms. This sort of abuse of notation will occur again later.

The off diagonal terms in D̄_{ϵ,t} play no role in computing the supertrace, so we may replace them by 0. Doing that, we have

Tr_{s}(f(D̄_{ϵ,t})) = Tr_{s}((P \frac{1}{2}ωP + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{(P \frac{1}{2} ωP)^{2}+O(t^{−γ})}) +

Tr_{s}((T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})}) + Tr_{s}(f(P_{ϵ}D_{t}P_{ϵ})).

Using the technique above, we get

Tr_{s}((P \frac{1}{2}ωP + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{(P \frac{1}{2} ωP)^{2}+O(t^{−γ})}) = Tr_{s}(f(P \frac{1}{2}ωP)) + O(t^{−γ}).

Thus

Tr_{s}(f(D̄_{ϵ,t})) = Tr_{s}(f(P \frac{1}{2}ωP)) + O(t^{−γ}) +

Tr_{s}((T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})}) + Tr_{s}(f(P_{ϵ}D_{t}P_{ϵ})).

Proposition 2.19 For any γ > 0, as t → ∞,

Tr_{s}((T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})}) = O(t^{−γ}).

Proof: Write

(T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})} = (Q_{ϵ} \frac{\sqrt{t}}{2}V Q_{ϵ} + A)e^{Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}+C}

where A and C are elements of M_{1} of the form Q_{ϵ}KQ_{ϵ} with K ∈ M_{1}. Note that we are again abusing notation and should be considering

Tr_{s}(Q_{ϵ}(Q_{ϵ} \frac{\sqrt{t}}{2}V Q_{ϵ} + A)e^{Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}+C}Q_{ϵ}).

The Volterra series for the exponential of a perturbed operator, [BGV92] p. 78, gives

(Q_{ϵ} \frac{\sqrt{t}}{2}V Q_{ϵ} + A)e^{Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}+C} =

∑_{k=0}^{∞} ∫_{Δ_{k}}(Q_{ϵ} \frac{\sqrt{t}}{2}V Q_{ϵ} + A)e^{x_{1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}Ce^{x_{2}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}C … Ce^{x_{k+1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}dx_{k} … dx_{1},

where Δ_{k} is the standard k simplex Δ_{k} = {(x_{1}, …, x_{k+1}) | x_{i} ≥ 0, ∑ x_{i} = 1}. As the operator C ∈ M_{1} the sum is finite, in particular k ≤ n = dim M. (Below, we shall improve this to k ≤ q). Fix k. At each point in Δ_{k} there is at least one x_{i} ≥ 1/(k + 1). We may assume that i = k + 1, since the general case is handled in the same way. We may also assume that all x_{i} > 0.

Write the integrand as

(Q_{ϵ} \frac{\sqrt{t}}{2}V Q_{ϵ} + A)e^{x_{1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}t^{− \frac{1}{2}} Ce^{x_{2}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}t^{− \frac{1}{2}} C … t^{− \frac{1}{2}} Ct^{\frac{k}{2}} e^{x_{k+1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}.

The Spectral Mapping Theorem implies that ∥Q_{ϵ} \frac{\sqrt{t}}{2}V Q_{ϵ}e^{x_{1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}∥_{s,s} ≤ 1/\sqrt{2x_{1}e} and that ∥e^{x_{i}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}∥_{s,s} ≤ 1 for all s. The operators A, and t^{− \frac{1}{2}} C also have bounded s, s norm for all s. To see this last, note that, although they are ostensibly order one differential operators, a direct computation shows that neither V ω + ωV nor V ĉ + ĉV involves differentiation, so both are order zero differential operators with uniformly bounded coefficients, for say t ≥ 1.

Recalling our abuse of notation, write t^{\frac{k}{2}} e^{x_{k+1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}Q_{ϵ} as

t^{\frac{k}{2}} Q_{ϵ}e^{x_{k+1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}} = t^{−m+ \frac{m}{a} + \frac{k}{2a}} (t^{− \frac{1}{a}} G_{ϵ})^{m+ \frac{k}{2}} Q_{ϵ}t^{m+ \frac{k}{2}} V^{2m+k}Q_{ϵ}e^{x_{k+1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}},

where m is a positive integer as large as we please. Now ∥(t^{− \frac{1}{a}} G_{ϵ})^{m+ \frac{k}{2}} ∥_{s,s} ≤ 1 for all s, where Gϵ^{\frac{1}{2}} is the Green’s operator for Q_{ϵ} \frac{i}{2}V Q_{ϵ}. The Spectral Mapping Theorem gives that

∥Q_{ϵ}t^{m+ \frac{k}{2}} V^{2m+k}Q_{ϵ}e^{x_{k+1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}∥_{−s,s} ≤ max_{z≥0} (1 + z^{2})^{s}(tz^{2})^{m+ \frac{k}{2}} e^{−x_{k+1} \frac{t}{4} z^{2}}.

For any r ≥ 0 and t ≥ 1

max_{z≥0} z^{2r}(tz^{2})^{m+ \frac{k}{2}} e^{−x_{k+1} \frac{t}{4} z^{2}} = t^{−r}[\frac{4(r + m + \frac{k}{2})}{x_{k+1}e}]^{r+m+ \frac{k}{2}} ≤

[\frac{4(r + m + \frac{k}{2})(k + 1)}{e}]^{r+m+ \frac{k}{2}} ≤ [\frac{4(r + m + \frac{n}{2})(n + 1)}{e}]^{r+m+ \frac{n}{2}} ,

since x_{k+1} ≥ 1/(k + 1) and k ≤ n. Thus for t ≥ 1, ∥Q_{ϵ}t^{m+ \frac{k}{2}} V^{2m+k}Q_{ϵ}e^{x_{k+1}Q_{ϵ} \frac{t}{4} V^{2}Q_{ϵ}}∥_{−s,s} is bounded independently of t and x_{k+1}. As ∫_{Δ_{k}} x_{1}^{− \frac{1}{2}} dx_{k} … dx_{1} is finite and k ≤ n, at worst

(T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})} is t^{−m+ \frac{m}{a} + \frac{n}{2a}} times a bounded smoothing operator whose −s, s norm is bounded independently of t and so at worst

Tr_{s}((T_{ϵ,t} + O(t^{− \frac{1}{2} + \frac{2}{a}}))e^{T_{ϵ,t}^{2}+O(t^{\frac{2}{a}})}) = O(t^{−m+ \frac{m}{a} + \frac{n}{2a}}).

Since n is fixed, a ≥ 6 and m is arbitrarily large, we are done. □

This completes the proof of Theorem 2.8 in the case that P_{ϵ} = 0 for some positive ϵ since D̄_{ϵ,t} = D_{t} in that case and we have shown that there is γ > 0 so that as t → ∞,

Tr_{s}(f(D̄_{ϵ,t})) = Tr_{s}(f(P \frac{1}{2}ωP)) + Tr_{s}(f(P_{ϵ}D_{t}P_{ϵ})) + O(t^{−γ}).

Note that we have not used Condition 2.7 up to this point.

2.6 Proof of Equation 2.10 when P_{ϵ} ≠ 0 for all ϵ > 0

We now deal with the space P_{ϵ}W and show that it contributes nothing as t → ∞. This is where we use Condition 2.7 to obtain estimates on the rate of decay as t → ∞ of the terms associated to this space. Set

γ = \frac{β}{a} − \frac{q}{2}

and recall that 6 < a < 2β/q, so γ > 0.

Proposition 2.20 As t → ∞, Tr_{s}(f(P_{ϵ}D_{t}P_{ϵ})) = O(t^{−γ}).

Proof: Recalling our abuse of notation, we have Tr_{s}(f(P_{ϵ}D_{t}P_{ϵ})) = Tr(AP_{ϵ}), where A = ηP_{ϵ}D_{t}P_{ϵ}e^{(P_{ϵ}D_{t}P_{ϵ})^{2}}.

Lemma 2.21 t^{− \frac{q}{2}} ∥A∥ is bounded independently of t for t large.

Proof: As η is bounded, we may ignore it. Note that

P_{ϵ}D_{t}P_{ϵ} = P_{ϵ} \frac{\sqrt{t}}{2}V P_{ϵ} + C_{1} and (P_{ϵ}D_{t}P_{ϵ})^{2} = P_{ϵ} \frac{t}{4}V^{2}P_{ϵ} + C_{2}

where C_{1} and t^{− \frac{1}{2}} C_{2} are nilpotent and bounded independently of t. Proceeding as in the proof of Proposition 2.19 will give the result provided we show that at most q C_{2} s occur in any element of the Volterra series for A. But the proof of the Lemma 2.1 implies that C_{2} is in fact a section of ∧^{∗}ν^{∗} ⊗ End(W). As ∧^{q+1}ν^{∗} = 0, we have the lemma. □

Now

Tr_{s}(f(P_{ϵ}D_{t}P_{ϵ})) = Tr(AP_{ϵ}) = Tr(P_{ϵ}AP_{ϵ}),

as P_{ϵ} = P_{ϵ}^{2} and P_{ϵ} commutes with η. Denote by A_{x} the action of A on W_{x} and let ω_{1}, ω_{2}, … be a basis of ∧^{∗}ν_{x}^{∗}. Recalling the comment after the proof of Lemma 2.1, we may write A_{x} = ∑_{j} ω_{j} ⊗ A_{j} where A_{j} ∈ End(W_{x}). Let v_{1}, …, v_{k} be an orthonormal basis of (∧^{∗}T^{∗}F_{s} ⊗ Ẽ)_{x}, where the subscript x ∈ L̃_{x}. Denoting the Schwartz kernel of P_{ϵ}AP_{ϵ} by H_{ϵ}, we have

Tr(H_{ϵ}^{x}(x, x)) = ∑_{i,j} ω_{j}⊗ < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > .

Now

| < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > | = | < A_{j}P_{ϵ}(δ_{v_{i}}), P_{ϵ}(δ_{v_{i}}) > | ≤
∥A_{j}P_{ϵ}(δ_{v_{i}})∥ ∥P_{ϵ}(δ_{v_{i}})∥ ≤ ∥A_{j}∥ ∥P_{ϵ}(δ_{v_{i}})∥^{2} ≤ ∥A∥ ∥P_{ϵ}(δ_{v_{i}})∥^{2}.

Recalling that K_{ϵ} is the Schwartz kernel of P_{ϵ}, we have

∑_{i} ∥P_{ϵ}(δ_{v_{i}})∥^{2} = ∑_{i} < P_{ϵ}(δ_{v_{i}}), P_{ϵ}(δ_{v_{i}}) > = ∑_{i} < P_{ϵ}^{2}(δ_{v_{i}}), δ_{v_{i}} > =

∑_{i} < P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > = ∑_{i} K_{ϵ}^{x}(v_{i}, v_{i}) = tr(K_{ϵ}^{x}(x, x)).

Let U and {ϕ_{U} | U ∈ U} be a good foliation cover and partition of unity subordinate to it. Consider a chart U ∈ U with transversal T_{U}. Let ω_{1}^{T} , ω_{2}^{T} , … be an orthonormal basis of ∧^{∗}T^{∗}T. We may assume that ω_{j} = π^{∗}(ω_{j}^{T}) for each x ∈ U, where π : U → T_{U} is the projection. On T_{U}, the Haefliger form Tr(P_{ϵ}AP_{ϵ}) is given by

∫_{U} ϕ_{U} tr(H_{ϵ}^{x}(x, x))dx = ∫_{U} ϕ_{U} ∑_{i,j} ω_{j}⊗ < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > dx =

∑_{j} ω_{j}^{T} ∫_{U} ϕ_{U} ∑_{i} < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > dx.

Now

| ∫_{U} ϕ_{U} ∑_{i} < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > dx| ≤ ∫_{U} ϕ_{U} ∑_{i} | < P_{ϵ}A_{j}P_{ϵ}(δ_{v_{i}}), δ_{v_{i}} > |dx ≤

∫_{U} ϕ_{U} ∑_{i} ∥A∥ ∥P_{ϵ}(δ_{v_{i}})∥^{2} = ∥A∥ ∫_{U} ϕ_{U} tr(K_{ϵ}^{x}(x, x))dx ≤ C∥A∥ϵ^{β}

for some constant C since ∫_{U} ϕ_{U} tr(K_{ϵ}^{x}(x, x))dx is the form Tr(P_{ϵ}) on T_{U}. Now ϵ = t^{− \frac{1}{a}}, so

∥A∥ϵ^{β} = t^{− \frac{q}{2}} ∥A∥t^{\frac{q}{2} − \frac{β}{a}} = t^{− \frac{q}{2}} ∥A∥t^{−γ}. As t^{− \frac{q}{2}} ∥A∥ is bounded the result follows. □

We have thus proven Equation 2.10 for D̄_{ϵ,t}, namely we have shown that as t → ∞

Tr_{s}(f(D̄_{ϵ,t})) = Tr_{s}(f(P \frac{1}{2}ωP)) + O(t^{−γ}).

To finish the proof of Theorem 2.8 in all generality, we have

Proposition 2.22 As t → ∞, Tr_{s}(D_{t}e^{D_{t}^{2}}) = Tr_{s}(D̄_{ϵ,t}eD̄_{ϵ,t}^{2}) + O(t^{−γ}).

Proof: Now

(2.23) Tr_{s}(D̄_{ϵ,t}eD̄_{ϵ,t}^{2}) − Tr_{s}(D_{t}e^{D_{t}^{2}}) = Tr_{s}(D_{t}(eD̄_{ϵ,t}^{2} − e^{D_{t}^{2}})) − Tr_{s}(A_{ϵ,t}eD̄_{ϵ,t}^{2}).

Note that

eD̄_{ϵ,t}^{2} = (P + Q_{ϵ})e^{((P +Q_{ϵ})D_{t}(P +Q_{ϵ}))^{2}}(P + Q_{ϵ}) + P_{ϵ}e^{(P_{ϵ}D_{t}P_{ϵ})^{2}}P_{ϵ}

and that

A_{ϵ,t} = (P + Q_{ϵ})(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ)P_{ϵ} + P_{ϵ}(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ)(P + Q_{ϵ}).

The proof of Lemma 2.21 shows that for any spectral projection S of V^{2}, the operator Se^{(SD_{t}S)^{2}}S is bounded (the bound may depend on t). P_{ϵ} = P_{ϵ}^{2} is a bounded smoothing operator. As the spectral projections of V^{2} commute with the operator η and (P + Q_{ϵ})P_{ϵ} = P_{ϵ}(P + Q_{ϵ}) = 0 it follows easily from the trace property that Tr_{s}(A_{ϵ,t}eD̄_{ϵ,t}^{2}) = 0.

For 0 ≤ z ≤ 1, set D_{t}(z) = zD_{t} + (1 − z)D̄_{ϵ,t} = D̄_{ϵ,t} + zA_{ϵ,t}. By Equation 3.10 of [Hei95], we have

\frac{d}{dz} e^{D_{t}^{2}(z)} = − ∫_{0}^{1}[e^{(1−s)D_{t}^{2}(z)}]\frac{d(D_{t}^{2}(z))}{dz}e^{sD_{t}^{2}(z)}ds.

Thus the first term on the right hand side of Equation 2.23 is given by

∫_{0}^{1} ∫_{0}^{1} Tr(ηD_{t}e^{(1−s)D_{t}^{2}(z)}(D_{t}(z)A_{ϵ,t} + A_{ϵ,t}D_{t}(z))e^{sD_{t}^{2}(z)})dsdz.

As ω commutes and V and ĉ anti-commute with η, ηD_{t} = −D_{t}η + ηω. Recalling that D_{t} = D_{t}(z) + (1 − z)A_{ϵ,t}, the above equals

∫_{0}^{1} ∫_{0}^{1} Tr(ηD_{t}(z)e^{(1−s)D_{t}^{2}(z)}A_{ϵ,t}D_{t}(z)e^{sD_{t}^{2}(z)})dsdz
+ ∫_{0}^{1} ∫_{0}^{1} Tr(η(1 − z)A_{ϵ,t}e^{(1−s)D_{t}^{2}(z)}A_{ϵ,t}D_{t}(z)e^{sD_{t}^{2}(z)})dsdz
+ ∫_{0}^{1} ∫_{0}^{1} Tr(−D_{t}(z)ηe^{(1−s)D_{t}^{2}(z)}D_{t}(z)A_{ϵ,t}e^{sD_{t}^{2}(z)})dsdz
+ ∫_{0}^{1} ∫_{0}^{1} Tr(−(1 − z)A_{ϵ,t}ηe^{(1−s)D_{t}^{2}(z)}D_{t}(z)A_{ϵ,t}e^{sD_{t}^{2}(z)})dsdz
+ ∫_{0}^{1} ∫_{0}^{1} Tr(ηωe^{(1−s)D_{t}^{2}(z)}D_{t}(z)A_{ϵ,t}e^{sD_{t}^{2}(z)})dsdz

= ∫_{0}^{1} ∫_{0}^{1} Tr(A_{ϵ,t}D_{t}(z)e^{sD_{t}^{2}(z)}ηD_{t}(z)e^{(1−s)D_{t}^{2}(z)})dsdz
+ ∫_{0}^{1} ∫_{0}^{1} Tr(A_{ϵ,t}e^{(1−s)D_{t}^{2}(z)}A_{ϵ,t}D_{t}(z)e^{sD_{t}^{2}(z)}η(1 − z))dsdz
− ∫_{0}^{1} ∫_{0}^{1} Tr(A_{ϵ,t}e^{sD_{t}^{2}(z)}D_{t}(z)ηe^{(1−s)D_{t}^{2}(z)}D_{t}(z))dsdz
− ∫_{0}^{1} ∫_{0}^{1} Tr(A_{ϵ,t}ηe^{(1−s)D_{t}^{2}(z)}D_{t}(z)A_{ϵ,t}e^{sD_{t}^{2}(z)}(1 − z))dsdz
+ ∫_{0}^{1} ∫_{0}^{1} Tr(A_{ϵ,t}e^{sD_{t}^{2}(z)}ηωe^{(1−s)D_{t}^{2}(z)}D_{t}(z))dsdz

= Tr(A_{ϵ,t} ∫_{0}^{1} ∫_{0}^{1} f(s, z)dsdz) ≡ Tr(A_{ϵ,t}F (s, z)).

As in Lemma 2.21, we may write

D_{t}(z) = \frac{\sqrt{t}}{2}V + C_{1} and D_{t}^{2}(z) = \frac{t}{4}V^{2} + C_{2}

where C_{1} and t^{− \frac{1}{2}} C_{2} are nilpotent and bounded independently of t. The proofs of Proposition 2.19 and Lemma 2.21 extend to show that t^{− \frac{q}{2}} ∥F (s, z)∥ is bounded independently of t. Now

Tr(A_{ϵ,t}F (s, z)) =

Tr([(P + Q_{ϵ})(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ)P_{ϵ} + P_{ϵ}(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ)(P + Q_{ϵ})]F (s, z)) =

Tr(P_{ϵ}F (s, z)(P + Q_{ϵ})(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ) + P_{ϵ}(\frac{1}{2}ω + \frac{1}{2\sqrt{t}}ĉ)(P + Q_{ϵ})F (s, z)).

As t^{− \frac{q}{2}} [F (s, z)(P + Q_{ϵ})(\frac{1}{2}ω + \frac{1}{2\sqrt{t}} ĉ) + (\frac{1}{2}ω + \frac{1}{2\sqrt{t}} ĉ)(P + Q_{ϵ})F (s, z)] is bounded independently of t for t large, the method of proof of Proposition 2.20 gives the result and completes the proofs of the Proposition and Theorem 2.8. □

2.7 The Riemann-Roch-Grothendieck Theorem

Recall the definition of the Chern type characteristic classes of the flat vector bundle E over M with metric h^{E}, see [KT74] and [BL95]. For k a positive odd integer, consider the differential form on M given by

c_{k}(E, h^{E}) = (2πi)^{− \frac{k−1}{2}} 2^{−2k}tr_{s}([(∇^{E})^{∗} − ∇^{E}]^{k}).

This k form is closed and its cohomology class, denoted c_{k}(E) is independent of h^{E}. If E admits a covariantly constant Hermitian metric, then c_{k}(E) = 0. We define the Chern type characteristic classes c_{k}(H^{∗}(F; E)) of the ‘flat bundle’ H^{∗}(F; E) to be the Haefliger cohomology classes of the closed Haefliger forms (2πi)^{− \frac{k−1}{2}} 2^{−2k}Tr_{s}((P ωP)^{k}), i.e.

c_{k}(H^{∗}(F; E)) = [(2πi)^{− \frac{k−1}{2}} 2^{−2k}Tr_{s}((P ωP)^{k})].

We have the following corollary of Theorem 2.8 which is the promised Riemann-Roch-Grothendieck type theorem for flat bundles over a foliated manifold.

Theorem 2.24 (Riemann-Roch-Grothendieck) Let E be a flat Hermitian vector bundle over a foliation F of a compact manifold M. Assume that Condition 2.7 holds. Then for any positive odd integer k

c_{k}(H^{∗}(F; E)) = ∫_{F} e(F)c_{k}(E) in H_{c}^{k}(F).

If the dimension of F is odd or if E admits a Hermitian metric which is equivariant with respect to ∇^{E}, then c_{k}(H^{∗}(F; E)) = 0 for all k. If c_{k}(E) = 0 for a given k, then c_{k}(H^{∗}(F; E)) = 0 also.

Proof: The form f(C′_{t}, h^{W}) is closed and its cohomology class is independent of t, so the forms f(∇^{H^{∗}(F;E)}, h^{H^{∗}(F;E)}) and ∫_{F} e(T F)f(∇^{E}, h^{E}) are closed and ∫_{F} e(T F)f(∇^{E}, h^{E}) =

f(∇^{H^{∗}(F;E)}, h^{H^{∗}(F;E)}) in cohomology. Equating terms of the same degree gives the result. If the dimension of F is odd, then e(F) = 0 and if E admits an equivariant Hermitian metric, c_{k}(E) = 0 for all k. □

3 Torsion for foliations

We begin this section with a transgression formula for the forms f(C′_{t}, h^{W}). Recall that N is the number operator on W. With the notation as in Section 2, for t > 0 set

f^{∧}(C′_{t}, h^{W}) = φTr_{s}(\frac{N}{2} f′(D_{t})) = φTr_{s}(\frac{N}{2} (1 + 2D_{t}^{2})e^{D_{t}^{2}}).

Theorem 3.1 [Theorem 3.20 of [BL95]] The form f^{∧}(C′_{t}, h^{W}) ∈ Ω_{c}^{∗}(F) is real, even and satisfies

\frac{∂}{∂t}f(C′_{t}, h^{W}) = \frac{1}{t}d^{H}f^{∧}(C′_{t}, h^{W}).

Proof: The proof that f^{∧}(C′_{t}, h^{W}) is real and even is essentially the same as that in Theo-

rem 2.8. A simple computation, (see Remark 2.12 of [BL95]), shows that \frac{∂}{∂t}D_{t} = [C_{t}, \frac{N}{2t} ]. Using this we have

\frac{∂}{∂t}f(C′_{t}, h^{W}) = \frac{∂}{∂t}(2πi)^{1/2}φTr_{s}(f(D_{t}))

= (2πi)^{1/2}φTr_{s}([C_{t}, \frac{N}{2t}]f′(D_{t}))

= (2πi)^{1/2}φTr_{s}([C_{t}, \frac{N}{2t}f′(D_{t})])

= \frac{1}{t}d^{H}(φTr_{s}(\frac{N}{2} f′(D_{t})).

The second equality follows from Proposition 3.5 of [Hei95], the third from [C_{t}, D_{t}] = 0, and the last from Proposition 2.4 and the fact that (2πi)^{1/2}φ d^{H} = d^{H}φ. □

We now define the torsion for a flat Hermitian vector bundle E over a foliation F for which Condition 2.7 holds.

Set

χ′(F; E) = Tr_{s}(\frac{N}{2} f′(P \frac{1}{2}ωP)) ∈ Ω_{c}^{∗}(F).

Theorem 3.2 (Theorem 3.21, [BL95]) As t → 0

(3.3) f^{∧}(C′_{t}, h^{W}) = \{ \begin{matrix}\frac{1}{4} dim(F)rk(E)χ(F) + O(t) & if dim F is even, \\ O(\sqrt{t}) & if dim F is odd. \end{matrix}

There is γ > 0 so that as t → ∞

f^{∧}(C′_{t}, h^{W}) = \frac{1}{2}χ′(F; E) + O(t^{−γ}). (3.4)

Proof: We translate the proof in [BL95] to our situation. Let F̂ be the foliation on M̂ = M × (0, ∞) whose leaves are L × {s} for s ∈ (0, ∞). All the objects for F̂ will be denoted by the same symbols used for F with a over them. Then Ĝ = G × (0, ∞) and we denote the projection by ρ : Ĝ → G. Let ĝ be the metric on T Ĝ ≃ ρ^{∗}T G ⊕ T R^{+} obtained from the metric ρ^{∗}(g) ⊕ dr^{2} by scaling it by 1/r on T F̂_{s} |_{G×{r}}. Then

d_{E} = d^{E} + dr\frac{∂}{∂r}, and

(d_{E})^{∗} = r^{−(N−dim F/2)})((d^{E})^{∗} + dr\frac{∂}{∂r})r^{N−dim F/2}.

(The dim F term arises from the dependence of the volume form on r.) Then

(d_{E})^{∗} = (d^{E})^{∗} + dr(\frac{∂}{∂r} + \frac{1}{r}(N − \frac{dim F}{2} )),

and

D̂_{t} = r^{−N/2}D_{rt}r^{N/2} + \frac{dr}{2r}(N − \frac{dim F}{2} ).

It follows that

f(Ĉ′_{t}, h_{W}) = f(C′_{rt}, h^{W}) + \frac{dr}{r}f^{∧}(C′_{rt}, h^{W}) − \frac{dim F}{4r} dr φ Tr_{s}(f′(D_{rt})).

Now

Tr_{s}(f′(D_{rt})) = Tr_{s}(e^{D_{rt}^{2}} + 2D_{rt}^{2}e^{D_{rt}^{2}})) = Tr_{s}(e^{D_{rt}^{2}}) + Tr_{s}(2D_{rt}^{2}e^{D_{rt}^{2}})).

By Theorem 2.3, φTr_{s}(e^{D_{rt}^{2}}) = rk(E)χ(F), and since both 2D_{rt} and D_{rt}e^{D_{rt}^{2}} are odd operators, Tr_{s}(2D_{rt}^{2}e^{D_{rt}^{2}}) = Tr_{s}([D_{rt}, D_{rt}e^{D_{rt}^{2}}]), which is zero by Lemma 2.6. Thus we have

f(Ĉ′_{t}, h_{W}) = f(C′_{rt}, h^{W}) + \frac{dr}{r}f^{∧}(C′_{rt}, h^{W}) − dr\frac{dim F}{4r} rk(E)χ(F). (3.5)

Equation 2.9 gives the time t = 0 asymptotics of f(Ĉ′_{t}, h_{W}). Let ρ_{M} : M̂ → M be the projection and note that e(T F̂) = ρ_{M}^{∗}(e(T F)) and ∇_{E} = ρ_{M}^{∗}(∇^{E}). It then follows easily that the t → 0 limit of f(Ĉ′_{t}, h_{W}) contains no dr, so the same is true of the right hand side

of Equation 3.5. Setting r = 1, we have that as t → 0, f^{∧}(C′_{rt}, h^{W}) − \frac{dim F}{4} rk(E)χ(F) is

O(t) if dim F is even, and O(\sqrt{t}) if dim F is odd. As χ(F) = 0 when dim F is odd, we have Equation 3.3.

Next, note that H^{∗}(F̂; Ê) = ρ_{M}^{∗}(H^{∗}(F; E)). A simple computation then shows that

P̂ω̂P̂ = P ωP + \frac{dr}{r}(N − \frac{dim F}{2} )

where N is now the number operator on H^{∗}(F̂; Ê). We then have

f(∇^{H^{∗}(_{F};_{E})}, h^{H^{∗}(_{F};_{E})}) = f(∇^{H^{∗}(F;E)}, h^{H^{∗}(F;E)}) + \frac{dr}{2r} [χ′(F; E) − \frac{dim F}{2} χ̂(F; E)], (3.6)

where χ̂(F; E) = Tr_{s}(f′(P \frac{1}{2}ωP)) ∈ Ω_{c}^{∗}(F). Equation 2.10 gives the time t = ∞ asymptotics of f(Ĉ′_{t}, h_{W}) which implies that the right hand sides of Equations 3.5 and 3.6 are the same modulo O(t^{−γ}). Setting r = 1 and comparing the dr terms we obtain that as t → ∞

f^{∧}(C′_{t}, h^{W}) = \frac{dim F}{4} rk(E)χ(F) + \frac{1}{2}χ′(F; E) − \frac{dim F}{4} χ̂(F; E) + O(t^{−γ}).

To finish the proof, we have

Lemma 3.7 χ̂(F; E) = rk(E)χ(F).

Proof: f′(z) = (1 + 2z^{2})e^{z^{2}} is an even function and P ωP is an odd endomorphism so Tr_{s}(f′(P tωP)) is independent of t. To see this note that

\begin{aligned}\frac{∂}{∂t}Tr_{s}(f′(P tωP)) & = & Tr_{s}((P ωP)f′′(P tωP)) \\ & = & \frac{1}{2}Tr_{s}([P ωP, f′′(P tωP)]) = 0. \end{aligned}

The first equality follows from from Proposition 3.5 of [Hei95], the second from the fact that f′′ and P ωP are both odd, and the last from Lemma 2.6.

Now regard P tωP as acting on H^{∗}(F; E) = Ker(V). Setting t = 0 we obtain χ̂(F; E) = Tr_{s}(I_{H^{∗}(F;E)}) = Tr_{s}(P). Now Tr_{s}(P) is the index of the leafwise operator V and by the extended Connes Foliation Index Theorem, [Con86] (i.e. without assuming the existence a transverse invariant measure), it is equal to ∫_{F} ch(E)e(F) = rk(E)χ(F) since E is flat. □

Definition 3.8 The torsion form T (F; E) ∈ Ω_{c}^{∗}(F) is given by

T (F; E) = − ∫_{0}^{∞}[f^{∧}(C′_{t}, h^{W}) − \frac{1}{2}χ′(F; E) + (\frac{1}{2}χ′(F; E) − \frac{dim F}{4} rk(E)χ(F))f′(\frac{i\sqrt{t}}{2})]\frac{dt}{t}.

Theorem 3.2 and the fact that f′(\frac{i\sqrt{t}}{2}) = e^{−t/4} − \frac{t}{2}e^{−t/4} imply that T (F; E) is well defined.

Theorem 3.9 The torsion form T (F; E) is real, even and

d^{H}(T (F; E)) = −f(∇^{H^{∗}(F;E)}, h^{H^{∗}(F;E)}) + ∫_{F} e(T F)f(∇^{E}, h^{E}).

Proof: As f^{∧}(C′_{t}, h^{W}) is real and even, we need only show that χ′(F; E) is also. Now ω = (∇^{W})^{∗}−∇^{W} is an element of Ω^{1}(M, End(W)). As f′(z) = (1+2z^{2})e^{z^{2}} is an even holomorphic function, it follows that χ′(F; E) is an even Haefliger form. In addition, (P ωP)^{∗} = P^{∗}ω^{∗}P^{∗} = −P ωP so χ′(F; E)^{∗} = χ′(F; E) and χ′(F; E) is real valued.

Now d^{H} χ̂(F; E) = d^{H}(rk(E)χ(F)) = 0, and χ̂(F; E) = Tr_{s}(f′(P \frac{1}{2}ωP)). As χ′(F; E) = Tr_{s}(\frac{N}{2} f′(P \frac{1}{2}ωP)), we have d^{H}(χ′(F; E)) = 0. The theorem now follows from Theorem 2.8 and Theorem 3.1. □

The Hodge Theorem for complete Riemannian manifolds, [HL90], the Appendix, implies that H^{∗}(F; E) being 0 does not depend on the metrics on M and E. Thus we have the following important Theorem 3.10 [Corollary 3.25, [BL95]] Assume that dim F is odd, H^{∗}(F; E) = 0 and Condition 2.7 holds. Then T (F; E) is a closed form whose Haefliger cohomology class depends only on the foliation F and flat bundle E.

Now we define the Ray-Singer torsion for E over F and show that it is the order zero part of T (F; E).

Let Δ = −V^{2} be the leafwise Laplacian associated to d^{F} (the leafwise de Rham operator on the leaves of F_{s}). As V is skew adjoint, ker(Δ) = ker(V), so P is the projection onto the kernel of Δ. Denote by e^{−tΔ} the heat operator associated to Δ. Both P and e^{−tΔ} are of trace class.

In the classical case of a flat bundle over a compact n-dimensional manifold, one shows that the integral

ξ(s) = \frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}tr_{s}(N(e^{−tΔ} − P))dt

converges for s >> 0 because of two facts:

• for each i, the pointwise trace tr(e^{−tΔ_{i}}) has an asymptotic expansion as t → 0^{+} of the form tr(e^{−tΔ_{i}}) ∼ ∑_{m=0}^{∞} t^{\frac{m−n}{2}} a_{m,i} where a_{m,i} are bounded functions on M and Δ_{i} is the restriction of Δ to the i forms;

• Δ_{i} is non-negative and its positive spectrum is bounded away from zero, so that tr(e^{−tΔ_{i}} − P_{i}) = O(e^{−tδ}) as t ↦ ∞, for some δ > 0.

The analytic torsion T is then defined to be T = −\frac{1}{2} \frac{dξ}{ds}(0).

In the case of non-compact manifolds of bounded geometry, the first fact is still true, but the second is not in general. To compensate for this, we do two things. First, we assume that the strong foliation Novikov-Shubin invariants of E over F are positive, which implies that as t ↦ ∞ (3.11) Tr_{s}(N(e^{−tΔ} − P)) = O(t^{−γ}) for some γ > 0.

Second, we redefine ξ(s) as

ξ(s) = \frac{1}{Γ(s)} ∫_{0}^{ϵ} t^{s−1}Tr_{s}(N(e^{−tΔ} − P))dt + s ∫_{ϵ}^{∞} t^{−1}Tr_{s}(N(e^{−tΔ} − P))dt,

for some ϵ > 0. Then ξ(s) is a well defined analytic function for s large, and as in the classical case, it extends to the complex plane as a meromorphic function which is analytic at s = 0. See [Lot92], [CGT82]. The Ray-Singer torsion T (F, E) of E over F is defined to be

T (F, E) = −\frac{1}{2} \frac{dξ}{ds}(0),

and it is independent of ϵ.

Theorem 3.12 [Theorem 3.29,[BL95]] The order zero term of T (F; E), denoted T^{(0)}(F; E) satisfies

T^{(0)}(F; E) = T (F; E).

Proof: Set c = \frac{1}{2} dim(F)rk(E)χ(F) − χ′(F; E) and g(a) = (1 + 2a)e^{a}. Then

T^{(0)}(F; E) = ∫_{0}^{∞} h(t)\frac{dt}{t},

where

h(t) = −[Tr_{s}(\frac{N}{2} g(\frac{−tΔ}{4})) − \frac{χ′(F; E)}{2} − \frac{c}{2}g(−\frac{t}{4})].

Using the fact that h(t) is at least O(\sqrt{t}) as t → 0 we have that for any ϵ > 0,

∫_{0}^{∞} h(t)\frac{dt}{t} = \frac{d}{ds} |_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{ϵ} h(t)t^{s−1}dt + s ∫_{ϵ}^{∞} h(t)\frac{dt}{t}], (3.13)

so the right hand side of Equation 3.13 is independent of ϵ. Now

Tr_{s}(\frac{N}{2} g(\frac{−tΔ}{4}) |_{ker Δ}) = Tr_{s}(\frac{N}{2} g(0)) = Tr_{s}(\frac{N}{2} P) = \frac{1}{2}χ′(F; E)

where an argument similar to that in the proof of Lemma 3.7 gives the last equality. Set e^{−tΔ′} = e^{−tΔ} − P. We then have that 3.13 is equal to

−\frac{d}{ds} |_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{ϵ} t^{s−1}(Tr_{s}(\frac{N}{2} e^{−tΔ′}) + t\frac{d}{dt}Tr_{s}(Ne^{−tΔ′}) − \frac{c}{2}g(−t))dt

+s ∫_{ϵ}^{∞} Tr_{s}(\frac{N}{2} e^{−tΔ′}) + t\frac{d}{dt}Tr_{s}(Ne^{−tΔ′}) − \frac{c}{2}g(−t)\frac{dt}{t}].

We claim that this is equal to

\frac{d}{ds} |_{s=0}[−\frac{1}{2}ξ(s) + sξ(s)] − c.

Recalling the definition of ξ(s), and using the independence from ϵ and the fact that

\frac{d}{ds} |_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{∞} t^{s−1}g(−t)dt] = −2,

we see that this follows from the

Lemma 3.14

−\frac{d}{ds} |_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{ϵ} t^{s} \frac{d}{dt}Tr_{s}(Ne^{−tΔ′})dt + s ∫_{ϵ}^{∞} \frac{d}{dt}Tr_{s}(Ne^{−tΔ′})dt] = \frac{d}{ds} |_{s=0}sξ(s).

Proof: Integrating by parts we have

−\frac{d}{ds} |_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{ϵ} t^{s} \frac{d}{dt}Tr_{s}(Ne^{−tΔ′})dt + s ∫_{ϵ}^{∞} \frac{d}{dt}Tr_{s}(Ne^{−tΔ′})dt]

= −\frac{d}{ds} |_{s=0}[\frac{1}{Γ(s)} ∫_{0}^{ϵ} −st^{s−1}Tr_{s}(Ne^{−tΔ′})dt + \frac{1}{Γ(s)}t^{s}Tr_{s}(Ne^{−tΔ′})|_{0}^{ϵ} + sTr_{s}(Ne^{−tΔ′}) |_{ϵ}^{∞}]

= \frac{d}{ds} |_{s=0}[s\frac{1}{Γ(s)} ∫_{0}^{ϵ} t^{s−1}Tr_{s}(Ne^{−tΔ′})dt + s^{2} ∫_{ϵ}^{∞} t^{−1}Tr_{s}(Ne^{−tΔ′})dt − Tr_{s}(Ne^{−ϵΔ′})(\frac{ϵ^{s}}{Γ(s)} − s)],

and

\frac{d}{ds} |_{s=0}(\frac{ϵ^{s}}{Γ(s)} − s) = 0.

□

To finish the theorem we need Lemma 3.15 ξ(0) = c. Proof: The proof of Theorem 7.10 of [BZ92] involves only local analysis and so remains valid in our situation. Thus we have that as t → 0,

Tr_{s}(Ne^{−tΔ}) = \{ \begin{matrix}\frac{1}{2} dim(F)rk(E)χ(F) + O(t) & if dim F is even, \\ c_{1}/\sqrt{t} + O(\sqrt{t}) & if dim F is odd, \end{matrix}

for some constant c_{1}. Since Tr_{s}(NP) = χ′(F; E) we have

ξ(s) = \frac{1}{Γ(s)} ∫_{0}^{ϵ} t^{s−1}[Tr_{s}(Ne^{−tΔ}) − χ′(F; E)]dt + s ∫_{ϵ}^{∞} t^{−1}Tr_{s}(N(e^{−tΔ} − P))dt.

If dim F is even, the first term equals

\frac{s}{Γ(s + 1)} ∫_{0}^{ϵ} t^{s−1}[c + O(t)]dt = \frac{cϵ^{s}}{Γ(s + 1)} + \frac{s}{Γ(s + 1)} ∫_{0}^{ϵ} t^{s−1}O(t)dt.

If dim F is odd, χ(F) = 0 and we get instead

\frac{s}{Γ(s + 1)} ∫_{0}^{ϵ} t^{s−1}[c+ \frac{c_{1}}{\sqrt{t}} +O(\sqrt{t})]dt = \frac{cϵ^{s}}{Γ(s + 1)} + \frac{sc_{1}ϵ^{s− \frac{1}{2}}}{(s − \frac{1}{2})Γ(s + 1)} + \frac{s}{Γ(s + 1)} ∫_{0}^{ϵ} t^{s−1}O(\sqrt{t})dt.

In both cases, ξ(s) = c + sA(s) where the function A is holomorphic around s = 0. □

4 Examples

We begin by noting that Condition 2.7 was needed only to handle Haefliger forms and not Haefliger functions. Thus in Examples 4.1 and 4.2 below, we do not need Condition 2.7.

4.1 Classical case

As noted above, Ray and Singer defined analytic torsion for acyclic flat bundles over a compact manifold M. Cheeger and Müller independently proved that it is the same as the classical Reidemeister torsion. The Ray-Singer torsion is a special case of the torsion for foliations. The foliation of M with only one leaf, namely M itself, has T F = T M and G = M × M. As a transversal we choose a single point in M, and the Haefliger cohomology of this foliation is just the usual cohomology of the point. Theorem 3.12 shows that the foliation torsion is the Ray-Singer torsion.

4.2 The L^{2} covering case

In [Lot92] Lott defined and studied an analog for covering spaces of the Ray-Singer torsion. This is essentially the same special case of the torsion for foliations considered above. Instead of using the holonomy graph, one uses the homotopy graph H = M × M̃. The trace in this situation is the Atiyah L^{2} covering space trace.

4.3 Torsion for families

In [BL95], Bismut and Lott construct a families torsion invariant for flat bundles over a fiber bundle M → B with fiber Z. As pointed out above, the results here extend their results. In particular, take the foliation of M given by the fibers of the bundle. Then G is the fiber product of M with itself, and the Haefliger cohomology of the foliation is just the usual cohomology of B. Condition 2.7 holds automatically as P_{ϵ} = 0 for small enough ϵ.

Proposition 4.1 The foliation torsion form of a fiber bundle is the torsion form of Bismut-Lott.

Proof: As D_{t} is a local operator (i.e. it depends only on the structures on a given leaf), we may assume that the fiber bundle M is a product bundle M = B × L. Then G = B × L_{s} × L_{r} = M × L_{r}, where we may take the foliation F_{s} to be given by the original foliation F on the M = B × L_{s} factors. On M × L_{r}, we have the bundle Ẽ = r^{∗}(E). Since the pull backs of the metrics and connections for E over M to Ẽ over M × L_{r} do not depend on L_{r}, it is immediate that the superconnection d^{G} = d^{M×L_{r}} = d^{M} + d^{L_{r}} and that its adjoint is (d^{M})^{∗} + d^{L_{r}}. Thus for each leaf L of F on M and for each x ∈ L, the operator D_{t} on the leaf L_{x} = s^{−1}(x) of the foliation F_{s} of G is the same as the operator D_{t} on L used to define the Bismut-Lott torsion. The same is true for the other objects used to define the foliation torsion. Note that the proof of Lemma 3.7 applied to χ′(F; E) shows that in the fiber bundle case, our χ′(F; E) agrees with Bismut-Lott’s χ′(F; E), as does our χ(F). Finally, integration over the fiber of the foliation is the same as integration over the fiber of the bundle. A review of the constructions of the two torsion forms completes the proof. □

If we use the homotopy graph H, which is a fiber bundle over M whose fiber is the universal cover Z̃ of Z, we obtain a families torsion invariant for families of covering spaces, i.e. a combination of the L^{2} covering case with the families case. Of course, we must require that Condition 2.7 is satisfied. This is a special case of the situation considered by Gong and Rothenberg in [GR96].

4.4 Torsion for Seifert fibrations of lens spaces

Let q_{1}, …, q_{n} be pairwise relatively prime integers and consider the action of S^{1} on C^{n+1}−{0} given by

(z_{0}, …, z_{n}) → (e^{2πiq_{0}t}z_{0}, …, e^{2πiq_{n}t}z_{n})

On C^{n+1} − {0} we will freely change among the three coordinate systems (z_{0}, …, z_{n}), (x_{0}, y_{0}, …, x_{n}, y_{n}) and (r_{0}, θ_{0}, …, r_{n}, θ_{n}). This action of S^{1} is given by the vector field

X = ∑_{0}^{n} q_{j}(x_{j}∂/∂y_{j} − y_{j}∂/∂x_{j}) = ∑_{0}^{n} q_{j}∂/∂θ_{j}.

Both X and the one form

ω = ∑_{0}^{n} q_{j}(x_{j}dy_{j} − y_{j}dx_{j}) = ∑_{0}^{n} q_{j}r_{j}^{2}dθ_{j}.

are invariant under the action of S^{1}. The pseudosphere

S_{q}^{2n+1} = {(r_{0}, θ_{0}, …, r_{n}, θ_{n}) | ∑_{0}^{n} q_{j}^{2}r_{j}^{2} = 1}

is fixed by the action of S^{1} so X is tangent to it and defines a one dimensional foliation F on it. Restricted to S_{q}^{2n+1}, ω is a dual form to X and so defines a normal bundle ν for F. Off the set S = {(z_{0}, …, z_{n}) ∈ S_{q}^{2n+1} | all but one of the z_{i} = 0}, the action of S^{1} is free and defines a principal S^{1} bundle structure, and the form ω is a connection form on this bundle. The curvature of ω is just Ω = dω, and we may compute this form on S_{q}^{2n+1} by computing it on C^{n+1} − {0} and restricting. This gives us

Ω = ∑_{0}^{n} 2q_{j}r_{j}dr_{j} ∧ dθ_{j} | S_{q}^{2n+1}.

The form ω^{⊥} = ∑_{0}^{n} q_{j}^{2}r_{j}dr_{j} is dual to the normal vector field of S_{q}^{2n+1} in C^{n+1} − {0} and a simple computation shows that

Ω^{n} ∧ ω ∧ ω^{⊥} = n!2^{n}q_{0} ⋯ q_{n}dx_{0} ∧ dy_{0} ∧ ⋯ ∧ dx_{n} ∧ dy_{n}

on S_{q}^{2n+1} in C^{n+1} − {0}. Thus the form Ω^{n} ∧ ω is a volume form on S_{q}^{2n+1}. We shall use this fact below to conclude that certain Haefliger cocycles define non trivial cohomology classes.

Let p be an integer relatively prime to each of the q_{i}. The action of Z_{p} ⊂ S^{1} on C^{n+1}−{0} given by

(z_{0}, …, z_{n}) → (e^{2πiq_{0}/p}z_{0}, …, e^{2πiq_{n}/p}z_{n})

is then free and it preserves S_{q}^{2n+1} and the foliation F, so F descends to a one dimensional foliation, also denoted F, of the generalized lens space L = S_{q}^{2n+1}/Z_{p}. The action of Z_{p} is the time 1/p flow of X, so X, ω and Ω also descend to L. As p is relatively prime to each of the q_{i}, the flat C bundle E corresponding to the map sending a generator of π_{1}(L) to a primitive p th root of unity has trivial leafwise L^{2} cohomology.

We now compute the foliation torsion for the flat bundle E and the foliation F of L. Note that P_{ϵ} = 0 for sufficiently small ϵ, as the graph of the foliation is a fiber bundle with all fibers compact. On {S_{q}^{2n+1} − S} → {S_{q}^{2n+1} − S}/Z_{p} = L_{1}, we have exactly the situation considered in [BL95] Proposition 4.13. The Bismut-Lott torsion form on L_{1}/F is given by the pushdown of the form

T = \frac{1}{2} ∑_{j=0}^{n} \frac{(2j + 1)!}{2^{2j}(j!)^{2}}( ∑_{m≠0} \frac{1}{m^{j}|m|}ξ^{m})(\frac{pΩ}{4π^{2}} )^{j}

on L_{1} where ξ is a primitive p th root of unity. Note that the form T extends to a smooth form on L. It follows from Proposition 4.1, that

T (F; E) = ∫_{F} T ∧ \frac{p}{2π} ω,

since this is true on L_{1} as the lengths of the fibers of L_{1} are 2π/p and ω is the volume form on the fibers.

Proposition 4.2 Any closed oriented k + 1 dimensional submanifold C of L which is saturated by the leaves of F determines a well defined map ∫_{C} : H_{c}^{k}(F) → C.

Proof: Any element α ∈ Ω_{c}^{k}(T) may be written as α = ∫_{F} α_{1} for some α_{1} ∈ Ω^{k+1}(L)⊗C. Set

∫_{C} α = ∫_{C} α_{1} where the integration on the right is the usual integration of forms over oriented submanifolds. As C is saturated by the leaves of F, it follows that this is independent of the choice of α_{1}, and that for any element h in the holonomy pseudogroup, ∫_{C}(α − h^{∗}α) = 0.

Thus C determines a well defined map : ∫_{C} : Ω_{c}^{k}(F) → C. If α ∈ Ω_{c}^{k−1}(T), and α = ∫_{F} α_{1},

then dα = ∫_{F} dα_{1}, and it is immediate that ∫_{C} dα = 0. Thus C determines a well defined

map ∫_{C} : H_{c}^{k}(F) → C. □

Denote by T^{(2k)}(F; E) the order 2k part of T (F; E), and let C_{2k+1} ⊂ L be

C_{2k+1} = {(z_{0}, …, z_{n}) ∈ S_{q}^{2n+1} | z_{k+1} = ⋯ = z_{n} = 0}/Z_{p}.

A simple computation shows that

∫_{C_{2k+1}} T^{(2k)}(F; E) = D(k)\frac{p^{k−1}}{q_{0} ⋯ q_{k}} ∑_{m≠0} \frac{1}{m^{k}|m|}ξ^{m}

where D(k) is a constant which depends only on k.

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