Equivariant torsion and G-CW-complexes

Ulrich Bunke^{∗} November 14, 1997

Contents

1 Introduction 1 2 Restriction of simple structures 8 3 Equivariant torsion 11 4 Computations 17 5 Reduction to Cartan subgroups 21

∗Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, GERMANY, Email:bunke@uni-math.gwdg.de

1 Introduction

1.1

In this note we consider equivariant Reidemeister and analytic torsion invariants of closed oriented G-manifolds, where G is any compact Lie group.

Equivariant analytic torsion for closed oriented odd-dimensional G-manifolds for arbitrary compact Lie groups G and equivariant Reidemeister torsion of closed oriented G-manifolds for finite G were introduced in [7] and further studied in [10] and [1]. In the present paper we generalize the definition of equivariant Reidemeister torsion to general compact Lie groups G and address the question of equality with equivariant analytic torsion.

If G is finite, then equivariant Reidemeister torsion is in fact an invariant of Gequivariant locally constant sheaves F of finite-dimensional Hilbert spaces over G-CW-complexes. We extend this invariant to general compact Lie groups G.

If M is a closed G-manifold (G a compact Lie group), then there is a natural equivalence class of G-homotopy equivalences f : X → M called simple structure (see section 2.2 for details), where X is a G-CW-complex. If F is a G-equivariant locally constant sheaf F of finite-dimensional Hilbert spaces, then the equivariant Reidemeister torsion of (M, F) is defined using (X, f^{∗}F), and it is independent of choices.

Viewing a G-CW complex X as a filtered G-space, we express the equivariant Reidemeister torsion of (X, F) in terms of the equivariant Reidemeister torsion of the restriction of F to the G-cells and a contribution of the spectral sequence induced by the filtration. There is a clear separation into an invariant which only depends on restrictions of F to the G-cells, and an invariant which depends on the way the cells are glued together. If G is connected, then the latter invariant is trivial.

We compute the equivariant Reidemeister torsion and the equivariant analytic torsion in terms of contributions of G-cells. The contribution of a G-cell can be further evaluated by restricting to one-dimensional subgroups of G. In particular we compute the equivariant Reidemeister torsion and the equivariant analytic torsion of compact symmetric spaces by topological means and recover the result of [6]

1.2

Let G be a compact Lie group, and M be a closed odd-dimensional oriented G-manifold. Let F → M be a G-equivariant flat hermitean vector bundle and F be the associated G-equivariant locally constant sheaf of finite-dimensional Hilbert spaces.

If we choose a G-equivariant Riemannian metric g^{M}, then one defines the equivariant analytic torsion ρ_{an}(M, g^{M}, F) : G → C (see [7], §X) as a spectral invariant of the Laplace operator Δ_{g^{M}} acting on F -valued forms:

ρ_{an}(M, g^{M}, F)(g) := \frac{d}{ds}_{|s=0} \frac{1}{Γ(s)} ∫_{0}^{∞}(Tr_{s}Nge^{−tΔ_{g^{M}}} − χ′(M, F)(g))t^{s−1}dt ,

where χ′(M, F) := ∑_{i=0}^{∞}(−1)^{i}iTrg_{|H^{i}(M,F)}, N denotes the Z-grading of the bundle of F -valued forms, and the integral converges for Re(s) >> 0 and has a meromorphic continuation to all of C.

By definition ρ_{an}(M, g^{M}, F) is a class function on G. If F is acyclic, i.e. H^{∗}(M, F) = 0, then ρ_{an}(M, g^{M}, F) is independent of g^{M}, and we write ρ_{an}(M, g^{M}, F) =: ρ_{an}(M, F).

1.3

Let G be finite, let M be a closed oriented G-manifold, and let F be a G-equivariant locally constant sheaf of finite-dimensional Hilbert spaces. We choose a G-Hilbert module structure on H^{∗}(M, F). Using a smooth G-equivariant triangulation of M one can define equivariant Reidemeister torsion ρ(M, F) : G → C which is again a class function on G (see [7], Sec. 5). Equivalently, there is a natural simple structure f : X → M and we can define ρ(M, F) := ρ(X, f^{∗}F). We present the details of this definition in Section 3.

Assume that M is odd-dimensional. If F is acyclic, then by [7], Prop.16, we have the equality of class functions

ρ(M, F) = ρ_{an}(M, F) .  (1)

In case that F is not acyclic we fix a G-invariant Riemannian metric g^{M}. Let H(M, g^{M}, F) denote the space of harmonic F -valued forms. Then H(M, g^{M}, F) is a G-Hilbert module. On H^{∗}(M, F) we choose the G-Hilbert module structure such that the de Rham isomorphism H(M, g^{M}, F) →^{∼} H^{∗}(M, F) becomes an isometry. Again by [7], Prop.16, we have the equality

ρ(M, F) = ρ_{an}(M, g^{M}, F)  (2)

of class functions.

1.4

Let G be any compact Lie group, M be a closed oriented G-manifold, and F → M be a G-equivariant flat hermitean vector bundle. We fix a G-Hilbert module structure on H^{∗}(M, F).

In the present subsection we define equivariant Reidemeister torsion of (M, F) and discuss the validity of (1) and (2).

Let F G := {g ∈ G | (∃n ∈ N | g^{n} = 1)} denote the set of elements of finite order. F G is a dense subset of G which is invariant under conjugation. Let C(F G) := {f : F G → C | f(g^{h}) = f(g), ∀g ∈ F G, h ∈ G} denote the space of all real-valued functions on F G, which are invariant under conjugation, where g^{h} := hgh^{−1}.

Let Or_{f}(G) denote the full subcategory of the orbit category (see [8], 8.16) consisting of all homogeneous spaces G/Γ, where Γ ⊂ G is finite.

We have a contravariant functor C : Or_{f}(G) → C − vect associating to the object G/Γ the space of class functions C(Γ) on Γ. If f : G/Γ → G/Γ′ is a morphism in Or_{f}(G), then there is a g ∈ G such that {gγg^{−1} | γ ∈ Γ} = Γ^{g} ⊂ Γ′ and f(hΓ) = hg^{−1}Γ′. The functor C associates to f the map C(f) : C(Γ′) → C(Γ^{g}) → C(Γ), where the first arrow res_{Γ^{g}}^{Γ′} : C(Γ′) → C(Γ^{g}) is the restriction of class functions and the second is induced by the map Γ → Γ^{g}, γ ↦ gγg^{−1}.

There is a natural bijection

C(F G) →^{∼} lim_{←^{Or_{f} (G)}} C(Γ) ,  (3)

which is induced by the restrictions res_{Γ}^{G} : C(F G) → C(Γ), Γ ⊂ G finite.

For Γ ⊂ G let res_{Γ}^{G}M denote the Γ-manifold obtained from M by restricting the G-action to Γ. If V is a G-module, then let res_{Γ}^{G}V denote the Γ-module obtained by restriction. Note that H^{∗}(res_{Γ}^{G}M, F) = res_{Γ}^{G}H^{∗}(M, F) canonically, and the latter has a natural Γ-Hilbert module structure.

If G/Γ ∈ Or_{f}(G), then ρ(res_{Γ}^{G}M, F) ∈ C(Γ) is well defined by 1.3. By Proposition 3.5 the collection {ρ(res_{Γ}^{G}M, F)}_{G/Γ∈Or_{f} (G)} is a section of the functor C and thus defines an element

ρ̃(M, F) ∈ lim_{←^{Or_{f} (G)}} C(Γ) .

Definition 1.1 Let ρ(M, F) ∈ C(F G) be the element which corresponds to ρ̃(M, F) under the bijection (3).

With this definition equality of equivariant Reidemeister and analytic torsion for arbitrary Lie groups becomes a formal consequence of the corresponding result for finite groups.

Lemma 1.2 Let G be a compact Lie group, M be a closed odd-dimensional oriented G manifold, g^{M} be a G-invariant Riemannian metric and F → M be a G-equivariant flat hermitean vector bundle. We equip H^{∗}(M, F) with the G-Hilbert module structure such that the de Rham isomorphism H^{∗}(M, g^{M}, F) ≅ H^{∗}(M, F) becomes an isometry. Then

ρ(M, F) = ρ_{an}(M, g^{M}, F)_{|F G} .

Proof.Let g ∈ F G and G/Γ ∈ Or_{f}(G) be such that g ∈ Γ. Then obviously ρ_{an}(M, g^{M}, F)(g) = ρ_{an}(res_{Γ}^{G}M, g^{M}, F)(g). By (2) we have

ρ(M, F)(g) =^{def} ρ(res_{Γ}^{G}M, F)(g) = ρ_{an}(res_{Γ}^{G}M, g^{M}, F)(g) = ρ_{an}(M, g^{M}, F)(g) .

□

1.5

Note that equivariant Reidemeister torsion depends on the choice of a G-Hilbert module structure on the cohomology H^{∗}(M, F). In the present subsection we show how one can define an invariant that is independent of this choice.

Let G^{0} ⊂ G denote the component of the identity of G. Then we have an exact sequence

0 → G^{0} → G →^{q} π_{0}(G) → 0 .

Note that the restriction q_{|F G} → π_{0}(G) is still surjective. Thus the pull-back in the sequence below defining Ĉ(F G) is injective.

0 → C(π_{0}(G)) →^{q^{∗}} C(F G) → Ĉ(F G) → 0

For any G/Γ ∈ Or_{f}(G) let Γ^{0} := Γ ∩ G^{0}. Then we have exact sequences

0 → Γ^{0} → Γ →^{q} Γ/Γ^{0} → 0

0 → C(Γ/Γ^{0}) →^{q^{∗}} C(Γ) → Ĉ(Γ) → 0 ,

where Ĉ(Γ) is defined by the second sequence.

We can consider the functor Ĉ : Or_{f}(G) → C − vect, which associates to G/Γ ∈ Or_{f}(G) the space Ĉ(Γ). If f : G/Γ → G/Γ′ is a morphism in Or_{f}(G), then the map Ĉ(f) : Ĉ(G/Γ′) → Ĉ(G/Γ) is represented by C(f) : C(Γ) → C(Γ′) which maps C(Γ/Γ^{0}) to C(Γ′/(Γ′)^{0}).

Since the natural map C(π_{0}(G)) → lim_{←^{Or_{f} (G)}} C(Γ/Γ^{0}) is an isomorphism we conclude from

0 →  C(π_{0}(G))  →  C(F G)  →  Ĉ(F G)  →  0
↓ ∥  ↓ ∥  ↓ Î
0 → lim_{←^{Or_{f} (G)}} C(Γ/Γ^{0}) → lim_{←^{Or_{f} (G)}} C(Γ) → lim_{←^{Or_{f} (G)}} Ĉ(Γ) → lim1 ; ←^{Or_{f} (G)} C(Γ/Γ^{0})

that Î is injective.

Note that G^{0} acts trivially on H^{∗}(M, F). Thus if G/Γ ∈ Or_{f}(G), then Γ^{0} acts trivially on H^{∗}(res_{Γ}^{G}M, F), too. By Lemma 3.1 the class ρ̂(res_{Γ}^{G}M, F) ∈ Ĉ(Γ) of ρ(res_{Γ}^{G}M, F) ∈ C(Γ) is independent of the choice of a Γ-Hilbert module structure on H^{∗}(res_{Γ}^{G}M, F).

The collection {ρ̂(res_{Γ}^{G}M, F)}_{G/Γ∈Or_{f} (G)} defines a section of the functor Ĉ and therefore an element

˜ρ̂(M, F) ∈ lim_{←^{Or_{f} (G)}} Ĉ(Γ) .

It is easy to see that ˜ ρ̂(M, F) is in the range of Î.

Definition 1.3 Let ρ̂(M, F) ∈ Ĉ(F G) be the unique element such that Î(ρ̂(M, F)) = ˜ρ̂(M, F).

The class ρ̂_{an}(M, g^{M}, F) ∈ Ĉ(F G) of ρ_{an}(M, g^{M}, F)_{|F G} is independent of the choice of the G-invariant Riemannian metric g^{M} (see [7], §X). We thus write ρ̂_{an}(M, F) := ρ̂_{an}(M, g^{M}, F). The proof of the following Lemma is similar to that of Lemma 1.2.

Lemma 1.4 Let G be a compact Lie group, M be a closed odd-dimensional G manifold, and F → M be a G-equivariant flat hermitean vector bundle. Then

ρ̂(M, F) = ρ̂_{an}(M, F) .  (4)

1.6

It is natural to ask what kind of differential-topological information about the G-manifold M and the flat bundle F is encoded in the invariants ρ(M, F) and ρ̂(M, F). While ρ(M, F) contains global information about M it turns out that ρ̂(M, F) only depends on the type of G-cells of X and the restriction of f^{∗}F to the cells, where f : X → M represents the prefered simple structure of the smooth closed G-manifold M. In particular it is independent of the way the cells are glued together.

We now formulate the result in detail. A G-space G/H × D^{n} is called a n-dimensional G-cell of type H, where H ⊂ G is a closed subgroup. Let X be a finite G-CW-complex and F be a G-equivariant locally constant sheaf of finite-dimensional Hilbert spaces over X. For any G-cell E = G/H_{E} × D^{n} ↪ X of dimension dim(E) := n let F_{E} → G/H_{E} denote the restriction of F to G/H_{E} × {0}. Then ρ̂(G/H_{E}, F_{E}) ∈ Ĉ(F G) is defined.

Proposition 1.5 (Corollary 4.4) Let X be a finite G-CW-complex, and F be a Gequivariant locally constant sheaf of finite-dimensional Hilbert spaces over X. Then we have

ρ̂(X, F) = ∑(−1)^{dim(E)}ρ̂(G/H_{E}, F_{E}) ,

where the sum is taken over all G-cells of X.

1.7

The equivariant torsion ρ̂(M, F) is determined by its restrictions to all Cartan subgroups T of G. If we apply Proposition 1.5 to res_{T}^{G}M, then we can compute the equivariant torsion of M in terms of the T -cells. In Section 5.2 we study the equivariant torsion of a T -cell. It vanishes iff the isotropy group has codimension zero or greater than one, and it is explicitly computable, if the isotropy group has codimension one.

Let f : X → res_{T}^{G}M represent the prefered simple structure. Consider t ∈ T. Let I be the collection of T -cells E ≅ T/S_{E} × D^{n_{E}} of X with dim(T/S_{E}) = 1. For E ∈ I let J_{E}(t) be the collection of connected components E_{i}, i ∈ J_{E}(t), of E (note that E_{i} ≅ S^{1} × D^{n_{E}}), such that tE_{i} = E_{i}. Let m = dim(F) and U_{i} ∈ U(m) be the holonomy of f^{∗}F_{|E_{i}}. Then U_{i} is determined uniquely by the choice of a base point o_{i} ∈ E_{i}, identification of the fibre F_{o_{i}} with C^{m}, and the choice of an orientation of E_{i}. The matrix U_{i} can be written as e^{2πıa_{i}} for a selfadjoint a_{i} ∈ Mat(m, C). The element t acts as rotation of the circle-part of E_{i} by the angle 2πτ_{i}. Moreover there are unitary isomorphisms λ_{i} of F_{o_{i}} given by the action of t composed with parallel transport back to o_{i} in direction opposite to the orientation. Note that λ_{i} and a_{i} commute. Then

Lemma 1.6 The equivariant torsion ρ̂(res_{T}^{G}M, F) is given by the class of

T ∋ t ↦ ∑_{E∈I,i∈I_{E}}(−1)^{n_{E}}Tr ψ(λ_{i}, a_{i}, τ_{i}) ,

where an explicit formula for ψ is given in 5.4.

For the purpose of illustration in 5.6 we compute the equivariant Reidemeister torsion of odd-dimensional symmetric spaces. Employing (4) we essentially recover the results of [6] about equivariant analytic torsion of symmetric spaces.

2 Restriction of simple structures

In this section we recall some basic results in equivariant topology.

2.1

The G-space G/H × D^{n} is called a n-dimensional G-cell of type H, where H ⊂ G is a closed subgroup. A finite relative G-CW-complex is a pair of G-spaces (X, A) together with a finite filtration by G-spaces

A = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{N} = X, ∪_{n=−1}^{N}X_{n} = X ,

and a collection of G-subspaces e_{i}^{n} ⊂ X_{n}, i ∈ I_{n}, n ≥ 0, ♯I_{n} < ∞, with the following properties

(1) : A/G is Hausdorff

(2) : X has the weak topology with respect to the filtration {X_{n}}

(3) : for n ≥ 0 there are G-push outs

\begin{matrix}⊔_{i∈I_{n}}G/H_{i} × S^{n−1} & ⟶^{⊔_{i}q_{i}} & X_{n−1} \\ ↓ & & ↓ \\ ⊔_{i∈I_{n}}G/H_{i} × D^{n} & ⟶^{⊔_{i}Q_{i}} & X_{n} \end{matrix} ,

such that e_{i}^{n} = Q_{i}(G/H_{i} × int D^{n}) (see [8] Def. 1.2.).

2.2

If X is a G-space, then a simple structure on X is given by a pair (Z, f), where Z is a finite G-CW-complex and f : Z → X is a G-homotopy equivalence. A second pair (Z′, f′) defines the same simple structure on X, if f′_{∗}τ^{G}((f′)^{−1} ∘ f) = 0 holds true in the Whitehead group W h^{G}(X), where (f′)^{−1} denotes any homotopy inverse of f′ (see [8], 4.27).

Let

\begin{aligned}X_{0} & → & X_{1} \\ ↓ & & ↓ \\ X_{2} & → & X \end{aligned}

be a push-out of G-spaces and each of the X_{i}, i = 0, 1, 2 be equipped with a simple structure. Then X has a prefered simple structure [8], p75.

2.3

If X is a closed smooth G-manifold (possibly with boundary), then it has a prefered simple structure ([8], 4.36). It is obtained by induction over the number of orbit types. If X has one orbit type, then X/G is a smooth manifold which has a smooth triangulation. Lifting this triangulation to X we obtain a G-CW-decomposition of X representing the prefered simple structure. If X has several orbit types then we write X as a push-out of two G-manifolds with less orbit types. We apply the induction hypothesis and 2.2 in order to obtain the prefered simple structure of X.

2.4

Let G be a compact Lie group and X be a finite G-CW complex. If H ⊂ G is a closed subgroup, then in general there is no natural H-CW structure on res_{H}^{G}X. But we have the following result.

Proposition 2.1 (1) : Let X be a finite G-CW complex. If H ⊂ G is a closed subgroup, then there exists a prefered simple structure f : Z → res_{H}^{G}X.

(2) : Let X be a finite G-CW complex. If K ⊂ H ⊂ G are closed subgroups, f : Z → res_{H}^{G}X and g : Y → res_{K}^{H}Z represent the prefered simple structures given in (1), then f ∘ g : Y → res_{K}^{G}X represents the prefered simple structure, too.

(3) : Let M be a smooth closed G-manifold, and let f : X → M represent the prefered simple structure. Let H ⊂ G be a closed subgroup, and let g : Z → res_{H}^{G}X represent the prefered simple structure. Then f ∘ g : Z → res_{H}^{G}M represents the prefered simple structure of the closed smooth H-manifold res_{H}^{G}M.

(4) : Let X be a finite G-CW complex, H ⊂ G be a closed subgroup, and let f : Z → res_{H}^{G}X represent the prefered simple structure. Let g ∈ G, H^{g} := gHg^{−1}, Z^{g} be the H^{g}-CW complex which is obtained from Z by letting u ∈ H^{g} act by g^{−1}ug, and define f^{g} : Z^{g} → res_{H^{g}}^{G}X by f^{g} = g ∘ f. Then f^{g} : Z^{g} → res_{H^{g}}^{G}X represents the prefered simple structure.

Proof. (1) The homogeneous spaces G/H for all closed subgroups L ⊂ G are smooth H-manifolds. Thus we have prefered simple structures on the G-cells of X considered as H-spaces. Writing X as a push-out over its G-cells and using 2.2 we obtain a prefered simple structure on res_{H}^{G}X. Note that the technical assumptions [8], 7.3 and 7.23 for this procedure are satisfied (see [8], 7.27, see also [3],[4],[5]).

(2) It suffices to show this for the homogeneous spaces G/L. In this case we can apply (3) (Transitivity of the restriction was also announced in [5]).

(3) This is [8], Lemma 7.4.5.

(4) It again suffices to verify this assertion for the homogeneous spaces G/L. In this case we can apply [5], Lemma 1.4. □

3 Equivariant torsion

3.1

Let Γ be a finite group, and let R(Γ) denote the representation ring of Γ with real coefficients. If π is a finite-dimensional representation of Γ, then χ_{π} denotes its character. We have χ_{π} ∈ C(Γ), and the map R(Γ) ∋ π ↦ χ_{π} ∈ C(Γ) induces an isomorphism of C-vector spaces X : R(Γ) ⊗_{R} C ≅ C(Γ).

3.2

Consider a finite group Γ. Let f : V → W be an ismorphism of finite-dimensional Γ-Hilbert modules. If π is an irreducible representation of Γ, then let V (π), W (π) denote the π-isotypic components and f(π) : V (π) → W (π) the induced isomorphism. We define [[f]] :∈ R(Γ) by

[[f]](π) := \frac{1}{2 dim(π)} log | det f(π)^{∗}f(π)| .

Let

C : … → C^{p} →^{c^{p}} C^{p+1} → …

be an acyclic finite cochain complex of finite-dimensional Γ-Hilbert modules. Then there exists a chain contraction κ^{∗} : C^{∗} → C^{∗−1}, and c^{ev} + κ^{ev} : C^{ev} → C^{odd} is an isomorphism, where C^{ev} := ⊕_{k∈Z}C^{2k}, C^{odd} := ⊕_{k∈Z}C^{2k+1}. We define

ρ(C) := X [[c^{ev} + κ^{ev} : C^{ev} → C^{odd}]] .

Note that ρ(C) does not depend on the choice of the chain contraction κ.

Let f : C → D be a chain homotopy equivalence. Then we consider the complex cone(f) with cone(f)^{n} := C^{n} ⊕ D^{n−1} and the differential

( \begin{matrix}c^{n} & 0 \\ f^{n} & −d^{n−1} \end{matrix} ) .

We define t(f) = ρ(cone(f)) ∈ C(Γ). Note that if g : C → D is homotopy equivalent to f, then t(f) = t(g). If g : D → E is a second chain homotopy equivalence, then t(g ∘ f) = t(f) − t(g).

If C is a finite complex of finite-dimensional Γ-Hilbert modules, then we consider the complex H(C) with p’th space H^{p}(C) and trivial differential. If f : C → D is a homotopy equivalence, then we obtain a homotopy equivalence f_{∗} : H(C) → H(D).

Let C, D be finite complexes of finite-dimensional Γ-Hilbert modules with prefered Γ-Hilbert module structures on H^{∗}(C), H^{∗}(D). A homotopy equivalence f : C → D is called simple if t(f) + t(f_{∗}) = 0.

We call two finite complexes of finite-dimensional Γ-Hilbert modules C, D with prefered Γ-Hilbert module structures on H^{∗}(C), H^{∗}(D) equivalent, if there exists a simple homotopy equivalence f : C → D. We write [C] for the equivalence class.

If C is a finite complex of finite-dimensional Γ-Hilbert modules with prefered Γ-Hilbert module structure on H^{∗}(C), and if i : H(C) → C is any embedding such that i_{∗} = id, then we set ρ(C) := −t(i). If C and D are equivalent, then ρ(C) = ρ(D), hence we can write ρ([C]) := ρ(C), where C is any representative of [C].

Let Γ^{0} ⊂ Γ be a subgroup such that Γ^{0} acts trivially on H(C). Let Ĉ(Γ) := C(Γ)/C(Γ/Γ^{0}) and ρ̂(C) be the class of ρ(C) in Ĉ(Γ).

Lemma 3.1 ρ̂(C) is independent of the choice of the Γ-Hilbert module structure on H^{∗}(C).

Proof. Let H_{j}(C), j = 1, 2, be the complex H(C) equipped with two Γ-Hilbert module structures. Let ρ_{j}(C) be the corresponding torsion. Then we have ρ_{2}(C) = ρ_{1}(C) − t(id : H_{2}(C) → H_{1}(C)). It is easy to see that t(id : H_{2}(C) → H_{1}(C)) ∈ C(Γ/Γ^{0}). □

3.3

Let Γ be a finite group and X be a Γ-space of the homotopy type of a finite Γ-CW-complex. Let F be a Γ-equivariant locally constant sheaf of finite-dimensional Hilbert spaces. We fix a Γ-Hilbert module structure on H^{∗}(X, F).

Let (Z, f) represent a simple structure on X. Then we form the cellular cochain complex C(Z, f^{∗}F), which is a finite complex of finite-dimensional Γ-Hilbert modules. We equip H^{∗}(C(Z, f^{∗}F)) with the Γ-Hilbert module structure such that the canonical map f^{∗} : H^{∗}(X, F) → H^{∗}(C(Z, f^{∗}F)) becomes an isometry.

If (Z′, f′) and (Z, f) represent the same simple structure of X, then C(Z, f^{∗}F) and C(Z′, (f′)^{∗}F) are equivalent chain complexes. If X is a Γ-space with distinguished simple structure and Γ-Hilbert module structure on H^{∗}(X, F), then we write [C(X, F)] := [C(Z, f^{∗}F)], where (Z, f^{∗}F) is any representative of the distinguished simple structure of X. We define

ρ(X, F) := ρ([C(X, F)]) ∈ C(Γ) .

Let Γ^{0} ⊂ Γ act trivially on H^{∗}(X, F). Then the class ρ̂(X, F) ∈ Ĉ(Γ) does not depend on the choice of the Γ-Hilbert module structure on H^{∗}(X, F).

3.4

Let Γ be a finite group and Γ′ ⊂ Γ. If Z is a Γ-CW complex, then since Γ is finite res_{Γ′}^{Γ}Z carries a natural Γ′-CW structure. Moreover id : res_{Γ′}^{Γ}Z → res_{Γ′}^{Γ}Z represents the prefered simple structure given in Proposition 2.1 (1).

Let X be a Γ-space of the homotopy type of a finite Γ-CW complex. If (Z, f) represents a simple structure for X, then (res_{Γ′}^{Γ}Z, f) represents a simple structure of res_{Γ′}^{Γ}X.

We choose a Γ-Hilbert module structure on H^{∗}(X, F) which we also use for H^{∗}(res_{Γ′}^{Γ}X, F) = res_{Γ′}^{Γ}H^{∗}(X, F).

Lemma 3.2

res_{Γ′}^{Γ}ρ(X, F) = ρ(res_{Γ′}^{Γ}X, F) .

Proof. We use (res_{Γ′}^{Γ}Z, f) to represent the prefered simple structure of res_{Γ′}^{Γ}X. Then

C(res_{Γ′}^{Γ}Z, f^{∗}F) = res_{Γ′}^{Γ}C(Z, f^{∗}F)
[C(res_{Γ′}^{Γ}Z, f^{∗}F)] = [res_{Γ′}^{Γ}C(Z, f^{∗}F)] .  (5)

Let h : V → W be an isomorphism of finite-dimensional Hilbert-Γ-modules and res_{Γ′}^{Γ}h : res_{Γ′}^{Γ}V → res_{Γ′}^{Γ}W. Then for any irreducible representation τ of Γ′ we have

[[res_{Γ′}^{Γ}h]](τ) = \frac{1}{2 dim(τ)} log | det(res_{Γ′}^{Γ}h(τ)^{∗}res_{Γ′}^{Γ}h(τ)|

= ∑_{π∈Γ̂} \frac{[π : τ]}{2 dim(π)} log | det h^{∗}(π)h(π)|
= ∑_{π∈Γ̂}[π : τ][[h]](π)

Note that res_{Γ′}^{Γ}χ_{π} = ∑_{τ∈Γ̂′}[π : τ]χ_{τ}. Thus

X ([[res_{Γ′}^{Γ}h]]) = ∑_{τ∈Γ̂′}[[res_{Γ′}^{Γ}h]](τ)χ_{τ}  (6)
= ∑_{τ∈Γ̂′} ∑_{π∈Γ̂}[π : τ][[h]](π)χ_{τ}
= ∑_{π∈Γ̂}[[h]](π)res_{Γ′}^{Γ}χ_{π}
= res_{Γ′}^{Γ}X ([[h]]) .

The Lemma now follows from (5) and (6). □

3.5

Let G be a compact Lie group, X be a finite G-CW-complex and F be a G-equivariant locally constant sheaf of finite-dimensional Hilbert spaces. We choose a G-Hilbert module structure on H^{∗}(X, F) which induces a Γ-Hilbert module structure on H^{∗}(res_{Γ}^{G}X, F) = res_{Γ}^{G}H^{∗}(X, F) for all G/Γ ∈ Or_{f}(G).

If G/Γ ∈ Or_{f}(G), then res_{Γ}^{G}X has a prefered simple structure by Proposition 2.1 (1) and ρ(res_{Γ}^{G}X, F) ∈ C(Γ) is defined. If h ∈ G, then let Γ^{h} := hΓh^{−1}, G/Γ^{h} ∈ Or_{f}(G), and set g^{h} := hgh^{−1} for g ∈ G.

Lemma 3.3 If g ∈ Γ and h ∈ G, then ρ(res_{Γ}^{G}X, F)(g) = ρ(res_{Γ^{h}}^{G}X, F)(g^{h}).

Proof. Let (Z, f) represent the prefered simple structure of res_{Γ}^{G}X. Then by Proposition 2.1 (4) (we employ the notation introduced there) the pair (Z^{h}, f^{h}) represents the prefered simple structure of res_{Γ^{h}}^{G}X. We have an isomorphism of complexes of Γ^{h}-Hilbert modules C(Z, f^{∗}F)^{h} = C(Z^{h}, (f^{h})^{∗}F), where the underlying space of C(Z, f^{∗}F)^{h} is C(Z, f^{∗}F) and Γ^{h} acts by g^{h} ↦ g. Similarly we have isomorphisms of complexes Γ^{h}-Hilbert modules H(res_{Γ}^{G}X, F)^{h} = H(res_{Γ^{h}}^{G}X, F). If i, i^{h}, j denote the inclusions i : H(res_{Γ}^{G}X, F) ↪ C(Z, f^{∗}F), i^{h} : H(res_{Γ}^{G}X, F)^{h} ↪ C(Z, f^{∗}F)^{h}, j : H(res_{Γ^{h}}^{G}X, F) ↪ C(Z^{h}, (f^{h})^{∗}F), then we have t(i)(g) = t(i^{h})(g^{h}) = t(j)(g^{h}). This proves the Lemma. □

3.6

We keep the assumptions of 3.5 Then the Lemmas 3.3 and 3.2 imply

Corollary 3.4 The collection {ρ(res_{Γ}^{G}X, F)}_{G/Γ∈Or_{f} (G)}, defines a section of the functor C (see Subsection 1.4).

Let now M be a closed G-manifold and F → M be a G-equivariant flat hermitean vector bundle. We fix G-Hilbert module structures on H^{∗}(M, F). If G/Γ ∈ Or_{f}(G), then res_{Γ}^{G}M has a prefered simple structure. We employ this structure in order to define ρ(res_{Γ}^{G}M, F) as explained in Section 3.3.

Proposition 3.5 The collection {ρ(res_{Γ}^{G}M, F)}_{G/Γ∈Or_{f} (G)}, defines a section of the functor C.

Proof. Let f : X → M represent the prefered simple structure. Then by Proposition 2.1, (3), we have ρ(res_{Γ}^{G}M, F) = ρ(res_{Γ}^{G}X, f^{∗}F). Thus the Proposition is implied by Corollary 3.4. □

4 Computations

4.1

Let G be a compact Lie group and X be a finite G-CW-complex. Let F be a G-equivariant locally constant sheaf of finite-dimensional Hilbert spaces. We choose a G-Hilbert module structure on H^{∗}(X, F). Being a G-CW complex X has a natural filtration ∅ = X_{−1} ⊂ X_{0} ⊂ X_{1} ⊂ … ⊂ X_{N} = X. Consider G/Γ ∈ Or_{f}(G).

Lemma 4.1 There exists a representative f : Z → res_{Γ}^{G}X of the prefered simple structure such that

(1) : Z is filtered by Γ-CW subcomplexes ∅ = Z_{−1} ⊂ Z_{0} ⊂ Z_{1} ⊂ … ⊂ Z_{N} = Z and f_{|Z_{p}} : Z_{p} → res_{Γ}^{G}X_{p} represents the prefered simple structure for all p ∈ {0, 1, …, N}.

(2) If

… ⊂ F_{p+1}C(Z, f^{∗}F) ⊂ F_{p}C(Z, f^{∗}F) ⊂ …

denotes the decreasing filtration of the associated cochain complexes (we write F_{p} := F_{p}C(Z, f^{∗}F)), then F_{p}/F_{p+1} is the cochain complex associated to a representative of the prefered simple structure of ⊔_{i∈I_{p}}res_{Γ}^{G}G/H_{i} × (D^{p}, S^{p−1}) and the local system Q^{∗}F, where Q is given by

Q := ⊔_{i∈I_{p}}Q_{i} : ⊔_{i∈I_{p}}G/H_{i} × D^{p} → X_{p}

(recall that I_{p} is the indexing set of the p-dimensional G-cells of X and Q_{i} denote chacteristic maps).

Proof. The construction of f : Z → res_{Γ}^{G}X goes by induction and is based on [8], 4.29-4.32. For X_{−1} the assertion is trivial. Assume that we have constructed the simple structure (Z_{n−1}, f_{n−1}) of res_{Γ}^{G}X_{n−1} together with the filtration by Γ-subspaces (Z_{n−1})_{m}. Then we have to construct a simple structure (Z_{n}, f_{n}) of res_{Γ}^{G}X_{n} together with the filtration by Γ-subspaces (Z_{n})_{m}.

Let X_{n} be given by the G-push out

\begin{matrix}⊔_{i∈I_{n}}G/H_{i} × S^{n−1} & ⟶^{⊔_{i}q_{i}} & X_{n−1} \\ ↓ & & ↓ \\ ⊔_{i∈I_{n}}G/H_{i} × D^{n} & ⟶^{⊔_{i}Q_{i}} & X_{n} \end{matrix} .

We choose representatives h_{i} : V_{i} → res_{Γ}^{G}G/H_{i} of the prefered simple structures. Then h_{i} × id : U_{i} := V_{i} × D^{n} → res_{Γ}^{G}G/H_{i} × D^{n} is a simple structure on the cell res_{Γ}^{G}G/H_{i} × D^{n}. We choose a Γ-equivariant cellular map p : ⊔_{i∈I_{n}}U_{i} × S^{n−1} → Z_{n−1} such that f_{n−1} ∘ p ∼_{Γ} ⊔_{i∈I_{n}}q_{i} ∘l_{i} and ∼_{Γ} stands for Γ-homotopic. We now replace (Z_{n−1}, f_{n−1}) by (Cyl(p), f′_{n−1}), which represents the same simple structure on X_{n−1} (f′_{n−1} has still to be constructed). The filtration of Cyl(p) = ⊔_{i∈I_{n}}U_{i} × S^{n−1} × I ∪_{p} Z_{n−1} is given by Cyl(p)_{m} = (Z_{n−1})_{m} ⊂ Cyl(p) for m ≤ n − 2. Let pr : Cyl(p) → Z_{n−1} be the projection. In order to construct f′_{n−1} we consider the following Γ-homotopy commutative diagram

⊔_{i∈I_{n}}res_{Γ}^{G}G/H_{i} × S^{n−1}⊔X_{n−2} →^{⊔_{i}q_{i}⊔j} res_{Γ}^{G}X_{n−1} ↑ ⊔_{i}l_{i}⊔f_{n−2}  ↑ f_{n−1} ∘ pr ⊔_{i∈I_{n}}U_{i} × S^{n−1}⊔Cyl(p)_{n−2}  ↪^{J}  Cyl(p) ,

where j : X_{n−2} ↪ X_{n−1} is the inclusion, l_{i} := (h × id)_{|U_{i}×S^{n−1}}, and J_{|⊔_{i∈In}U_{i}×S^{n−1}} is the natural identification with the closed subspace ⊔_{i∈I_{n}}U_{i} ×S^{n−1} ×{0} ⊂ ⊔_{i∈I_{n}}U_{i} ×S^{n−1} ×I. Since J is a cofibration we can find f′_{n−1} ∼_{Γ} f_{n−1} ∘ pr such that f_{n−2} ∘ J_{|Cyl(p)_{n−2}} = (f′_{n−1})_{|Cyl(p)_{n−2}} and the following diagram commutes:

⊔_{i∈I_{n}}res_{Γ}^{G}G/H_{i} × D^{n} ↩ ⊔_{i∈I_{n}}res_{Γ}^{G}G/H_{i} × S^{n−1} →^{⊔_{i}q_{i}} res_{Γ}^{G}X_{n−1} ↑  ↑  ↑ f′_{n−1} ⊔_{i∈I_{n}}U_{i} × D^{n}  ↩  ⊔_{i∈I_{n}}U_{i} × S^{n−1}  ↪^{J} Cyl(p) .

Let Z_{n} be the Γ-push out

⊔_{i∈I_{n}}U_{i} × S^{n−1} → Cyl(p)
↓  ↓
⊔_{i∈I_{n}}U_{i} × D^{n} → Z_{n}

and f_{n} : Z_{n} → res_{Γ}^{G}X_{n} be the natural map of push outs. Then f_{n} : Z_{n} → res_{Γ}^{G}X_{n} represents the prefered simple structure, and Z_{n} is filtered by Γ-subspaces (Z_{n})_{m} such that (f_{n})_{|(Z_{n})_{m}} : (Z_{n})_{m} → res_{Γ}^{G}X_{m} represents the prefered simple structure for all m ≤ n. This finishes the proof of (1). Assertion (2) is an easy consequence of the construction. □

4.2

For i ∈ I_{p} define

Q̄_{i} : G/H_{i} ≅ G/H_{i} × {0} ↪ G/H_{i} × D^{p} →^{Q_{i}} X

and set F_{i} := Q̄_{i}^{∗}F. For any p ∈ N_{0} and i ∈ I_{p} we fix G-Hilbert module structures on H^{∗}(G/H_{i}, F_{i}). This induces G-Hilbert module structures on the cohomology complexes H(G/H_{i}, F_{i}). If C is a cochain complex, then let C[p] be the cochain complex with C[p]^{n} := C^{n+p} obtained from C. We equip H^{∗}(F_{p}/F_{p+1}) with Γ-Hilbert module structures such that the natural isomorphism H(F_{p}/F_{p+1}) ≅ ⊕_{i∈I_{p}}H(G/H_{i}, F_{i})[p] becomes an isometry. Then we have

[F_{p}/F_{p−1}] = ⊕_{i∈I_{p}}[C(U_{i}, h_{i}^{∗}F_{i})[p]] .

We are going to express ρ(res_{Γ}^{G}X, F) in terms of ρ(res_{Γ}^{G}G/H_{i}, F_{i}) and a contribution of the spectral sequence E := (E_{r}^{p,q}, d_{r}) associated to the filtration of X. For trivial Γ this was worked out in [9]. But [9], Lemmas 4.6 and 4.7, extend immediately to the case of a finite group Γ. We recall the result. Let

Z_{r}^{p,q} := im(H^{p+q}(F_{p}/F_{p+r}) → H^{p+q}(F_{p}/F_{p+1}))
B_{r}^{p,q} := im(H^{p+q−1}(F_{p−r+1}/F_{p}) → H^{p+q}(F_{p}/F_{p+1}))
E_{r}^{p,q} := Z_{r}^{p,q}/B_{r}^{p,q}
Z_{∞}^{p,q} := im(H^{p+q}(F_{p}) → H^{p+q}(F_{p}/F_{p+1}))
B_{∞}^{p,q} := im(H^{p+q−1}(C(Z, f^{∗}F)) → H^{p+q}(F_{p}/F_{p+1}))
E_{∞}^{p,q} := Z_{∞}^{p,q}/E_{∞}^{p,q}
F^{p,q} := im(H^{p+q}(F_{p}) → H^{p+q}(C(Z, f^{∗}F))) .

There are natural isomorphisms ψ^{p,q} : F^{p,q}/F^{p+1,q−1} → E_{∞}^{p,q}.

Note that H^{∗}(C(Z, f^{∗}F)) ≅^{f^{∗}} H^{∗}(X, F) has a prefered Γ-Hilbert module structure. We equip Z_{r}^{p,q}, B_{r}^{p,q}, E_{r}^{p,q}, Z_{∞}^{p,q}, B_{∞}^{p,q}, E_{∞}^{p,q}, F^{p,q} with the corresponding (sub)quotient Γ-Hilbert module structures. For any p, q, r we have a complex of finite Γ-Hilbert modules

E_{r}^{p,q} : … → E_{r}^{p+nr,q−(r−1)n} → E_{r}^{p+(n+1)r,q−(r−1)(n+1)} → …,

and H^{n}(E_{r}^{p,q}) = E_{r+1}^{p+nr,q−(r−1)n} has a prefered Γ-Hilbert module structure. Thus ρ(E_{r}^{p,q}) is well defined. Furthermore note that ρ(F_{p}/F_{p+1}) = (−1)^{p} ∑_{i∈I_{p}} ρ(res_{Γ}^{G}G/H_{i}, F_{i}). The following Proposition can be proved by repeating the argument of the proof of [9], Thm. 4.4.

Proposition 4.2

ρ(res_{Γ}^{G}X, F) = ∑_{p}(−1)^{p} ∑_{i∈I_{p}} ρ(res_{Γ}^{G}G/H_{i}, F_{i})+∑_{r≥1} ∑_{p=0}^{r−1} ∑_{q}(−1)^{p+q}ρ(E_{r}^{p,q})−∑_{p,q}(−1)^{p+q}X [[ψ^{p,q}]] .

Note that all three terms of the right-hand side may depend on the choice of the prefered Γ-Hilbert module structures on H^{∗}(G/H_{i}, F_{i}), while the left hand side does not.

4.3

Let X be a finite G-CW-complex and F a G-equivariant locally constant sheaf of finitedimensional Hilbert spaces. Let G/Γ ∈ Or_{f}(G). Then ρ̂(res_{Γ}^{G}X, F) ∈ Ĉ(Γ) is well defined.

Proposition 4.3

ρ̂(res_{Γ}^{G}X, F) = ∑_{p}(−1)^{p} ∑_{i∈I_{p}} ρ̂(res_{Γ}^{G}G/H_{i}, F_{i})

Proof. We fix G-Hilbert module structures on H^{∗}(X, F) and H^{∗}(G/H_{i}, F_{i}). Let q : Γ → Γ/Γ^{0} be the projection. We have to show that

ρ(E_{r}^{p,q}) ∈ q^{∗}C(Γ/Γ^{0}), X [[ψ^{p,q}]] ∈ q^{∗}C(Γ/Γ^{0}) .  (7)

Viewing X as a filtered G-space we see that there is a spectral sequence Ẽ:= (Ẽ_{r}^{p,q}, d_{r}) of G-modules with

Z̃_{r}^{p,q} := im(H^{p+q}(X_{p+r−1}, X_{p−1}) → ⊕_{i∈I_{p}}H^{p+q}(G/H_{i}, F_{i})) ,

B̃_{r}^{p,q} := im(H^{p+q−1}(X_{p−1}/X_{p−r}) → ⊕_{i∈I_{p}}H^{p+q}(G/H_{i}, F_{i})) ,
Ẽ_{r}^{p,q} := Z̃_{r}^{p,q}/B̃_{r}^{p,q}
Z̃_{∞}^{p,q} := im(H^{p+q}(X, X_{p−1}, F) → ⊕_{i∈I_{p}}H^{p+q}(G/H_{i}, F_{i}))
B̃_{∞}^{p,q} := im(H^{p+q−1}(X, F)) → ⊕_{i∈I_{p}}H^{p+q}(G/H_{i}, F_{i}))
Ẽ_{∞}^{p,q} := Z̃_{∞}^{p,q}/Ẽ_{∞}^{p,q}
F̃^{p,q} := im(H^{p+q}(X, X_{p−1}, F) → H^{p+q}(X, F))

and G-equivariant maps ψ̃^{p,q} : F̃^{p,q}/F̃^{p+1,q−1} → Ẽ_{∞}^{p,q} such that E = res_{Γ}^{G}Ẽ, ψ^{p,q} = res_{Γ}^{G}ψ̃^{p,q}. Consider the exact sequence

0 → G^{0} → G →^{q} π_{0}(G) → 0 .

Observe that the representation of G on Ẽ and F̃^{p,q} factors over q. In particular Γ^{0} is represented trivially. Thus (7) follows. □

Fix G-Hilbert module structures on H^{∗}(X, F) and H^{∗}(G/H_{i}, F_{i}). Then Proposition 4.3 has the

Corollary 4.4

ρ̂(X, F) = ∑_{p}(−1)^{p} ∑_{i∈I_{p}} ρ̂(G/H_{i}, F_{i}) .

5 Reduction to Cartan subgroups

5.1

Let G be a compact Lie group, X be a finite G-CW complex, and let F → X be an equivariant flat hermitean vector bundle.

Recall that a Cartan subgroup T of G is a topologically cyclic closed subgroup such that the Weyl group N_{G}(T)/T is finite, where N_{G}(T) is the normalizer of T in G. If T ⊂ G is a Cartan subgroup then it is isomorphic to the product of a torus and a finite cyclic group (see [2], p.177 ff, for more details about Cartan subgroups).

If g ∈ G, then there exists a Cartan subgroup containing g. The conjugacy classes of Cartan subgroups are in natural bijection with the cyclic subgroups of π_{0}(G).

Let {T_{i}}_{i∈C} be a set of representatives of conjugacy classes of Cartan subgroups of G. Then restriction defines inclusions

⊕_{i∈C}res_{T_{i}}^{G} : C(F G) → ⊕_{i∈C}C(F T_{i}), ⊕_{i∈C} r̂es_{T_{i}}^{G} : Ĉ(F G) → ⊕_{i∈C} Ĉ(F T_{i}) .

In order to determine f ∈ Ĉ(F G) it is thus sufficient to compute r̂es_{T_{i}}^{G}f ∈ Ĉ(F T_{i}) for all i ∈ C.

Let T be any Cartan subgroup of G and let f : Z → res_{T}^{G}X represent the prefered simple structure. Then r̂es_{T}^{G}ρ̂(X, F) = ρ̂(Z, f^{∗}F).

5.2

We further study the contribution of the T -cells of Z. Let S ⊂ T be any closed subgroup, and let F be a T -equivariant local system on T/S. Then ρ(T/S, F) ∈ Ĉ(F T) is well defined.

Lemma 5.1 If dim(T/S) ≠ 1, then ρ̂(T/S, F) ∈ Ĉ(F T) = 0.

Proof. Assume first that T/S is even-dimensional. Then T/S is orientable. Let t ∈ F T generate the finite group H ⊂ Aut(T/S). Then H acts by orientation-preserving diffeomorphisms on T/S. We employ [10], Prop. 3.23, which says that ρ̂(res_{H}^{T}T/S, F) can be derived from the Poincaré torsion ρ_{pd}^{H}(T/S, F) of the H-manifold res_{H}^{T}T/S (see loc.cit.

Definition 3.19). Since T is abelian, we see that H acts freely or trivially on T/S. In both cases ρ_{pd}^{H}(T/S, F) vanishes (see loc.cit. Prop. 3.20), and thus r̂es_{H}^{T} ρ̂(T/S, F) = 0. Since t was arbitrary we conclude that ρ̂(T/S, F) = 0.

Now we consider the case that T/S is odd-dimensional. We fix a T -invariant Riemannian metric g^{T/S} on T/S and equip H^{∗}(T/S, F) with the T -module structure such that the de Rham isomorphism becomes an isometry. Then ρ(T/S, F) ∈ C(F T) is well-defined.

By Lemma 1.2 we have ρ(T/S, F) = ρ_{an}(T/S, g^{T/S}, F)_{|F T}. If dim(T/S) > 1, then we find two everywhere linearly independend unit-length Killing vector fields in C^{∞}(T/S, T (T/S)). Using the induced decomposition of the de Rham complex by a standard argument ρ_{an}(T/S, g^{T/S}, F) = 0. This implies the lemma. □

5.3

Let S ⊂ T be any closed subgroup such that dim(T/S) = 1, and let F be a T -equivariant locally constant sheaf of Hilbert spaces. Since T is abelian and ρ(T/S, F) is additive with respect to F without loss of generality we can assume that F is one-dimensional.

We fix an orientation on T/S. The space T/S is a disjoint union of oriented circles C_{1}, …, C_{s}. Let e^{2πıa_{i}} ∈ S^{1} denote the holonomy of F on C_{i}. We choose the Riemannian metric g^{T/S} such that the circles have length 1.

Let t ∈ T. We distinguish two cases

a) that tC_{i} = C_{i} for all i

b) that tC_{i} ≠ C_{i} for all i. In case b) we have ρ_{an}(T/S, g^{T/S}, F)(t) = 0. In case a) we have ρ_{an}(T/S, g^{T/S}, F)(t) = ∑_{i=1}^{s} ρ_{an}(C_{i}, g^{C_{i}}, F_{i})(t), where F_{i} is the restriction of F to C_{i}.

Thus we are reduced to the following situation. Let C_{i} be a circle, F → C_{i} be a one-dimensional flat hermitean bundle over C_{i}, and t be an automorphism of F → C_{i}, which acts as rotation on C_{i}. We parametrize C_{i} = R/Z with coordinate x. Then t(x) = x + τ_{i}, τ_{i} ∈ R/Z. Sections of F_{|C_{i}} are identified with functions f : R → C satisfying f(x + 1) = e^{2πıa_{i}}f(x). Then (tf)(x) = λ_{i}f(x − τ_{i}) for certain λ_{i} ∈ S^{1}.

We have ρ(C_{i}, g^{C_{i}}, F)_{an}(t) = ψ(λ_{i}, a_{i}, τ_{i}), and ψ will be determined in Subsection 5.4. We obtain ρ_{an}(T/S, g^{T/S}, F)(t) = ∑_{i=1}^{s} ψ(λ_{i}, a_{i}, τ_{i}). If F is higher-dimensional, then λ_{i} and a_{i} become commuting diagonalizable matrices, and we have ρ_{an}(T/S, g^{T/S}, F)(t) = ∑_{i=1}^{s} Tr ψ(λ_{i}, a_{i}, τ_{i}).

5.4

In this Subsection we derive a formula for ψ(λ, a, τ). Let C be a circle, F → C be a one-dimensional flat hermitean bundle over C, and t be an automorphism of F → C, which acts as rotation on C.

We parametrize C = R/Z with coordinate x. Then t(x) = x + τ, τ ∈ R/Z. Sections of F are identified with functions f : R → C satisfying f(x + 1) = e^{2πıa}f(x). Then (tf)(x) = λf(x − τ) for certain λ ∈ S^{1}.

We identify one-forms on C with functions using the basis dx ∈ C^{∞}(C, T^{∗}C). Let Δ_{F} = −(d/dx)^{2} be the Laplace operator. The eigenvectors of Δ_{F} are f_{n}(x) = exp(2πı(n+ a)x), and the corresponding eigenvalue is μ_{n} = 4π^{2}(n + a)^{2}. The action of t on the eigenspace spanned by f_{n} is multiplication by λe^{−2πı(n+a)τ}.

First assume that a ∈ (0, 1). Let

F_{n}(s) := \frac{λe^{−2πı(n+a)τ}}{Γ(s)} ∫_{0}^{∞} t^{s−1}e^{−4π^{2}(n+a)^{2}t}dt = \frac{λe^{−2πı(n+a)τ}}{4^{s}π^{2s}(n + a)^{2s}} ,

then

ψ(λ, a, τ) = −\frac{d}{ds}_{|s=0} ∑_{n∈Z} F_{n}(s) ,

where we employ a meromorphic continuation of the sum ∑_{n∈Z} F_{n}(s) which converges for Re(s) > 1.

Let ϕ(y, a, s) := ∑_{n=0}^{∞} \frac{e^{2πıyn}}{(n+a)^{s}}. Then

ψ(λ, a, τ) = −\frac{d}{ds}_{|s=0} \frac{λ}{4^{s}π^{2s}} (e^{−2πıτa}ϕ(−τ, a, 2s) + e^{−2πıτ(a−1)}ϕ(τ, 1 − a, 2s)) .

If a = 0 and τ ∈ (0, 1), then with ϕ(y, s) := ∑_{n=1}^{∞} \frac{e^{2πıyn}}{n^{s}} we obtain

ψ(λ, 0, τ) = −\frac{d}{ds}_{|s=0} \frac{λ}{4^{s}π^{2s}} (ϕ(τ, 2s) + ϕ(−τ, 2s)) .

Using [7], Prop. 31, one can show that

ψ(λ, 0, τ) = λD + λ (\frac{Γ(τ)′}{Γ(τ)} + \frac{Γ(1 − τ)′}{Γ(1 − τ)} )

for some explicitly known constant D ∈ C. In the case that a = 0 and τ = 0 we have

ψ(λ, 0, 0) = −\frac{d}{ds}_{|s=0} \frac{2λζ_{R}(2s)}{4^{s}π^{2s}} ,

where ζ_{R} denotes the Riemann zeta function.

5.5

In this Subsection we combine the results of Subsections 5.1, 5.2, and 5.3. For simplicity we assume that G is a connected compact Lie group. Let X be a finite G-CW complex, and let F → X be a G-equivariant locally constant sheaf of finite-dimensional Hilbert spaces. Let T ⊂ G be a maximal torus, and let f : Z → res_{T}^{G}X represent the prefered simple structure.

Let I be the index set for the T -cells E = T/S_{E} × D^{n_{E}} of Z with dim(T/S_{E}) = 1. Since T is connected the quotient T/S is a circle S^{1}. If t ∈ T, then for each E ∈ I we fix an orientation of T/S_{E}, and we define the rotation number τ_{E}(t) ∈ R/Z, the constant λ_{E}(t), and the holonomy e^{2πıa_{E}} of F_{E}. We obtain the following Proposition.

Proposition 5.2 ρ̂(X, F) is uniquely determined by its restriction to T, which is represented by the function

F T ∋ t ↦ ∑_{E∈I}(−1)^{n_{E}}Tr ψ(λ_{E}(t), a_{E}, τ_{E}(t)) .

5.6

In this section we compute the equivariant Reidemeister torsion of odd-dimensional symmetric spaces. We recover results of [6] using a completely different method. Let θ be the constant sheaf with fibre C, and let ρ̂(M) := ρ̂(M, θ) for any closed G-manifold M, where we equip θ with the obvious G-action.

Let G/K be a compact, irreducible, odd-dimensional symmetric space. We assume that G is connected.

Lemma 5.3 If ρ̂(G/K) ≠ 0, then rankG = rankK + 1, and

G/K = \{ \begin{matrix}SO(2m)/SO(2p − 1) × SO(2m − 2p + 1) or \\ SU(3)/SO(3) \end{matrix} .

Proof. Let T be a maximal torus of G and f : X → res_{T}^{G}(G/K) be a representative of the prefered simple structure. Using the construction given in [8], 4.36, we can choose X such that it has the same set of T -orbit types as G/K. If ρ̂(G/K) = ρ̂(X) ≠ 0, then by Proposition 5.2 there exists a one-dimensional T -orbit T gK ⊂ G/K. Then T^{g^{−}1} ∩ K is a rankG − 1-dimensional torus in K. Hence rankG ≥ rankK ≥ rankG − 1. If rankK = rankG, then G/K is even-dimensional. Thus rankK = rankG − 1. The second assertion follows from the classification of irreducible compact symmetric spaces. □

Now assume that G/K is a compact, irreducible, odd-dimensional symmetric space with rankK = rankG − 1.

Lemma 5.4 There is a one-to-one correspondence of one-dimensional T -orbits in G/K and W_{G}(T)/W_{K}(T) given by N_{G}(T) ∋ g ↦ T g^{−1}K, where W_{G}(T) := N_{G}(T)/T and W_{K}(T) := N_{K}(T)/T ∩ K.

Proof. We can assume that T ∩ K = S is a maximal torus of K. If T′ ⊂ G is a second maximal torus of G with T ∩ K = S, then T′ = T. Indeed, on the level of Lie algebras we have t′ = s ⊕ (k^{⊥})^{S} = t.

Let T gK be a one-dimensional T -orbit in G/K. Then T^{g^{−}1} ∩ K is a maximal torus in K, and hence T^{g^{−}1} ∩ K = S^{k} for a suitable k ∈ K. Replacing g by gk^{−1} we obtain T^{g^{−}1} ∩ K = S, and thus g^{−1} ∈ N_{G}(T). If g ∈ T ∪ K, then T gK = T K. Thus the correspondence of W_{G}(T)/W_{K}(T) with the set of one-dimensional orbits is established by g^{−1} ∈ N_{G}(T) ↦ T gK. □

Let s ⊂ t be the Lie algebra of S ⊂ T. Let a := t/s, and let L ⊂ a be the lattice of those [l] ∈ t/s, l ∈ t, which satisfy exp(l) ∈ S. We identify a ≅ R such that L is identified with Z. The exponential map yields an identification i : R/Z ≅ a/L ≅ T/S. If t ∈ T, then let α(t) ∈ R/Z be such that ti(x) = i(α(t) + x), ∀x ∈ R.

For τ ∈ R/Z, τ ≠ 0, define

ψ(τ) := D + (\frac{Γ(τ̂)′}{Γ(τ̂)} + \frac{Γ(1 − τ̂)′}{Γ(1 − τ̂)} ) ,

where τ̂ ∈ (0, 1) represents τ. If τ = 0, we put

ψ(τ) := −\frac{d}{ds}_{|s=0} \frac{2λζ_{R}(2s)}{4^{s}π^{2s}} .

Because of the equality (4) the following proposition recovers the computation of equivariant analytic torsion of compact symmetric spaces [6], Thm. 11, up to a constant function.

Proposition 5.5 Let G/K be a compact, irreducible, odd-dimensional symmetric space. If rankK ≠ rankG − 1, then ρ̂(G/K) = 0. If rankK = rankG − 1, then let T ⊂ G be a maximal torus such that S := T ∩ K is a maximal torus of K. Then the restriction of ρ̂(G/K) to T is represented by

F T ∋ t ↦ \frac{1}{♯W_{K}(T)} ∑_{w∈W_{G}(T)} ψ(α(t^{w})) .

Proof. Let f : X → res_{N_{G}(T)}^{G}(G/K) be a representative of the natural simple structure. By Lemma 5.4 there is exactly one isolated one-dimensional N_{G}(T)-orbit in G/K. Constructing X by the inductive procedure given in [8], 4.36, we can assume that X has exactly one cell E = N_{G}(T)/N_{K}(T) × D^{n_{E}} with one-dimensional N_{G}(T)-orbits. Moreover n_{E} = 0.

The T -space res_{T}^{N_{G}(T)}(X) has a natural T -CW structure (since N_{G}(T)/T is finite), and res_{T}^{N_{G}(T)}(E) is the only T -cell with one-dimensional T -orbits. But res_{T}^{N_{G}(T)}(E) is the disjoint union of spaces T/S^{g}, g ∈ W_{G}(T)/W_{K}(T). The proposition now follows from Proposition 5.2. □

Acknowledgement:The author thanks W. Lück for his interest in this work and many valuable hints.

References

[1] J. M. Bismut and W. Zhang. Milnor and Ray-Singer metrics on the equivariant determinant of a flat vector bundle. Geometric and Functional Analysis, 4(1994), 136–212.

[2] T. Broecker and T. tom Dieck. Representations of Compact Lie Groups. Springer Verlag, 1985.

[3] S. Illman. Smooth equivariant triangulations of G-manifolds for G a finite group. Math. Ann., 233(1978), 199–220.

[4] S. Illman. The equivariant triangulation theorem for actions of compact Lie groups. Math. Ann., 262(1983), 487–501.

[5] S. Illman. The restriction homomorphism Res_{H}^{G} : W h_{G}(X) → W h_{H}(X) for G a compact Lie group. J.Fac.Sci.Univ.Tokyo, 37(1990), 601–634.

[6] K. Köhler. Equivariant Reidemeister torsion on symmetric spaces. Math. Ann., 307(1997), 57–69.

[7] J. Lott and M. Rothenberg. Analytic torsion for group actions. J.Diff.Geom, 34(1991), 431–481.

[8] W. Lück. Transformation groups and algebraic K-theory. LNM 1408. Springer Verlag, 1989.

[9] W. Lück, T. Schick, and T. Thielmann. Torsion and fibrations. preprint, Mainz, 1995.

[10] W. Lück. Analytic and topological torsion for manifolds with boundary and symmetries. J.Diff.Geom, 37(1993), 263–322.