Higher analytic torsion of sphere bundles and continuous cohomology classes of the diffeomorphism

group of a sphere

Ulrich Bunke

December 5, 2001

Abstract

Using the higher analytic torsion form of Bismut and Lott we construct a characteristic class T (E) ∈ H^{∗}(B, ℝ) for smooth fiber bundles E → B with fiber isomorphic to S^{2n−1}. When E is the unit sphere bundle of a hermitian vector bundle V → B, then we calculate this class in terms of the Chern character ch(V).

The higher analytic torsion form of the trivial sphere bundle over the space of all unit-volume metrics on S^{2n−1} with its tautological vertical metric is closed and invariant under the group of diffeomorphisms Diff(S^{2n−1}). It leads to a continuous cohomology class T ∈ H_{c}^{∗}(Diff(S^{2n−1}), ℝ). Viewing S^{2n−1} as the boundary at infinity of the complex hyperbolic space ℂH^{n} we obtain an embedding SU(n, 1) ↪ Diff(S^{2n−1}). If Γ ⊂ SU(n, 1) is a torsion-free cocompact subgroup, then the compact Kähler manifold B := Γ\ℂH^{n} is a model of the classifying space BΓ. Under the identification H^{∗}(Γ, ℝ) ≅ H^{∗}(B, ℝ) the restriction of T to Γ corresponds to the characteristic class T (SB) of the sphere bundle of B. We conclude that 0 ≠ T_{4k} ∈ H_{c}^{4k}(Diff(S^{2n−1}), ℝ) as long as 0 < 2k ≤ n. Moreover, if r : Diff(S^{2n−1})^{δ} → Diff(S^{2n−1}) is the canonical map, where Diff(S^{2n−1})^{δ} is the diffeomorphism group with the discrete topology, then r^{∗}T_{4k} ≠ 0 for 0 < 2k ≤ n, too.

Contents

1 Introduction 1

2 Higher equivariant torsion of complex representation spheres 5

3 Construction of continuous cohomology classes of Diff(S^{2n−1}) 7

4 Nontriviality of the continuous cohomology classes 10

1 Introduction

Let E → B be a smooth fiber bundle with a closed manifold M as fiber, and let F → E be a complex vector bundle with a flat connection. In [2] Bismut and Lott defined the higher analytic torsion form T (E, F, T^{H}E, g^{v}, h^{F}) ∈ Ω^{ev}(B) which depends on the following additional structures:

• a vertical Riemannian metric g^{v},

• a horizontal distribution T^{H}E,

• a hermitian metric h^{F} on F.

\underline{Note}, that in the present paper we completely neglect the zero dimensional part of the torsion form and let T (E, F, T^{H}E, g^{v}, h^{F}) denote the higher part starting in degree two.

Under certain conditions the higher analytic torsion form is closed and defines a cohomology class T (E, F) ∈ H^{∗}(B, ℝ) which only depends on E → B and F and not on the additional structures. The class T (E, F) behaves naturally with respect to pull-back of bundles.

The basic challenge in the theory is to give a topological description of the cohomology class T (E, F). The definition of higher Reidemeister torsion by Igusa and Klein [9], Igusa [8] (there is also an alternative approach by Dwyer, Weiss and Williams [6]) yields topologically defined cohomology classes of B with similar properties as T (E, F). The relation between the topological and analytical torsion invariants is now understood in some cases (see Goette [7], Thm. 0.2).

On the computational side the higher Reidemeister torsion is still very complicated. In fact a direct calculation is only known in the case where E → B is a bundle with fibre S^{1} and F has finite holonomy along the fibres (see [8]).

The higher analytic torsion T (E, F) can at the present time be calculated in two cases. In the first case one assumes the existence of a fibrewise Morse function (see Bismut and Goette [1], Goette [7]). In the other case the assumption is that the structure group of the bundle E → B can be reduced to a compact group (see [3]). Note that there is some overlap between these cases (e.g. unit sphere bundles of hermitian vector bundles), and is was checked in [1] that the calculations agree.

The structure group G of the fibre bundle E → B is a subgroup of the group of diffeomorphisms Diff(M) of M. If G is compact, then it was shown by Bismut and Lott [2], that T (E, F, T^{H}E, g^{v}, h^{F}) can be expressed in terms of characteristic forms of the G-principal bundle P → B associated to E. Here we assume that T^{H}E is induced from a connection of P, g^{v} comes from a G-invariant Riemannian metric on M. Moreover, we assume that the action of G on M lifts to the restriction F_{M} of F to M and the hermitian metric h^{F_{M}} is G-invariant. We see that T (E, F, T^{H}E, g^{v}, h^{F}) is a closed form which represents a cohomology class T (E, F) ∈ H^{∗}(B, ℝ). Given a representation E = P ×_{G} M this class is independent of the choice of the connection on P, the G-invariant Riemannian metric on M, and the hermitean metric h^{F_{M}}. But in general it is not an invariant of the pair (E, F) alone.

The G-manifold M and F_{M} give rise to an invariant power series T(M, F_{M}) ∈ Î(G) on the Lie algebra of G. One obtains T (E, F) by applying the Chern-Weil homomorphism to T(M, F_{M}), i.e.

T (E, F) = CW_{P} (T(M, F_{M})) .  (1)

If F is trivial, then in [3] the power series T(M, F_{M}) was explicitly calculated in terms of a G-cell decomposition of M. Note that it is easy to extend this result to non-trivial F.

In order to assure that T (E, F, T^{H}E, g^{v}, h^{F}) is closed one usually assumes that M is odd-dimensional and that F_{M} is acyclic, i.e. H^{p}(M, F_{M}) = 0 for all p ≥ 0, where F_{M} is the locally constant sheaf of parallel sections of F_{M}. In the present paper we consider sphere bundles, i.e. bundles with fibre S^{2n−1}. For n ≥ 2 the bundle F_{M} is trivial and not acyclic. If F is the trivial flat hermitian bundle E × ℂ (which we omit in our notation from now on), then by Lemma 3.1 the form T (E, T^{H}E, g^{v}) is still closed provided the volume of the fibres induced by g^{v} is constant. By Lemma 4.3 the higher analytic torsion form defines a characteristic class T (E) ∈ H^{ev}(B, ℝ) for any bundle E → B with fibre S^{2n−1} which is independent of the additional choices.

Let G be a compact connected Lie group and ρ : G → U(n) be a unitary representation. Then G acts on the unit sphere S^{2n−1}. Let T ⊂ G be a maximal torus, 𝔱 be its Lie algebra, 𝔱_{ℂ}^{∗} be the complexified dual of 𝔱, and Λ ⊂ 𝔱_{ℂ}^{∗} the weight lattice. Restriction to 𝔱 identifies Î(G) with the space of Weil-invariant power series Î(T)^{W} on 𝔱. We compute T(S^{2n−1}) ∈ Î(T)^{W} in terms of the weights Δ(ρ) ⊂ Λ of the representation ρ. If α ∈ Δ(ρ), then m_{α} denotes the multiplicity of the weight α. We assume that 0 ∉ Δ(ρ).

In order to formulate the result in a compact way we introduce the following formal power series Q ∈ ℝ[x]:

Q(x) := ∑_{j=1}^{∞} \frac{(4j + 1)!}{2^{4j}((2j)!)^{2}} ζ_{R}(2j + 1) (\frac{x}{4π^{2}} )^{2j} ,  (2)

where ζ_{R} is the Riemann zeta function.

Theorem 1.1

T(S^{2n−1}) ∑_{α∈Δ(ρ)} m_{α}Q(α) .

Let B be a Kähler manifold, i.e. B is a Riemannian manifold with holonomy U(n). Let SB → B be the unit sphere bundle. Combining Theorem 1.1 in the case where ρ is the standard representation of U(n) on ℂ^{n} and Equation (1) we obtain

Theorem 1.2

T (SB) = ∑_{j=1}^{∞} \frac{(4j + 1)!}{2^{4j}(2j)!}ζ_{R}(2j + 1)ch_{4j}(T B) ,

where ch(T B) = ch_{0}(T B) + ch_{2}(T B) + ch_{4}(T B) + … denotes the Chern character of T B.

If we take B = ℂP^{n}, then for p ≥ 1 we have ch_{2p}(T B) = (−1)^{p} \frac{n+1}{p!}c_{1}(H)^{p}, where H is the tautological bundle. In particular, ch_{2p}(T B) ≠ 0 for p ≤ n. Since ζ_{R}(2j + 1) ≠ 0 for j ≥ 1 we see that T_{4j}(SB) ≠ 0 as long as 0 < 2j ≤ n.

We now consider the trivial fibre bundle

E = Met_{1}(S^{2n−1}) × S^{2n−1} → Met_{1}(S^{2n−1}) ,

where Met_{1}(S^{2n−1}) is the Fréchet manifold of all volume one Riemannian metrics on S^{2n−1}. The bundle E has a natural horizontal distribution T^{H}E and a tautological vertical Riemannian metric g^{v}. The group Diff(S^{2n−1}) acts on this bundle and leaves the additional structures invariant. The higher analytic torsion form T (E, T^{H}E, g^{v}) ∈ Ω^{ev}(Met_{1}(S^{2n−1})) is closed and Diff(S^{2n−1})-invariant. For k ∈ ℕ (using the fact that Met_{1}(S^{2n−1}) is contractible) we construct from the degree 2k component of T (E, T^{H}E, g^{v}) a continuous group cocycle c_{T,2k} on Diff(S^{2n−1}) which represents a continuous group cohomology class T_{2k} ∈ H_{c}^{2k}(Diff(S^{2n−1}), ℝ).

The main goal of the present paper is to prove the following theorem.

Theorem 1.3 The classes T_{4k} ∈ H_{c}^{4k}(Diff(S^{2n−1}), ℝ), 0 < 2k ≤ n, are nontrivial.

Except for n = 1 the continuous group cohomology of Diff(S^{2n−1}) has not been calculated yet. Thus Theorem 1.3 provides a lower bound on the dimensions of the continuous group cohomology of Diff(S^{2n−1}).

We shall sketch the main steps of the proof of Theorem 1.3. We view S^{2n−1} as the boundary at infinity of the complex hyperbolic space ℂH^{n}. We thus obtain an embedding of the isometry group SU(n, 1) of ℂH^{n} into Diff(S^{2n−1}). We then consider a discrete torsion-free cocompact subgroup Γ ⊂ SU(n, 1). We want to show that the restriction to Γ of T_{4k} is non-trivial for 0 < 2k ≤ n. The quotient B := Γ\ℂH^{n} is a model of the classifying space BΓ, and we thus have a natural identification H^{∗}(Γ, ℝ) ≅ H^{∗}(B, ℝ). It turns out that the restriction of T to Γ corresponds to the class T (SB) ∈ H^{∗}(B, ℝ) under this identification. Now B is a closed locally symmetric Kähler manifold and we can apply Theorem 1.2 in order to express T (SB) in terms of the Chern character ch(T B). The complex projective space ℂP^{n} is the compact dual symmetric space to ℂH^{n}. Since ch_{4k}(T ℂP^{n}) ≠ 0 for 0 ≤ 2k ≤ n we conclude by the proportionality principle that ch_{4k}(T B) ≠ 0 for 0 ≤ 2k ≤ n, too. Since for k > 0 the class T_{4k}(SB) is a nonzero multiple of ch_{4k}(T B) it must be nontrivial. This finishes the sketch of the proof of Theorem 1.3.

As a corollary of this proof we obtain:

Corollary 1.4 Let Diff(S^{2n−1})^{δ} denote the group of diffeomorphisms of S^{2n−1} with the discrete topology, and let r : Diff(S^{2n−1})^{δ} → Diff(S^{2n−1}) be the natural map. Then the classes r^{∗}T_{4k} ∈ H^{4k}(Diff(S^{2n−1})^{δ}, ℝ) are nontrivial for 0 < 2k ≤ n.

...............

The present paper is an updated version of a preprint which was circulating since 1997/1998. In the meantime it was cited in several related works so that we decided to publish it now. The author thanks S. Goette for fruitful and encouraging discussions.

2 HIGHER EQUIVARIANT TORSION OF COMPLEX REPRESENTATION SPHERES5

2 Higher equivariant torsion of complex representation spheres

In this section we specialize the results of [3] to complex representation spheres and prove Theorems 1.1 and 1.2. Let G be a connected compact Lie group. Higher equivariant torsion of a closed G-manifold M was introduced by Lott [10] using a normalization which is slightly different from the one employed in Bismut and Lott [2] and in the present paper. But the results of [10] and [3] remain true with the present normalization. The present normalization behaves even better since the local trace of the heat kernel integrable for small times.

We describe the two necessary changes in [3]. Let Ĩ(G) denote the space of invariant formal power series on the Lie algebra 𝔤 of G with trivial constant term. The definition [3], Def. 3.1, of the higher equivariant torsion T (M) ∈ Ĩ(G) has to be replaced by

T(M) := ϕ ∫_{0}^{∞} Tr_{s} \frac{N}{2} (1 + 2D_{t}^{2})e^{D_{t}^{2}} \frac{dt}{t} ,

where we employ the notation introduced in [3], Sec 3., and ϕ multiplies the component of homogeneity k by (\frac{1}{2πı})^{−k}. The other point is the computation for G = S^{1} and M = S^{1}. Here we employ the formula of Bismut and Lott, [2], Prop. 4.13. Let S^{1} ≅ ℝ/ℤ act on M = ℝ/\frac{1}{r}ℤ, r ∈ ℕ. Let α ∈ (𝔰^{1})_{ℂ}^{∗} be the generator of the weight lattice such that α(∂_{t}) = 2πı. Then

T(M) = ∑_{j=1}^{∞} \frac{(4j + 1)!}{2^{4j}((2j)!)^{2}} ζ_{R}(2j + 1) (\frac{rα}{4π^{2}} )^{2j} .  (3)

Let G be a connected compact Lie group and ρ : G → U(n) be an unitary representation. The unit sphere S^{2n−1} ⊂ ℂ^{n} becomes a G-manifold. Let T ⊂ G be a maximal torus, 𝔱 be its Lie algebra, and W := N_{G}(T)/T be the Weyl group. By restriction we identify the space Ĩ(G) with the space of Weil invariant power series Ĩ(T)^{W} on 𝔱 with constant trivial term. We have to calculate the higher equivariant torsion T(S^{2n−1}) ∈ Ĩ(T)^{W} of the T -manifold S^{2n−1}.

By Theorem 1.1 [3] we first must compute the contribution of the one-dimensional T -orbits to the equivariant Euler characteristic χ_{T} (S^{2n−1}). Then we have to apply the homomorphism T_{T} mapping the target group of the equivariant Euler characteristic to power series on 𝔱. The homomorphism T_{T} was defined in [3], Sec.7.

We assume that T has no fixed point on S^{2n−1}. Let Δ(ρ) ⊂ t_{ℂ}^{∗} be the set of weights of ρ. Then 0 ∉ Δ(ρ). A point ξ ∈ S^{2n−1} generates a one-dimensional T -orbit iff it is a weight vector. Let S_{α} ⊂ S^{2n−1} be the subset of weight vectors to α ∈ Δ(ρ). All points ξ ∈ S_{α} generate the same orbit type T/T_{α}, where T_{α} is the stabilizer of ξ. The quotient T \S_{α} is diffeomorphic to the complex projective space ℂP^{m_{α}−1}, where m_{α} is the multiplicity of the weight α. Note that ℂP^{m_{α}−1} has Euler characteristic m_{α}. We conclude that

χ_{T} (S^{2n−1}) = ∑_{α∈Δ(ρ)} m_{α}[T/T_{α}] .  (4)

Now T/T_{α} ≅ S^{1}. Using [3], Prop. 7.2, and Equation (3) we obtain

T_{T} ([T/T_{α}]) = T(T/T_{α}) = Q(α) ,  (5)

where Q is the formal power series (2). Theorem 1.1 immediately follows from the Equations (4) and (5). □

We now specialize to the case that G = U(n) and ρ = id. Let T ⊂ U(n) be the subgroup of diagonal matrices and identify 𝔱 with ℝ^{n} such that exp(x) = diag(e^{2πıx_{1}}, …, e^{2πıx_{n}}), where x = (x_{1}, …, x_{n}) ∈ ℝ^{n}. If {α_{i}}_{i=1}^{n} denotes the basis of 𝔱_{ℂ}^{∗} which is the dual of the standard basis of 𝔱 ≅ ℝ^{n} multiplied by 2πı, then Δ(ρ) = {α_{1}, …, α_{n}}, and each weight occurs with multiplicity one. By Theorem 1.1 the homogeneous component of degree 2j of T(S^{2n−1}) is

T_{2j} = \frac{(4j + 1)!}{2^{4j}((2j)!)^{2}} ζ_{R}(2j + 1)\frac{∑_{l=1}^{n} α_{l}^{2j}}{(4π^{2})^{2j}} ,  (6)

while the components of odd degree vanish.

Let now B be a n-dimensional Kähler manifold, and P → B be the U(n)-principal bundle associated to the tangent bundle T B. If R is the curvature of T B, then the Chern character ch(T Y) ∈ H^{ev}(B, ℝ) is represented by

Tr exp(−\frac{R}{2πı}) ∈ Ω^{ev}(B) .

We can write

ch_{4j}(T Y) = CW_{P} (\frac{∑_{l=1}^{n} α_{l}^{2j}}{(2j)!(4π^{2})^{2j}} ) .

Thus Theorem 1.2 follows from Equations (6) and (1).

3 CONSTRUCTION OF CONTINUOUS COHOMOLOGY CLASSES OF DIFF(S^{2N−1})7

3 Construction of continuous cohomology classes of Diff(S^{2n−1})

In this section we construct nontrivial continuous cohomology classes

T_{4j} ∈ H_{c}^{4j}(Diff(S^{2n−1}), ℝ) , 0 < 2j ≤ n ,

which are related to higher analytic torsion. In fact Diff(S^{2n−1}) is a smooth manifold, and we will use differentiable cochains. For more details on continuous cohomology we refer to Fuks [5].

Let Met_{1}(S^{2n−1}) be the manifold of all unit volume Riemannian metrics on S^{2n−1}. We consider the trivial fibre bundle E := Met_{1}(S^{2n−1}) × S^{2n−1} → Met_{1}(S^{2n−1}) with the obvious horizontal distribution T^{H}E. The bundle E has a tautological vertical Riemannian metric g^{v}. Let T (E, T^{H}E, g^{v}) ∈ Ω^{ev}(Met_{1}(S^{2n−1})) be the higher analytic torsion form of Bismut and Lott. The following Lemma implies that T (E, T^{H}E, g^{v}) is closed.

Lemma 3.1 Let E → B be a smooth fibre bundle with fibre S^{2n−1}, T^{H}E be a horizontal distribution and g^{v} be a vertical Riemannian metric. If the volume of the fibres is a constant function on B, then the higher analytic torsion form T (E, T^{H}E, g^{v}) is closed.

Proof. We cannot refer to Bismut and Lott [2], Cor. 3.26, since H^{∗}(S^{2n−1}, ℝ) is non-trivial in dimensions 0 and 2n − 1. But we can employ [2], Thm. 3.23.

Let H^{∗} → B be the bundle of fibrewise harmonic forms. By Hodge theory this bundle can be identified with the bundle of fibrewise cohomology and thus acquires a natural flat connection ∇^{H^{∗}}. As a subbundle of a bundle of Hilbert spaces it also comes equipped with a natural hermitian metric h^{H^{∗}}. Thus we can consider the characteristic form f(∇^{H^{∗}}, h^{H^{∗}}) ∈ Ω^{odd}(B) introduced in [2], Def. 1.7, where f(x) = xe^{x^{2}}. Since S^{2n−1} is odd dimensional the first term of the r.h.s. of [2], Thm. 3.23 (3.119), vanishes and we have

dT (E, T^{H}E, g^{v}) = −f(∇^{H^{∗}}, h^{H^{∗}}) .

We now show that f(∇^{H^{∗}}, h^{H^{∗}}) = 0. Note that H^{∗} is nontrivial for ∗ = 0 and ∗ = 2n − 1. In both cases we have parallel sections given by the constant function and the fibrewise volume form, respectively. The length of these sections at b ∈ B w.r.t. h^{H^{∗}} is just the volume of the fibre of E over b. Thus H^{∗} is trivialized by sections of constant length and f(∇^{H^{∗}}, h^{H^{∗}}) = 0. □

The action of the group Diff(S^{2n−1}) on S^{2n−1} induces an action on Met_{1}(S^{2n−1}) by (f, g) ↦ f(g) := (f^{−1})^{∗}g, f ∈ Diff(S^{2n−1}), g ∈ Met_{1}(S^{2n−1}). We consider the diagonal action of Diff(S^{2n−1}) on Met_{1}(S^{2n−1})×S^{2n−1} so that E becomes a Diff(S^{2n−1})-equivariant bundle. The additional structures T^{H}E and g^{v} are Diff(S^{2n−1})-invariant. We conclude that the torsion form T (E, T^{H}E, g^{v}) is invariant, too.

For k ∈ ℕ we construct a real valued continuous group cocycle c_{T,k} ∈ C_{c}^{k}(Diff(S^{2n−1}), ℝ) of Diff(S^{2n−1}). The construction depends on the choice of a base point g_{0} ∈ Met_{1}(S^{2n−1}). Let q : Met(S^{2n−1}) → Met_{1}(S^{2n−1}) be the projection from the manifold of all Riemannian metrics to the manifold of volume one metrics given by

q(g) := \frac{g}{vol_{g}(S^{2n−1})^{\frac{1}{2n−1}}} .

The projection q is Diff(S^{2n−1})-equivariant.

If f_{0}, …, f_{k} ∈ Diff(S^{2n−1}), then we define the map s_{k}(f_{0}, …, f_{k}) : Δ^{k} → Met_{1}(S^{2n−1}) by

s_{k}(f_{0}, …, f_{k}) := q (∑_{j=0}^{k} t_{j}f_{j}(g_{0})) ,

where t_{j} are barycentric coordinates of the k-dimensional standard simplex Δ^{k}. Then we define

c_{T,k}(f_{0}, …, f_{k}) := ∫_{Δ^{k}} s_{k}(f_{0}, …, f_{k})^{∗}T (E, T^{H}E, g^{v}) .  (7)

Lemma 3.2 c_{T,k} is a Diff(S^{2n−1})-invariant continuous (differentiable) group cocycle of Diff(S^{2n−1}), and the cohomology class T_{k} ∈ H_{c}^{k}(Diff(S^{2n−1}), ℝ) represented by c_{T,k} is independent of the choice of the base point g_{0}.

Proof. Let f ∈ Diff(S^{2n−1}). Then using the invariance of T (E, T^{H}E, g^{v})

(fc_{T,k})(f_{0}, …, f_{k}) = c_{T,k}(f^{−1}f_{0}, …, f^{−1}f_{k})
= ∫_{Δ^{k}} s_{k}(f^{−1}f_{0}, …, f^{−1}f_{k})^{∗}T (E, T^{H}E, g^{v})
= ∫_{Δ^{k}}(f^{−1} ∘ s_{k})(f_{0}, …, f_{k})^{∗}T (E, T^{H}E, g^{v})
= ∫_{Δ^{k}} s_{k}(f_{0}, …, f_{k})^{∗}T (E, T^{H}E, g^{v})
= c_{T,k}(f_{0}, …, f_{k}) .

3 CONSTRUCTION OF CONTINUOUS COHOMOLOGY CLASSES OF DIFF(S^{2N−1})9 Using Stokes Lemma and the fact that T (E, T^{H}E, g^{v}) is closed we obtain

(dc_{T,k})(f_{0}, …, f_{k+1}) = ∑_{i=0}^{k+1}(−1)^{i}c_{T,k}(f_{0}, …, f̂_{i}, …, f_{k+1})
= ∑_{i=0}^{k+1}(−1)^{i} ∫_{Δ^{i}} s_{k}(f_{0}, …, f̂_{i}, …, f_{i+1})^{∗}T (E, T^{H}E, g^{v})
= ∫_{∂Δ^{p+1}} s_{k+1}(f_{0}, …, f_{p+1})^{∗}T (E, T^{H}E, g^{v})
= ∫_{Δ^{k+1}} d(s_{k+1}(f_{0}, …, f_{k+1})^{∗}T (E, T^{H}E, g^{v}))
= ∫_{Δ^{k+1}} s_{k+1}(f_{0}, …, f_{k+1})^{∗}dT (E, T^{H}E, g^{v})
= 0 .
□

Let g′_{0} ∈ Met_{1}(S^{2n−1}) be another base point, and let c′_{T,k} be defined using g′_{0}. Then we define a Diff(S^{2n−1})-invariant chain u ∈ C_{c}^{k−1}(Diff(S^{2n−1}), ℝ) by

u(f_{0}, …, f_{k−1}) = ∫_{Δ^{k−1}×I} U_{k−1}(f_{0}, …, f_{k−1})^{∗}T (E, T^{H}E, g^{v}) ,

where U_{k−1}(f_{0}, …, f_{k−1}) : Δ^{p−1} × I → Met_{1}(S^{2n−1}) is given by

U_{k−1}(f_{0}, …, f_{k−1})(t, s) := q(∑_{i=0}^{k−1} t_{i}(sf_{i}(g_{0}) + (1 − s)f_{i}(g′_{0}))) .

Now we have

c′_{T,k}(f_{0}, …, f_{k}) − c_{T,k}(f_{0}, …, f_{k}) − du(f_{0}, …, f_{k})
= ∫_{∂(Δ^{k}×I)} U_{k}(f_{0}, …, f_{k})^{∗}T (E, T^{H}E, g^{v})
= ∫_{Δ^{k}×I} U_{k}(f_{0}, …, f_{k})^{∗}dT (E, T^{H}E, g^{v})
= 0 .
□

Definition 3.3 We define T_{k} ∈ H_{c}^{j}(Diff(S^{2n−1}), ℝ) to be the cohomology class represented by c_{T,k}, k ∈ ℕ. Furthermore, we set T := T_{2} + T_{4} + … ∈ H_{c}^{ev}(Diff(S^{2n−1}), ℝ). Note that T_{2k−1} = 0 for all k ∈ ℕ.

4 Nontriviality of the continuous cohomology classes

We now consider S^{2n−1} as the boundary at infinity of the complex hyperbolic space ℂH^{n}. The isometry group SU(n, 1) of ℂH^{n} acts on S^{2n−1}. We choose any discrete torsion-free cocompact subgroup Γ ⊂ SU(n, 1). It is a well known theorem of A. Borel that such subgroups exist. The group Γ acts freely on the contractible space ℂH^{n} and the quotient B := Γ\ℂH^{n} is a model for the classifying space BΓ. In particular, there is a natural identification of H^{∗}(Γ, ℝ) with H^{∗}(B, ℝ).

Proposition 4.1 Under the above identification of H^{∗}(Γ, ℝ) with H^{∗}(B, ℝ) the restriction of T to Γ is equal to T (SB).

Proof. We first make the identification H^{∗}(Γ, ℝ) ≅ H^{∗}(B, ℝ) explicit. Let T ∈ Ω^{j}(B) be a closed form representing a class [T] ∈ H^{j}(B, ℝ). We will construct a group cocycle C_{T,j} ∈ C^{j}(Γ, ℝ) which represents the corresponding cohomology class in H^{j}(Γ, ℝ).

Let x_{0}, …, x_{j} be points in ℂH^{n}, 0 ≤ j ≤ 2n. We want to construct a map σ_{j}(x_{0}, …, x_{j}) : Δ^{j} → ℂH^{n} which sends the vertices of Δ^{j} to the points x_{0}, … x_{j}, and such that for all f ∈ SU(n, 1) we have σ(f(x_{0}), …, f(x_{j})) = f(σ(x_{0}, …, x_{j})). Assume that we already have constructed σ_{l} for all l < j. Since ℂH^{n} has negative sectional curvature we can consider the center of mass m(x_{0}, …, x_{j}) in which the function ℂH^{n} ∋ x ↦ ∑_{i=0}^{j} dist(x, x_{i})^{2} assumes it minimum. For any pair of points y, z ∈ ℂH^{n} let γ(y, z, .) : [0, 1] → ℂH^{n} be the unique geodesic from y to z. If t ∈ Δ^{j} we let u(t) ∈ {0, …, j} be the smallest index such that t_{u(t)} ≤ t_{i} for all i ∈ {0, 1, …, j}. We define

σ_{j}(x_{0}, …, x_{j})(t)

:= γ(m(x_{0}, …, x_{j}), σ_{j−1}(x_{0}, …, x̂_{u(t)}, …, x_{j})(\frac{t_{0}}{1 − t_{u(t)}} , …, ; \frac{t_{u(t)}}{1 − t_{u(t)}} , …, \frac{t_{j}}{1 − t_{u(t)}} ), (j + 1)t_{u(t)}) .

We choose a base point x ∈ ℂH^{n}. The cocycle C_{T,j} can now be written as

C_{T,j}(f_{0}, …, f_{j}) := ∫_{Δ^{j}} σ_{j−1}(f_{0}(x), …, f_{j}(x))^{∗}T̃, f_{0}, …, f_{j} ∈ Γ ,

where T̃ ∈ Ω^{j}(ℂH^{n}) denotes the lift of T from B to ℂH^{n}.

The subgroup SU(n) ⊂ SU(n, 1) is a maximal compact subgroup. Let g_{0} be the volume one metric on S^{2n−1} which is fixed by SU(n). We identify the orbit SU(n, 1)(g_{0}) ⊂ Met_{1}(S^{2n−1}) with the symmetric space SU(n, 1)/SU(n) = ℂH^{n} and thus obtain an inclusion ℂH^{n} ↪ Met_{1}(S^{2n−1}) which is SU(n, 1)-equivariant.

The restriction of T (E, T^{H}E, g^{v}) to ℂH^{n} is a SU(n, 1)-invariant closed form T̃. It induces a closed form T ∈ Ω^{∗}(B). Let C_{T,k} denote the corresponding group cocycle defined above using the base point g_{0} ∈ ℂH^{n} and the degree k-component of T. By c_{T,k}^{Γ} we denote the restriction to Γ of the cocycle c_{T,k} defined in (7) using the same base point g_{0}.

Lemma 4.2 The group cocycles c_{T,k}^{Γ} and C_{T,k} of Γ represent the same cohomology class in H^{k}(Γ, ℝ).

Proof. We define a Γ-invariant k − 1-cochain b ∈ C^{k−1}(Γ, ℝ) such that db = C_{T,k} − c_{T,k}^{Γ}. Let f_{0}, …, f_{k−1} ∈ Γ. We define a map a_{k−1}(f_{0}, …, f_{k−1}) : I × Δ^{k−1} → Met_{1}(S^{2n−1}) by

a_{k−1}(f_{0}, …, f_{k−1})(r, t) := q [rσ_{k−1}(f_{0}(g_{0}), …, f_{k−1}(g_{0}))(t) + (1 − r)s_{k−1}(f_{0}, …, f_{k−1})(t)] .

Then we put

b(f_{0}, …, f_{k−1}) := ∫_{I×Δ^{k−1}} a_{k−1}(f_{0}, …, f_{k−1})^{∗}T (E, T^{H}E, g^{v}) .

Now we compute

c_{T,k}^{Γ}(f_{0}, …, f_{k}) − C_{T,k}(f_{0}, …, f_{k}) − db(f_{0}, …, f_{k})
= ∫_{∂(I×Δ^{k})} a_{k}(f_{0}, …, f_{k})^{∗}T (E, T^{H}E, g^{v})
= ∫_{I×Δ^{k}} a_{k}(f_{0}, …, f_{k})^{∗}dT (E, T^{H}E, g^{v})
= 0 .
□

The restriction of E to ℂH^{n} is a trivial SU(n, 1)-equivariant fibre bundle over ℂH^{n} with S^{2n−1} as fibre equipped with a horizontal distribution and a vertical Riemannian metric. The quotient by Γ is a fibre bundle Ē → B with fibres S^{2n−1}, horizontal distribution T^{H} Ē and vertical Riemannian metric g^{v}. Since the higher analytic torsion form is functorial with respect to pull-back we have T = T (Ē, T^{H} Ē, g^{v}).

In order to finish the proof of the proposition it suffices to prove the equality of cohomology classes [T] = T (SB). In fact, by Lemma 4.2 the class [T] represents the restriction of T to Γ under the identification H^{∗}(Γ, ℝ) ≅ H^{∗}(B, ℝ).

We construct a diffeomorphism of fibre bundles F : SB → Ē. Since SB = Γ\SℂH^{n}, where SℂH^{n} is the sphere bundle of ℂH^{n}, and Ē = Γ\(ℂH^{n} × S^{2n−1}) w.r.t. the diagonal action it suffices to produce a Γ-equivariant diffeomorphism F̃ : SℂH^{n} → ℂH^{n} × S^{2n−1}. Let v ∈ SℂH^{n}. Then v defines a geodesic ray from the base point x ∈ ℂH^{n} of v which hits the sphere at infinity S^{2n−1} in a point b. We define F̃(v) := (x, b).

On SB we have two sets of additional structures. On the one hand we have the horizontal distribution T^{H}SB which is induced from the Levi-Civita connection and the vertical Riemannian metric g_{1}^{v} induced by the locally symmetric Kähler metric of B normalized such that the fibres have unit volume. On the other hand we have the locally integrable horizontal distribution F^{−1}(T^{H} Ē) and the vertical Riemannian metric F^{∗}g^{v} induced from Ē.

In order to prove the proposition it remains to prove the following lemma which assures that the class T (SB) is well defined independently of the choice of the additional structures.

Lemma 4.3 Let E → B be any smooth fibre bundle with fibre S^{2n−1} and (T^{H}E, g^{v}) and ((T^{H}E)′, (g^{v})′) two sets of additional structures such that the fibres have volume one with respect to both, g^{v} and (g^{v})′. Then we have the equality of cohomology classes

[T (E, T^{H}E, g^{v})] = [T (E, (T^{H}E)′, (g^{v})′)] .

Proof. We can connect T^{H}E with (T^{H}E)′ by a smooth path of horizontal distributions and g^{v} with (g^{v})′ by a smooth path of Riemannian metrics which induce volume one on the fibres. We thus obtain a horizontal distribution and a vertical Riemannian metric on [0, 1] × E → [0, 1] × B which restrict to the given ones on the boundaries. The associated higher analytic torsion form is a closed form (by Lemma 3.1) on [0, 1] × B which restricts to T (E, T^{H}E, g^{v}) and T (E, (T^{H}E)′, (g^{v})′) at the boundaries. This observation implies the Lemma. □

As we already observed in the introduction that by Theorem 1.2 the class T (SB)_{4k} is a non-zero multiple of ch_{4k}(T B) for k ∈ ℕ and ch_{4k}(T Y) ≠ 0 as long as 0 ≤ 2k ≤ n. Hence, by Proposition 4.1 the classes T_{4k} ∈ H_{c}^{4k}(Diff(S^{2n−1}), ℝ) are nontrivial for 0 < 2k ≤ n. This finishes the proof of Theorem 1.3. □

\underline{Remarks:}

• The restrictions of the classes T_{4k−2} to Γ vanish.

• One can produce non-trivial classes in H_{c}^{4k−2}(Diff(S^{1})_{0}, ℝ), k ∈ ℕ, using non-trivial local systems (see e.g. [4]).

References

[1] J. M. Bismut and S. Goette. Families torsion and Morse functions. Preprint 2000.

[2] J. M. Bismut and J. Lott. Flat vector bundles, direct images, and higher real analytic torsion. J. AMS, 8 (1995), 291–363.

[3] U. Bunke. Equivariant higher analytic torsion and equivariant Euler characteristic. American J. of Math, 122 (2000), 377-401.

[4] U. Bunke. Higher analytic torsion and cohomology of diffeomorphism groups. Preprint available at math.DG/9712101.

[5] D.B.Fuks. Cohomology of infinite-dimensional Lie algebras. Nauka, Moscow, 1984.

[6] W. Dwyer, M. Weiss, and B. Williams. A parametrized index theorem for algebraic K-theory Euler characteristic. Preprint, 1995.

[7] S. Goette. Morse theory and higher torsion invariants. Preprint, 2001.

[8] K. Igusa Higher Franz-Reidemeister Torsion Manuscript, 2001

[9] K. Igusa and J. R. Klein. The Borel regulator map on pictures, II. An example from Morse theory. K-theory, 7 (1993), 225–267.

[10] J. Lott. Equivariant analytic torsion for compact Lie group actions. J. Funct. Anal., 125 (1994), 438–451.

.........................

Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany, bunke@uni-math.gwdg.de

AMS 2000 Math. Subject Classification : 58D05; 58G10