”Analytic and Topological Torsion for Manifolds with Boundary and Symmetry” by Wolfgang Lück

0. Introduction

Let G be a finite group acting on a Riemannian manifold M by isometries. We introduce analytic torsion

ρ_{an}^{G}(M, M_{1}; V) ∈ R ⊗_{Z} Rep_{R}(G)

PL-torsion

ρ_{pl}^{G}(M, M_{1}; V) ∈ K_{1}(RG)^{Z/2}

Poincaré torsion

ρ_{pd}^{G}(M, M_{1}; V) ∈ K_{1}(RG)^{Z/2}

and Euler characteristic

χ^{G}(M, M_{1}; V) ∈ Rep_{R}(G)

for ∂M the disjoint union of M_{1} and M_{2} and V an equivariant coefficient system. The analytic torsionis defined in terms of the spectrum of the Laplace operator, the PL-torsionis based on the cellular chain complex and Poincaré torsionmeasures the failure of equivariant Poincaré duality in the PL-setting, which does hold in the analytic context. Denote by R̂ep_{R}(G) the subgroup of Rep_{R}(G) generated by the irreducible representations of real or complex type. We define an isomorphism

Γ_{1} ⊕ Γ_{2} : K_{1}(RG)^{Z/2} ⟶ (R ⊗_{Z} Rep_{R}(G)) ⊕ (Z/2 ⊕_{Z} R̂ep_{R}(G))

and show under mild conditions

ρ_{an}^{G}(M, M_{1}; V) = Γ_{1}(ρ_{pl}^{G}(M, M_{1}; V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M, M_{1}; V)) + \frac{ln(2)}{2} · χ(∂M; V)

and

Γ_{2}(ρ_{pl}^{G}(M, M_{1}; V)) = Γ_{2}(ρ_{pd}^{G}(M, M_{1}; V)) = 0

For trivial G this reduces to the equation in R

ρ_{an}(M, M_{1}; V) = ln(ρ_{pl}(M, M_{1}; V)) + \frac{ln(2)}{2} · χ(∂M) · dim_{R}(V)

Torsion invariants are important invariants which relate topology to algebraic K-theory and hence to number theory (see Milnor [25]). The s-Cobordism Theorem and the classification of lens spaces are important examples. Motivated by fruitful connections between topology and analysis, e.g. the Atiyah-Singer Index Formula, Ray and Singer [29] asked whether (Reidemeister) PL-torsion can be interpreted analytically, namely, by the spectral theory of the Laplace operator. They defined analytic torsion and gave some evidence for the conjecture that analytic and PL-torsion agree. This was independently proved by Cheeger [8] and Müller [26]. Analytic torsion is used and investigated in various contexts (see e.g. Bismut-Freed [4], [5], Fried [14], Quillen [27], Schwarz [32], Witten [35]).

The PL-torsion is powerful as it is a very fine invariant and there are good tools like sum and product formulas for its computation. In particular one can chop a manifold into ”elementary” pieces, determine the PL-torsion of the pieces and use a sum formula to compute the PL-torsion of M. Notice that these pieces have boundaries even if M is closed. In order to get a sum formula for analytic torsion also, it is necessary to investigate the relation between analytic and PL-torsion also for manifolds with boundary. Inspecting the proofs of Cheeger [8] and Müller [26] one recognizes that they do not extend to the case where M has a boundary. Moreover, an easy calculation for D^{1} shows that their result is not true for D^{1}. Now the key observation due to Cheeger (see [8], page 320) is that the equivariant spectrum of the Laplace operator on M ∪_{∂M} M with the Z/2-action given by the flip and the spectrum of the Laplace operator on M for both Dirichlet and Neumann boundary conditions determine one another. Hence the problem of comparing analytic torsion and PL-torsion for manifolds with boundary can be reduced to the case of a closed manifold with a Z/2-action. Notice that the flip on M ∪_{∂M} M reverses the orientation. Inspecting the proof of Müller [26] again it turns out that his methods carry over to closed orientable Riemannian G-manifolds with orientation preserving and isometric G-action for a finite group G. This is carried out in Lott-Rothenberg [19] and we will exploit their work. However, we will use a different setting which seems to be more appropiate for our purposes here and for more general situations (mainly an L^{2}-version for proper actions of infinite groups on non-compact manifolds we will treat in forthcoming papers).

Let (M; M_{1}, M_{2}) be a m-dimensional (compact) Riemannian G-manifold triad (with G acting by isometries). There is a canonical group extension

[figure: 0; π(M); i; D^{G}(M); q; G; 0]

and a D^{G}(M)-action on the universal covering M̃ extending the action of the fundamental group and covering the G-action. Consider an equivariant coefficient system V, i.e. an orthogonal representation of D^{G}(M). Such a V may be thought of as an equivariant flat G-vector bundle over M or as an equivariant flat connection. We want to allow a twisting of our invariants by such equivariant coefficient systems because analytic torsion is important for the study of moduli spaces of flat connections (see e.g. Quillen [27], Witten [35]). Put certain boundary conditions of Dirichlet type on M_{1} and of Neumann type on M_{2}. Then the Laplace operator Δ^{p} is elliptic, self-adjoint and non-negative definite and is compatible with the G-action. The eigenspace E_{λ}^{G}(M, M_{1}; V)^{p} of Δ^{p} for the eigenvalue λ is a real G-representation. We define the equivariant Zeta-function by meromorphic extension of

ζ_{p}^{G}(M, M_{1}; V)(s) := ∑_{λ>0} λ^{−s} · [E_{λ}^{G}(M, M_{1}; V)^{p}] ∈ C ⊗_{Z} Rep_{R}(G)

It is analytic in zero and we define analytic torsion in section 1 by

ρ_{an}^{G}(M, M_{1}; V) := ∑_{p=0}^{m}(−1)^{p} · p · \frac{d}{ds}ζ_{p}^{G}(M, M_{1}; V)|_{s=0} ∈ R ⊗_{Z} Rep_{R}(G)

For G = 1 this agrees up to a factor 2 with the definition of Ray-Singer [29].

A finite RG-Hilbert complex C is a finite-dimensional finitely generated RG-chain complex C together with a R-Hilbert structure compatible with the G-action on each C_{n}. Given a RG-chain equivalence f : C → D, we define in section 2 its Hilbert torsion

ht(f) ∈ K_{1}(RG)^{Z/2}

Let C be a RG-Hilbert complex. Suppose that its homology H(C) has the structure of a finite RG-Hilbert complex with respect to the trivial differential. There is a RG-chain map i : H(C) → C uniquely determined up to RG-chain homotopy by the property H(i) = id. Define the Hilbert-Reidemeister torsion of C

hr(C) := ht(i) ∈ K_{1}(RG)^{Z/2}

If G is trivial, hr(C) is the square of Milnor’s torsion defined for C and H(C) equipped with any orthonormal bases. There is also a cochain version.

Using the Hodge-decomposition theorem and the cellular bases we get RG -Hilbert structures on H_{∗}(M, M_{1}; V) and C_{∗}(M, M_{1}; V). We define the PL-torsion

ρ_{pl}^{G}(M, M_{1}; V) := hr(C_{∗}(M, M_{1}; V)) ∈ K_{1}(RG)^{Z/2}

Let ∩[M] : C^{m−∗}(M, M_{1}; V) ⟶ C_{∗}(M, M_{1}; V) be the Poincaré RG-chain equivalence. Its Hilbert torsion is the Poincaré torsion

ρ_{pd}^{G}(M, M_{1}; V) ∈ R ⊗_{Z} Rep_{R}(G)

This invariant is always zero for trivial G. This follows from the proof of Poincaré duality based on the dual cell decomposition of a triangulation. In the equivariant case the dual cell structure is not compatible with the G-action and ρ_{pd}^{G}(M, M_{1}; V) measures the failure. The equivariant Euler characteristic is defined by

χ^{G}(M, M_{1}; V) = ∑_{p=0}^{m}(−1)^{p} · [H^{p}(M, M_{1}; V)] ∈ Rep_{R}(G)

We will put the technical condition on the equivariant coefficient system V that it is coherent to a G-representation. This is always satisfied if G is trivial or if M^{H} is non-empty and connected for all H ⊂ G. Now we can state the main result of this paper.

Theorem 4.5 (Torsion Formula for Manifolds with Boundary and Symmetry) Let M be a Riemannian G-manifold whose boundary is the disjoint union M_{1} ∐ M_{2}. Let V be an equivariant coefficient system which is coherent to a G-representation. Assume that the metric is a product near the boundary. Then

ρ_{an}^{G}(M, M_{1}; V) = Γ_{1}(ρ_{pl}^{G}(M, M_{1}; V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M, M_{1}; V)) + \frac{ln(2)}{2} · χ(∂M; V)

If G is trivial, this reduces to the equation of real numbers

ρ_{an}(M, M_{1}; V) = ln(ρ_{pl}(M, M_{1}; V)) + \frac{ln(2)}{2} · χ(∂M) · dim_{R}(V)

Cheeger states in [8], page 320 without proof a formula relating analytic and P L-torsion for a manifold with boundary (without group action). His formulas are not as precise as ours since Cheeger claims only that the correction term can be computed locally at the boundary, whereas we can identify it with the Euler characteristic.

The proof of the main theorem is organized as follows. In section 1, 2, and 3 we show product and double formulas and Poincaré duality. We investigate how these invariants depend on the Riemannian metric and relate PL-torsion to the equivariant Whitehead torsion of a G-homotopy equivalence. Then Theorem 4.5 is verified in section 4 as follows. We first give the proof under the extra conditions i.) M is orientable ii.) G acts orientation preserving and iii.) ∂M is empty. If dim(M) is even, the assertion follows from Poincaré duality. The Poincaré duality formulas for analytic and PL-torsion differ just by the Poincaré torsion. This is the reason for the appearence of Poincaré torsion in the formula relating analytic and PL-torsion. If dim(M) is odd, we reduce the claim to the case of trivial coefficients V = R and then apply Lott-Rothenberg [19]. We remove condition ii.) by the various product formulas and explicit calculations for S^{1} with the involution given by complex conjugation. We get rid of i.) using the orientation covering. Finally we remove ii.) by the double formula which relates the invariants for the G × Z/2-manifold M ∪_{∂M} M to the invariants of the G-manifolds (M, ∅) and (M, ∂M). The double formulas for the analytic torsion and the PL-torsion differ by a certain Euler characteristic term of the boundary, as in the analytic case the boundary is a zero set and does not affect the RG-Hilbert structure whereas in the PL-case the cells of the boundary do contribute to the RG-Hilbert structure. This difference in the double formulas causes the Euler characteristic term in the formula of the Theorem above. The appearence of a correction term in the case of a manifold with boundary is not very surprising if one thinks of the index formula for manifolds with boundary where the η-invariant comes in (see Atiyah-Patodi-Singer [1], [2], [3]).

In section 5 we investigate some special cases. We derive from the sum formula in the PL-case a sum formula for the analytic torsion. This is remarkable because it is in general difficult to derive the spectrum of the Laplace operator on M ∪_{f} N for an isometric diffeomorphism f : ∂M ⟶ ∂N from the spectra of its restrictions to M, N and ∂M. We express the various torsion invariants for spheres and disks of G-representations in terms of their characters. We construct an injective homomorphism based on Poincaré torsion

ρR^{G} : Rep_{R}(G) ⟶ Z ⊕ (⊕_{(H)}K_{1}(R[W H])^{Z/2})

This reproves the Theorem of de Rham that two orthogonal G-representations V and W are linearly RG-isomorphic if and only if their unit spheres SV and SW are G-diffeomorphic. We use the sum formula for Poincaré torsion to establish a local formula for Poincaré torsion. It computes the Poincaré torsion of M in terms of the Poincaré torsion of the tangent representations of points with non-trivial isotropy group and the universal equivariant Euler characteristic of M.

The author wants to thank the Deutsche Forschungsgemeinschaft for the financial support and the department of mathematics of The University of Chicago for the hospitality during the stay from October 88 to March 89 and in February 90, when this paper was worked out. In particular the discussions with Prof. Rothenberg were very fruitful. The paper is organized as follows.

0. Introduction

1. Analytic Torsion

2. Torsion Invariants for Chain Complexes

3. PL-Torsion

4. Comparision of Analytic and PL-Torsion

5. Some Computations References

1. Analytic Torsion

Let G be a finite group. A Riemannian G-manifold M is a compact smooth manifold with differentiable G-action and invariant Riemannian metric. If ∂M is the disjoint union of M_{1} and M_{2}, we want to define the analytic torsion of (M, M_{1}) with certain coefficients V. We begin with explaining the coefficients.

Let X be a G-space with universial covering p : X̃ ⟶ X. The group of covering translations is denoted by Δ(p). Let D^{G}(p) be the discrete group

1.1 D^{G}(p) := {(f̃, g) | f : X̃ → X̃, g ∈ G, p ∘ f̃ = l(g) ∘ p}

where l(g) : X → X is multiplication with g. There is an obvious exact sequence

1.2  [figure: 0; Δ(p); i(p); D^{G}(p); q(p); G; 0]

and an operation of D^{G}(p) on X̃ making the following diagram commute

1.3

[figure: Δ(p); ×; X̃; X̃; i(p) × id; id; D^{G}(p); ×; X̃; X̃; q(p) × p; p; G; ×; X; X]

They are natural in p.In the sequel we identify Δ(p) with π = π_{1}(X) and write D^{G}(X) instead of D^{G}(p). The coefficients will be orthogonal D^{G}(X)-representations.

Let M be a Riemannian G-manifold of dimension m. Define the orientation homomorphism

1.4 w^{G}(M) : D^{G}(M) ⟶ {±1} as follows. For some base point x ∈ M an element (f̃, g) ∈ D^{G}(M) is given by a homotopy class of paths w from x to gx. The composition of the fibre transport of the tangent bundle T M along w and the differential of l(g^{−1}) at gx give an isotopy class of automorphisms of T M_{x}. If it is orientation preserving resp. reversing, we put w^{G}(M)(f̃, g) to be 1 resp. −1. An equivalent definition uses the observation that H^{m}(Hom_{Zπ}(C_{∗}(M̃), Zπ)) is an infinite cyclic group by Poincaré duality and f̃ induces an homomorphism f̃_{∗} : π ⟶ π, a Z[f̃_{∗}]-equivariant map C_{∗}(f̃) : C_{∗}(M̃) ⟶ C_{∗}(M̃) and hence an automorphism of this infinite cyclic group by H^{m}(Hom_{Zπ}(C_{∗}(f̃), Z[f_{∗}^{−1}])). Then w^{G}(M)(f̃, g) is its degree.

If V is a D^{G}(M)-representation, let ^{w}V be the w = w^{G}(M)-twisted D^{G}(M)-represen-tation given by w(e) · ev for e ∈ D^{G}(M) and v ∈ V. The vector bundle M̃ ×_{π} V over M becomes a G-vector bundle by g(x̃, v) = (x̃g̃^{−1}, g̃v) for any lift g̃ ∈ D^{G}(M) of g ∈ G. The deRham complex Λ^{∗}(M; V) of differential forms with coefficients in M̃ ×_{π} V is a RGcochain complex. A choice of a local orientation on T M̃x̃ for some x̃ ∈ M̃ together with the Riemannian metric determines a volume form dM ∈ Λ^{n}(M;^{w} R). Using the inner product on V we get the product ∧ : Λ^{p}(M; V) ⊗ Λ^{q}(M;^{w} V) ⟶ Λ^{p+q}(M;^{w} R). The Hodge star operator

1.5 ∗ : Λ^{p}(M; V) ⟶ Λ^{m−p}(M;^{w} V)

is defined by ω ∧ (∗η) =< ω, η > ·dM, where <, > is induced from the Riemannian metric. Since dM is G-invariant, the map ∗ is RG-linear. The Riemannian metric induces an inner product on Λ^{p}(M; V) by << ω, η >>:= ∫_{M} < ω, η > dM. Then ∗ is an isometry satisfying ∗ ∘ ∗ = (−1)^{p(m−p)}id. The adjoint δ^{p} : Λ^{p}(M; V) ⟶ Λ^{p−1}(M; V) of the differential d^{p} is (−1)^{mp+p+1} ∗ d^{m−p}∗. Define the Laplace operator

1.6 Δ^{p} : Λ^{p}(M; V) ⟶ Λ^{p}(M; V)

by d^{p−1}δ^{p} + δ^{p+1}d^{p}.

Let M be a Riemannian G-manifold whose boundary ∂M is the disjoint union M_{1} ∐ M_{2}. We allow that M_{1} or M_{2} or both are empty. Consider an orthogonal D^{G}(M)-representation V. Given a p-form ω ∈ Λ^{p}(M; V), let ω_{tan} be the p-form on ∂M coming from restriction with T i : T ∂M ⟶ T M for the inclusion i. Let ω_{nor} be the (p − 1)-form ∗_{∂M}(∗_{M}ω)_{tan}. We will consider the boundary conditions

1.7 b(M, M_{1}) : ω_{tan} = 0, (δω)_{tan} = 0 on M_{1} ω_{nor} = 0, (dω)_{nor} = 0 on M_{2}

In the sequel we write 1.8 Λ_{1}^{p}(M, M_{1}; V) = {ω ∈ Λ^{p}(M; V) | ω_{tan} = 0 on M_{1} and ω_{nor} = 0 on M_{2}}

Λ_{2}^{p}(M, M_{1}; V) = {ω ∈ Λ^{p}(M; V) | ω satisfies b(M, M_{1})}

H_{harm}^{p}(M, M_{1}; V) = {ω ∈ Λ^{p}(M; V) | Δω = 0, ω satisfies b(M, M_{1})}

The space H_{harm}^{p}(M, M_{1}; V) is called space of harmonic forms . Denote by

1.9 A(M; V) : Λ^{∗}(M; V) ⟶ C_{s}^{∗}(M; V)

the V -twisted deRham map which is the composition

Λ^{∗}(M; V) = Λ^{∗}(M; M̃ ×_{π} V) ⟶^{p^{∗}} Λ^{∗}(M̃; M̃ × V)^{π} ⟵^{j^{π}} (Λ^{∗}(M̃) ⊗_{R} V)^{π} ⟶^{(A(M̃)⊗Rid)^{π}}

(Hom_{R}(C_{∗}^{s}(M̃), R) ⊗_{R} V)^{π} ⟶^{Φ^{π}} Hom_{R}(C_{∗}^{s}(M̃), V)^{π} = Hom_{Rπ}(C_{∗}^{s}(M̃), V) =: C_{s}^{∗}(M; V)

Here C_{∗}^{s}(M̃) is the singular chain complex. The map p^{∗} is induced from the projection p : M̃ × V ⟶ M̃ ×_{π} V. The isomorphism j : Λ^{∗}(M̃) ⊗_{R} V ⟶ Λ^{∗}(M̃; M̃ × V) sends s ⊗ v given by a section s of ⋀^{p} T^{∗}M̃ and v ∈ V to the section x ↦ s(x) ⊗ v. We denote by A(M̃) the ordinary deRham map sending a p-form ω to the singular cosimplex σ ↦ ∫ σ^{∗}ω. The isomorphism Φ maps ϕ ⊗ v to the R-map C_{∗}^{s}(M̃) ⟶ V, σ ↦ ϕ(σ)v.

We denote by L^{2}Λ^{p}(M; V) the Hilbert completion of Λ^{p}(M; V) under the inner product << ω, η >>:= ∫_{M} ω ∧ ∗η. For later purposes we state the following result whose proof can be found in Müller [26] page 239.

Theorem 1.10 (Hodge-decomposition theorem) a.) H_{harm}^{p}(M, M_{1}; V) = ker(d) ∩ ker(δ) ∩ Λ_{1}^{p}(M, M_{1}; V) b.) The R-modules ker(Δ) ∩ Λ_{1}^{p}(M, M_{1}; V) and H_{harm}^{p}(M, M_{1}; V) are finitely generated. c.) We have the orthogonal decomposition

Λ_{1}^{p}(M, M_{1}; V) = H_{harm}^{p}(M, M_{1}; V) ⊕ d(Λ_{1}^{p−1}(M, M_{1}; V)) ⊕ δ(Λ_{1}^{p+1}(M, M_{1}; V))

L^{2}Λ^{p}(M, M_{1}; V) = H_{harm}^{p}(M, M_{1}; V) ⊕ clos(d(Λ_{1}^{p−1}(M, M_{1}; V))) ⊕ clos(δ(Λ_{1}^{p−1}(M, M_{1}; V)))

d.) The inclusion i : H_{harm}^{p}(M, M_{1}; V) ⟶ ker(d) ∩ Λ_{1}^{p}(M, M_{1}; V) composed with deRham map has image contained in the space of cocycles in C_{s}^{p}(M, M_{1}; V). We obtain an isomorphism

\overline{i} : H_{harm}^{p}(M, M_{1}; V) ⟶ H_{s}^{p}(M, M_{1}; V)

The Laplace operator Δ^{p} : Λ_{2}^{p}(M; V) ⟶ Λ_{2}^{p}(M; V) is an elliptic self-adjoint differential operator. Its spectrum is a pure point spectrum consisting of non-negative real numbers. For λ ≥ 0 we put

1.11 E_{λ}^{G}(M, M_{1}; V)^{p} = {ω ∈ Λ_{2}^{p}(M; V) | Δ^{p}ω = λω}

Because G acts on (M, M_{1}) by isometries, Δ is compatible with the RG-structure. Since E_{λ}^{G}(M, M_{1}; V)^{p} is finitely generated, it defines an element in the real representation ring Rep_{R}(G). We define the equivariant Zeta-function

1.12 ζ_{p}^{G}(M, M_{1}; V)(s) := ∑_{λ>0} λ^{−s} · [E_{λ}^{G}(M, M_{1}; V)^{p}] ∈ C ⊗_{Z} Rep_{R}(G) ζ^{G}(M, M_{1}; V) := ∑_{p=0}^{m}(−1)^{p} · p · ζ_{p}^{G}(M, M_{1}; V)

for s ∈ C with Real(s) > dim(M)/2. Because Rep_{R}(G) is a finitely generated free abelian group with the isomorphism classes of irreducible representations as base, we may identify C ⊗_{Z} Rep_{R}(G) with C^{r} for r = rk_{Z}(Rep_{R}(G)). Hence it makes sense to speak of convergence in C ⊗_{Z} Rep_{R}(G). Restriction to the trivial subgroup defines an homomorphism res : C ⊗_{Z} Rep_{R}(G) ⟶ C ⊗_{Z} Rep_{R}({1}) = C. The image of ζ_{p}^{G}(M, M_{1}; V) under this map is just the non-equivariant Zeta-function which converges absolutely for Real(s) > dim(M)/2 (see Gilkey [15], page 79). This implies that ζ_{p}^{G}(M, M_{1}; V) converges absolutely for Real(s) > dim(M)/2.

Lemma 1.13 The equivariant Zeta-function ζ_{p}^{G}(M, M_{1}; V) has a meromorphic extension to C. It is analytic in zero and its derivative at zero lies in R ⊗_{Z} Rep_{R}(G).

We defer the proof of Lemma 1.13 to section 4.

Definition 1.14 The analytic torsion

ρ_{an}^{G}(M, M_{1}; V) ∈ R ⊗_{Z} Rep_{R}(G)

is defined by ρ_{an}^{G}(M, M_{1}; V) := ∑_{p=0}^{m}(−1)^{p} · p · \frac{d}{ds}ζ_{p}^{G}(M, M_{1}; V)|_{s=0}

Example 1.15 Fix a positive real number μ. Equip R with the standard metric and the unit circle S^{1} with the Riemannian metric for which R ⟶ S^{1}, t ↦ exp(2πiμ^{−1}t) is isometric. Then S^{1} has volume μ. Let Z/2 act on S^{1} by complex conjugation. The Laplace operator Δ^{1} : Λ^{1}R ⟶ Λ^{1}R maps f(t)dt to −f′′(t)dt. By checking the μ-periodic solutions of f′′(t) = −λf(t) one shows that Δ^{1} on S^{1} has eigenspaces E_{λ}(S^{1})^{1} = span_{R}(f_{n}dt, g_{n}dt) for λ = (2πμ^{−1}n)^{2}, n ≥ 1, E_{λ}(S^{1})^{1} = span_{R}(dt) for λ = 0 and E_{λ}(S^{1}) = {0} otherwise, if f_{n}(exp(2πiμ^{−1}t)) = cos(2πμ^{−1}nt) and g_{n}(exp(2πiμ^{−1}t)) = sin(2πμ^{−1}nt). The Z/2-action on S^{1} induces the Z/2-action on E_{(2πμ^{−1}n)^{2}}(S^{1}) sending f_{n} to f_{n} and g_{n} to −g_{n}. Denote by ζ_{Rie}(s) = ∑_{n≥1} n^{−s} the Riemannian Zeta function . Let R be the trivial and R^{−} be the non-trivial 1-dimensional Z/2-representation. We get

ζ_{1}^{Z/2}(S^{1}; R) = (\frac{2π}{μ} )^{−2s} · ζ_{Rie}(2s) · ([R] + [R^{−}])

As ζ_{Rie}(0) = −\frac{1}{2} and ζ′_{Rie}(0) = −\frac{ln(2π)}{2} hold (see Titchmarsh [34]), we obtain

1.16 ρ_{an}^{Z/2}(S^{1}; R) = ln(μ) · ([R] + [R^{−}])

By restriction to the trivial subgroup we derive

1.17 ρ_{an}(S^{1}; R) = 2 · ln(μ)

Example 1.18 Equip D^{1} = [0, 1] with the standard metric scaled by μ > 0. The volume form is then μdt. The Laplace operator Δ^{1} : Λ^{1}D^{1} ⟶ Λ^{1}D^{1} maps f(t)dt to μ^{−2}f′′(t)dt. We obtain E_{λ}(D^{1}; R) = span_{R}(sin(πnt)dt), E_{λ}(D^{1}, ∂D^{1}; R) = span_{R}(cos(πnt)dt) for λ = (\frac{π}{μ}n)^{2} for n ∈ Z, n ≥ 1 and E_{λ}(D^{1}; R) = E_{λ}(D^{1}, ∂D^{1}; R) = {0} otherwise. Hence we get

ζ_{1}(D^{1}; R) = ζ_{1}(D^{1}, ∂D^{1}; R) = (\frac{π}{μ})^{−2s} · ζ_{Rie}(2s)

This implies

1.19 ρ_{an}(D^{1}, R) = ρ_{an}(D^{1}, ∂D^{1}; R) = ln(2μ)

Proposition 1.20 (Poincaré Duality for Analytic Torsion) Let M be a m-dimensio-nal Riemannian G-manifold with orientation homomorphism w = w^{G}(M). If V is an orthogonal D^{G}(M)-representation and ∂M the disjoint union M_{1} ∐ M_{2}, we have

ρ_{an}^{G}(M, M_{1}; V) + (−1)^{m} · ρ_{an}^{G}(M, M_{2};^{w} V) = 0

\underline{\underline{Proof}} : Recall E_{λ}^{G}(M, M_{1}; V)^{p} = \{ω ∈ Λ_{2}^{p}(M, M_{1}; V) | Δω = λω\} and Λ_{2}^{p}(M, M_{1}; V) = {ω ∈ Λ^{p}(M; V) | ω satisfies b(M, M_{1})}, where the boundary conditions b(M, M_{1}) were defined in 1.7. Put E′_{λ}(M, M_{1}; V)^{p} := {ω ∈ Λ_{2}^{p}(M, M_{1}; V) | dδω = λω} and E′′_{λ}(M, M_{1}; V)^{p} := {ω ∈ Λ_{2}^{p}(M, M_{1}; V) | δdω = λω}. For λ ≠ 0 we obtain an orthogonal RG-sum decomposition 1.21 λ^{−1}dδ ⊕ λ^{−1}δd : E_{λ}^{G}(M, M_{1}; V)^{p} ⟶ E′_{λ}(M, M_{1}; V)^{p} ⊕ E′′_{λ}(M, M_{1}; V)^{p}

and inverse isometric RG-isomorphisms 1.22 λ^{−1/2}δ : E′_{λ}(M, M_{1}; V)^{p+1} ⟶ E′′_{λ}(M, M_{1}; V)^{p}

λ^{−1/2}d : E′′_{λ}(M, M_{1}; V)^{p} ⟶ E′_{λ}(M, M_{1}; V)^{p+1} The Hodge star operator ∗ induces an isometric RG-isomorphism

1.23 ∗ : E_{λ}^{G}(M, M_{1}; V)^{p} ⟶ E_{λ}^{G}(M, M_{2};^{w} V)^{p}

Now the following computation finishes the proof : ζ^{G}(M, M_{1}; V) = = ∑_{p=0}^{m} ∑_{λ>0}(−1)^{p} · p · λ^{−s} · [E_{λ}^{G}(M, M_{1}; V)^{p}] = ∑_{p=0}^{m} ∑_{λ>0}(−1)^{p} · p · λ^{−s} · [E_{λ}^{G}(M, M_{2};^{w} V)^{p}] = (−1)^{m} · ∑_{p=0}^{m} ∑_{λ>0}(−1)^{m−p} · p · λ^{−s} · [E_{λ}^{G}(M, M_{2};^{w} V)^{p}] = (−1)^{m+1} · ∑_{p=0}^{m} ∑_{λ>0}(−1)^{m−p} · (m − p) · λ^{−s} · [E_{λ}^{G}(M, M_{2};^{w} V)^{p}]

+(−1)^{m} · m · ∑_{p=0}^{m} ∑_{λ>0}(−1)^{m−p} · λ^{−s} · [E_{λ}^{G}(M, M_{2};^{w} V)^{p}] = (−1)^{m+1} · ζ^{G}(M, M_{2};^{w} V)

+(−1)^{m} · m · ∑_{p=0}^{m} ∑_{λ>0}(−1)^{m−p} · λ^{−s} · ([E′_{λ}(M, M_{1}; V)^{p+1}] + [E′_{λ}(M, M_{1}; V)^{p}]) = (−1)^{m+1} · ζ^{G}(M, M_{2};^{w} V)

Suppose that M is orientable and closed, its dimension m is even and G acts orientation preserving. As w is trivial, we get ρ_{an}^{G}(M; V) = 0.

Remark 1.24 We often put the condition on the Riemannian metric that it is a product near the boundary, i.e. there is an equivariant collar f : ∂M × [0, 1[ onto an invariant neighbourhood of the boundary such that f is isometric if we equip ∂M ⊂ M and U ⊂ M with the induced, [0, 1[ with the standard and ∂M × [0, 1[ with the product metric. This condition ensures that for two such Riemannian G-manifolds M and N and an isometric G-diffeomorphism f : M_{1} ⟶ N_{1} between open and closed submanifolds M_{1} ⊂ ∂M and N_{1} ⊂ ∂N there is the structure of an Riemannian G-manifold on M ∪_{f} N such that the obvious inclusions i_{M} : M ⟶ M ∪_{f} N and i_{N} : N ⟶ M ∪_{f} N are isometric G-imbeddings. Let V resp. W be an orthogonal D^{G}(M) resp. D^{G}(N)-representation. We denote by j_{M} : M_{1} ⟶ M and j_{N} : N_{1} ⟶ N the inclusions. Fix an orthogonal D^{G}(M_{1})-isomorphism f̄: j_{M}^{∗}V ⟶ f^{∗}j_{N}^{∗}W. Then there is an orthogonal D^{G}(M ∪_{f} N)-representation V ∪f̄W such that j_{M}^{∗}(V ∪f̄W) and V resp. j_{N}^{∗}(V ∪f̄W) and W agree. If G is trivial and M_{1} is connected, this follows from the Theorem of Seifert-van Kampen. In the general case one must apply a generalized version saying that the corresponding diagram of fundamental categories in the sense of [21] is a push out of categories. Alternatively, one may think of the representations as G-vector bundles and glue them together.

In particular we can choose f = id and f̄ = id and consider M ∪_{M_{1}} M and V ∪_{M_{1}} V. There is a canonical Z/2-structure on M ∪_{M_{1}} M obtained by switching the two copies of M. Hence we can consider M ∪_{M_{1}} M as a Riemannian G × Z/2-manifold. The Z/2-structure induces a Z/2-action on D^{G}(M ∪_{M_{1}} M) and D^{G×Z/2}(M ∪_{M_{1}} M) is the semi-direct product D^{G}(M ∪_{M_{1}} M) ×_{s} Z/2, provided that M_{1} is not empty. The orientation homomorphism w^{G×Z/2}(M ∪_{M_{1}} M) maps (u, ±1) ∈ D^{G×Z/2}(M ∪_{M_{1}} M) to ±1 · w^{G}(M ∪_{M_{1}} M)(u). One can extend V ∪_{M_{1}} V an orthogonal D^{G×Z/2}(M ∪_{M_{1}} M)-representation by (u, ±1) · v = u · v for u ∈ D^{G}(M ∪_{M_{1}} M) and v ∈ V, since u and (−1) · u ∈ D^{G}(M ∪_{M_{1}} M) for ±1 ∈ Z/2 operate on V ∪_{M_{1}} V in the same way.

The following result will allow us to reduce the case of a manifold with boundary to the closed one. We have the isomorphism

1.25 (R ⊗_{Z} Rep_{R}(G)) ⊗_{Z} Rep_{R}(H) ⟶^{⊗R} R ⊗_{Z} Rep_{R}(G × H)

(λ · [P]) ⊗_{Z} [Q]  ↦  λ · [P ⊗_{R} Q]

For later purposes we mention the pairing, also denoted by ⊗_{R}, we obtain from 1.25 for G = H and restriction to the diagonal.

1.26 (R ⊗_{Z} Rep_{R}(G)) ⊗_{Z} Rep_{R}(G) ⟶^{⊗R} R ⊗_{Z} Rep_{R}(G)

For the next result put H = Z/2 in 1.25.

Proposition 1.27 (Double Formula for Analytic Torsion) Let M be a Riemannian G-manifold such that the Riemannian metric is a product near the boundary. Suppose that ∂M is the disjoint union of M_{1} and M_{2}. Let V be an orthogonal D^{G}(M)-representation. Then we have :

a.) E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)^{p} = (E_{λ}^{G}(M; V)^{p} ⊗_{R} R) ⊕ (E_{λ}^{G}(M, M_{1}; V)^{p} ⊗_{R} R^{−})

b.) ζ_{p}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) = (ζ_{p}^{G}(M; V) ⊗_{C} C) ⊕ (ζ_{p}^{G}(M, M_{1}; V) ⊗_{C} C^{−})

c.) ρ_{an}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) = (ρ_{an}^{G}(M; V) ⊗_{R} [R]) + (ρ_{an}^{G}(M, M_{1}; V) ⊗_{R} [R^{−}])

\underline{\underline{Proof}} : Obviously b.) and c.) follow from a.).Let τ : M ∪_{M_{1}} M ⟶ M ∪_{M_{1}} M be the flip map. Define

1.28 E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)_{+}^{p} = {ω ∈ E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)^{p} | τ^{∗}ω = ω}

E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)_{−}^{p} = {ω ∈ E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)^{p} | τ^{∗}ω = −ω}

Let i : M ⟶ M ∪_{M_{1}} M be the inclusion onto the first summand. Obviously i^{∗} is compatible with d, ∗, δ and Δ. Since τ is an isometry and reverses the local orientation at points in M_{1}, the induced map τ^{∗} maps the volume form d(M ∪_{M_{1}} M) to −d(M ∪_{M_{1}} M). This implies τ^{∗} ∘ ∗ = − ∗ ∘τ^{∗}. As τ is the identity on M_{1}, we get (i^{∗}τ^{∗}ω)_{tan} = (i^{∗}ω)_{tan} on M_{1}. Hence i^{∗}ω satisfies the boundary conditions b(M, ∅) resp. b(M, M_{1}) (see 1.7) if τ^{∗}ω = ω resp. τ^{∗}ω = −ω holds. Thus we can define RG-maps

1.29 i^{+} : E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)_{+}^{p} ⟶ E_{λ}^{G}(M; V) ω ↦ i^{∗}ω

i^{−} : E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)_{−}^{p} ⟶ E_{λ}^{G}(M, M_{1}; V)^{p} ω ↦ i^{∗}ω

Obviuosly i^{+} is injective, as i^{∗}ω determines ω because of τ^{∗}ω = ω. We have to show that i^{∗} is surjective. Given ω ∈ E_{λ}^{G}(M; V), there is only one candidate as preimage, namely ω ∪_{M_{1}} ω. The problem is that ω ∪_{M_{1}} ω is smooth on (M ∪_{M_{1}} M) − M_{1} and a priori only continuous on M ∪_{M_{1}} M, but we need smoothness on M ∪_{M_{1}} M. The obvious inclusion induces an RG-isomorphism

1.30 j : E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)_{+}^{p} ⊕ E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)_{−}^{p} ⟶

E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)^{p}

For any η ∈ Λ^{p}(M ∪_{M_{1}} M; V) we get
<< ω ∪_{M_{1}} ω, η >>_{M∪_{M_{1}}M}=<< ω, i^{∗}η >>_{M} + << ω, i^{∗}τ^{∗}η >>_{M}

since an integral over M ∪_{M_{1}} M is the sum of its restrictions to the two copies of M. If τ^{∗}η = −η holds, then << ω ∪_{M_{1}} ω, η >>_{M∪_{M_{1}}M}= 0. Consider η ∈ E_{μ}^{G}(M ∪_{M_{1}} M, V ∪_{M_{1}} V)^{p} with τ^{∗}η = η for μ ≠ λ. Then i^{∗}η ∈ E_{μ}^{G}(M, V) and μ ≠ λ imply << ω ∪_{M_{1}} ω, η >>= 0. Notice that the Hilbert space L^{2}Λ^{p}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) has an orthonormal bases of smooth eigenvectors of Δ_{M∪_{M_{1}}M}. For E_{λ}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)_{+}^{p} choose an orthonormal bases {η_{1}, η_{2}, ...η_{r}}. Then the Fourier developement of ω ∪_{M_{1}} ω is by the computations above

ω ∪_{M_{1}} ω = ∑_{i=1}^{r} << ω ∪_{M_{1}} ω, η_{i} >> ·η_{i}

This equation holds in L^{2}Λ^{p}(M ∪_{M_{1}} M; V ∪_{M_{1}} V). As both sides are represented by continuous sections, the two sides of the equation agree as functions. The right side is smooth and hence also ω ∪_{M_{1}} ω. This finishes the proof of Proposition 1.27.

We define the equivariant Euler characteristic

1.31 χ^{G}(M, M_{1}; V) = ∑_{p=0}^{m}(−1)^{p} · [H_{p}(M, M_{1}; V)] ∈ Rep_{R}(G)

Proposition 1.32 (Product Formula for Analytic Torsion) Regard a Riemannian Gmanifold M whose boundary is the disjoint union M_{1} ∐ M_{2} and a Riemannian H-manifold N with empty boundary. Let V resp. W be an orthogonal D^{G}(M)- resp. D^{H}(N)-representation. Then M × N is a Riemannian G × H-manifold and V ⊗_{R} W an orthogonal D^{G×H}(M × N)- representation and we get using the pairing 1.25

ρ_{an}^{G×H}(M × N, M_{1} × N; V ⊗_{R} W)

= χ^{G}(M, M_{1}; V) ⊗_{R} ρ_{an}^{H}(N; W) + ρ_{an}^{G}(M, M_{1}; V) ⊗_{R} χ^{H}(N; W)

\underline{\underline{Proof}} : :We show the analogous statement for the Zeta-functions if Real(s) > dim(M)/2. We conclude for λ > 0 from 1.21 and 1.22

1.33 ∑_{p≥0}(−1)^{p} · [E_{λ}^{G}(M, M_{1}; V)^{p}] = 0

We derive from Theorem 1.10 1.34 χ^{G}(M, M_{1}; V) = ∑_{p≥0}(−1)^{p} · [E_{0}^{G}(M, M_{1}; V)]

Notice that Λ^{∗}(M; V) ⊗_{R} Λ^{∗}(N; W) is dense in Λ^{∗}(M × N; V ⊗_{R} W) and on this dense subspace we have Δ_{M×N} = Δ_{M} ⊗_{R} id + id ⊗_{R} Δ_{N}. The eigenvalues of Δ_{M} build a Hilbert bases for L^{2}Λ^{∗}(M; V). We conclude

1.35 E_{γ}^{G×H}(M × N, M_{1} × N; V ⊗_{R} W)^{i} = ⊕_{p+q=i} ⊕_{λ+μ=γ} E_{λ}^{G}(M, M_{1}; V)^{p} ⊗_{R} E_{μ}^{H}(N; W)^{q}

Now we compute

ζ^{G×H}(M × N; M_{1} × N; V ⊗_{R} W)

= ∑_{i≥0}(−1)^{i} · i ∑_{γ>0} γ^{−s} · [E_{γ}^{G×H}(M × N; M_{1} × N; V ⊗_{R} W)]

= ∑_{p,q} ∑_{λ+μ>0}(λ + μ)^{−s} · (−1)^{p+q} · (p + q) · [E_{λ}^{G}(M, M_{1}; V)^{p} ⊗_{R} E_{μ}^{H}(N; W)^{q}]

= ∑_{λ+μ>0}(λ + μ)^{−s} · (∑_{p≥0}(−1)^{p} · [E_{λ}^{G}(M, M_{1}; V)^{p}]) ⊗_{R} (∑_{q≥0}(−1)^{q} · q · [E_{μ}^{H}(N; W)^{q}])

+ ∑_{λ+μ>0}(λ + μ)^{−s} · (∑_{p≥0}(−1)^{p} · p · [E_{λ}^{G}(M, M_{1}; V)]) ⊗_{R} (∑_{q≥0}(−1)^{q} · [E_{μ}^{H}(N; W)^{q}])

= (∑_{p≥0}(−1)^{p} · [E_{0}^{G}(M, M_{1}; V)]) ⊗_{R} (∑_{q≥0}(−1)^{q} · q · ∑_{μ>0} μ^{−s} · [E_{μ}^{H}(N; W)^{q}])

+ (∑_{p≥0}(−1)^{p} · p · ∑_{λ>0} λ^{−s} · [E_{λ}^{G}(M, M_{1}; V)^{p}]) ⊗_{R} (∑_{q≥0}(−1)^{q} · [E_{0}^{H}(N; W)^{q}])

+ ∑_{λ,μ>0}(λ + μ)^{−s} · (∑_{p≥0}(−1)^{p} · p · [E_{λ}^{G}(M, M_{1}; V)^{p}]) ⊗_{R} (∑_{q≥0}(−1)^{q} · [E_{μ}^{H}(N; W)^{q}])

+ ∑_{λ,μ>0}(λ + μ)^{−s} · (∑_{p≥0}(−1)^{p} · [E_{λ}^{G}(M, M_{1}; V)^{p}]) ⊗_{R} (∑_{q≥0}(−1)^{q} · q · [E_{μ}^{H}(N; W)^{q}])

= χ^{G}(M, M_{1}; V) ⊗_{R} (∑_{q≥0}(−1)^{q} · q · ζ_{q}^{H}(N; W)) +

(∑_{p≥0}(−1)^{p} · p · ζ_{p}^{G}(M, M_{1}; V)) ⊗_{R} χ^{H}(N; W)

= χ^{G}(M, M_{1}; V) ⊗_{R} ζ^{H}(N; W) + ζ^{G}(M, M_{1}; V) ⊗_{R} χ^{H}(N; W)

1.36 If H is a subgroup of G, then there are obvious restriction and induction homomorphism for R ⊗_{Z} Rep_{R}(G). Restriction and induction is also defined for (M, M_{1}; V). The equivariant analytic torsion is compatible with restriction and induction.

2. Torsion Invariants for Chain Complexes

In order to define PL-torsion invariants for G-CW -complexes and Riemannian Gmanifolds it is convenient to do this for RG-chain complexes C = (C_{∗}, c_{∗}). We say that C is finite if C_{i} is finitely generated for all i and is zero for all but a finite number of i ∈ Z. Let f : C ⟶ D be a RG-chain equivalence of finite RG-chain complexes and let ϕ : C_{odd} ⊕ D_{ev} ⟶ D_{odd} ⊕ C_{ev} be a RG-isomorphism. Denote by Cone(f) the mapping cone of f whose differential is also denoted by c and given by

c_{n} = ( \begin{matrix}−c_{n−1} & 0 \\ f_{n−1} & d_{n} \end{matrix} ) : C_{n−1} ⊕ D_{n} ⟶ C_{n−2} ⊕ D_{n−1}

Choose a chain contraction γ of Cone(f), i.e. a map of degree 1 such that c ∘ γ + γ ∘ c = id. Then we obtain an isomorphism (c + γ) : Cone(f)_{odd} ⟶ Cone(f)_{ev} if Cone(f)_{odd} resp. Cone(f)_{ev} is the sum of all chain modules of odd resp. even dimension. If π denotes the obvious permutation map, we get an RG-isomorphism of a finitely generated RG-module

Cone(f)_{odd} ⟶^{(c+γ)} Cone(f)_{ev} ⟶^{π} C_{odd} ⊕ D_{ev} ⟶^{ϕ} D_{odd} ⊕ C_{ev} ⟶^{π} Cone(f)_{odd}

Denote its class in K_{1}(A) by

2.1 t^{G}(f, ϕ) = t(f, ϕ) ∈ K_{1}(A)

We recall that K_{1}(A) is the abelian group generated by automorphisms f : P ⟶ P of finitely generated RG-modules with the relations [f_{2}] = [f_{1}] + [f_{3}] for any exact sequence {0} ⟶ (P_{1}, f_{1}) ⟶ (P_{2}, f_{2}) ⟶ (P_{3}, f_{3}) ⟶ {0} and [g ∘ f] = [g] + [f] for f, g : P ⟶ P and [id : P ⟶ P] = 0. We refer to Lück (1989)[21], chapter 12, for more details about this invariant and the proofs that it is well-defined. The proofs of some results of this sections are omitted as they are very similar to the ones appearing there.

A RG-Hilbert complex is a RG-chain complex C together with a G-invariant R-Hilbert space structure on each C_{n}. Let f : C ⟶ D be a RG-chain equivalence of finite RG-Hilbert complexes . Fix an isometric RG-isomorphism ϕ : C_{odd} ⊕ D_{ev} ⟶ D_{odd} ⊕ C_{ev}. Its existence follows from the Polar Decomposition Theorem and the fact that C_{odd} ⊕ D_{ev} and D_{odd} ⊕ C_{ev} are RG-isomorphic. Given a RG-module P, let P^{∗} be Hom_{R}(P, R) equipped with the RGmodule structure g · f = f ∘ l(g^{−1}). The natural RG-map P ⟶ P^{∗∗} is bijective if and only if P is finitely generated. We obtain an involution

2.2 ∗ : K_{1}(A) ⟶ K_{1}(A) [f] ↦ [f^{∗}] Define the Hilbert torsion

2.3 ht^{G}(f) = ht(f) ∈ K_{1}(RG)^{Z/2}

by ht(f) = t(f, ϕ) + ∗t(f, ϕ). This is independent of the choice of ϕ. If ψ is another choice we have

(t(f, ϕ) + ∗t(f, ϕ)) − (t(f, ψ) + ∗t(f, ψ))

= (t(f, ϕ) − t(f, ψ)) + ∗(t(f, ϕ) − t(f, ψ))

= [ψ^{−1} ∘ ϕ] + ∗[ψ^{−1} ∘ ϕ]

= [ψ^{−1} ∘ ϕ] − [ψ^{−1} ∘ ϕ]

= 0

Proposition 2.4 If f and g : C ⟶ D are RG-chain homotopic, we get

ht(f) = ht(g)

Let C be a finite RG-Hilbert complex. Consider its homology H(C) as aRG-chain complex by the trivial differential. Suppose that additionally H(C) has the structure of a RG-Hilbert complex. Up to RG-chain homotopy there is precisely one RG-chain map i : H(C) ⟶ C satisfying H(i) = id. Define the Hilbert-Reidemeister torsion

2.5 hr^{G}(C) = hr(C) ∈ K_{1}(RG)^{Z/2}

by hr(C) := ht(i : H(C) ⟶ C).

Example 2.6 Let G be the trivial group. Let C be a finite R-Hilbert complex together with a R-Hilbert structure on H(C). Choose orthonormal bases for each C_{i} and H(C)_{i}. The torsion defined by Milnor (1966)[25] takes values in K̃_{1}(R) = R^{∗}/Z^{∗}. Its square is a positive real number which agrees with hr(C) ∈ K_{1}(R) = R^{∗}.

We collect the main properties of these invariants. Consider the commutative diagram of finite RG-Hilbert complexes whose rows are exact and whose vertical arrows are RG-chain equivalences

2.7  [figure: 0; C; i; D; p; E; 0]

[figure: f; g; h; 0; C′; i′; D′; p′; E′; 0]

We get an acyclic finite RG-Hilbert complex 0 ⟶ C_{n} ⟶^{i_{n}} D_{n} ⟶^{p_{n}} E_{n} ⟶ 0 concentrated in dimension 0, 1, 2 for each n ∈ Z. Let hr(C_{n}, D_{n}, E_{n}) be its Hilbert-Reidemeister torsion. Define

2.8 hr(C, D, E) := ∑_{n}(−1)^{n} · hr(C_{n}, D_{n}, E_{n}) ∈ K_{1}(RG)^{Z/2}

Proposition 2.9 (Additivity) ht(f) − ht(g) + ht(h) = hr(C, D, E) − hr(C′, D′, E′)

Let 0 ⟶ C ⟶^{i} D ⟶^{p} E ⟶ 0 be an exact sequence of finite RG-Hilbert complexes. Suppose that H(C), H(D) and H(E) come with RG-Hilbert structures. The long homology sequence H(C, D, E) inherits the structure of an acyclic finite RG-Hilbert complex. Analogously to Milnor [25] we get

Proposition 2.10 (Additivity) hr(C) − hr(D) + hr(E) = hr(C, D, E) − hr(H(C, D, E))

Proposition 2.11 (Composition Formula) If f : C ⟶ D and g : D ⟶ E are chain equivalences of RG-Hilbert complexes , we have

ht(g ∘ f) = ht(g) + ht(f)

Proposition 2.12 (Comparision Formula) If f : C ⟶ D is a RG-chain equivalence of finite RG-Hilbert complexes and H(C) and H(D) come with finite RG-Hilbert complex structures, then H(f) : H(C) ⟶ H(D)is a RG-chain equivalence of finite RG-Hilbert complexes and we get

ht(f) = hr(D) − hr(C) + ht(H(f))

\underline{\underline{Proof}} : Proposition 2.11

Given two finite groups G and H, there is a pairing

2.13 ⊗_{R} : K_{0}(RG) ⊗_{Z} K_{1}(RH) ⟶ K_{1}(RG × H)

[P] ⊗_{Z} [f : Q ⟶ Q] ↦ [id ⊗_{R} f : P ⊗_{R} Q ⟶ P ⊗_{R} Q]

Notice that K_{0}(RG) is Rep_{R}(G). Because P and P^{∗} are (not naturally) RG-isomorphic, we get induced pairings

2.14 ⊗_{R} : K_{0}(RG) ⊗_{Z} K_{1}(RH)^{Z/2} ⟶ K_{1}(RG × H)^{Z/2}

⊗_{R} : K_{0}(RG) ⊗_{Z} K_{1}(RG)^{Z/2} ⟶ K_{1}(RG)^{Z/2}

If C is a finite RG-chain complex, define its Euler characteristic

2.15 χ(C) := ∑_{i≥0}(−1)^{i} · [C_{i}] = ∑_{≥0}(−1)^{i} · [H_{i}(C)] ∈ K_{0}(RG)

Proposition 2.16 (Product Formula) a.) Let f : C′ ⟶ C resp. g : D′ ⟶ D be a RG- resp. RH-chain equivalence of finite RG-Hilbert complexes. Then

ht^{G×H}(f ⊗_{R} g) = χ^{G}(C) ⊗_{R} ht^{H}(g) + ht^{G}(f) ⊗_{R} χ^{H}(D)

b.) Let C resp. D be a finite RG- resp. RH-chain complex. Assume that H(C) resp. H(D) possesses a finite RG- resp. RH-Hilbert complex structure. Equip H(C ⊗_{R} D) with the finite RG × H- Hilbert structure for which the Künneth isomorphism H(C) ⊗_{R} H(D) ≅ H(C ⊗_{R} D) becomes an isometry. Then

hr^{G×H}(C ⊗_{R} D) = χ^{G}(C) ⊗_{R} hr^{H}(D) + hr^{G}(C) ⊗_{R} χ^{H}(D)

2.17 If H ⊂ G is a subgroup, there are obvious induction and restriction homomorphism. Both ht and hr are compatible with induction and restriction.

If C = (C_{∗}, c_{∗}) is a finite RG-chain complex, define its dual RG-chain complex C^{n−∗} by (C^{n−∗})_{r} := (C_{r})^{∗} and (c^{n−∗})_{r} := (c_{n−r+1})^{∗}. If the RG-module P has a finite RG-Hilbert structure given by an RG-isomorphism ϕ : P ⟶ P^{∗} satisfying ϕ^{∗} = ϕ, equip P^{∗} with the finite RG-Hilbert structure P^{∗} ⟶^{ϕ^{−1}} P ⟶ P^{∗∗}. Hence C^{n−∗} inherits a finite RG-Hilbert complex structure from C. Notice that the natural RG-map C ⟶ (C^{n−∗})^{n−∗} is an isometric RG-chain isomorphism.

Proposition 2.18 a.) If f : C ⟶ D is a RG-chain equivalence of RG-Hilbert complexes, we get

ht(f^{n−∗}) = (−1)^{n} · ht(f)

b.) Let C be a finite RG-Hilbert complex. Assume that H(C) has a finite RG-Hilbert complex structure. Then:

hr(C^{n−∗}) = (−1)^{n+1} · hr(C)

Let f : C ⟶ D be a RG-chain equivalence of finite RG-Hilbert complexes. Let κ(C_{i}) and κ̃(C_{i}) resp. κ(D_{i}) and κ̃(D_{i}) be two different RG-Hilbert structures on C_{i} resp D_{i}. We obtain RG-automorphisms

2.19 κ(C_{i})^{−1} ∘ κ̃(C_{i}) : C_{i} ⟶ C_{i}^{∗} ⟶ C_{i}

κ(D_{i})^{−1} ∘ κ̃(D_{i}) : D_{i} ⟶ D_{i}^{∗} ⟶ D_{i}

Proposition 2.20 We have :

ht(f : (C, κ(C)) ⟶ (D, κ(D))) − ht(f : (C, κ̃(C)) ⟶ (D, κ̃(D)))

= ∑_{i≥0}(−1)^{i} · [κ(C_{i})^{−1} ∘ κ̃(C_{i})] − ∑_{i≥0}(−1)^{i} · [κ(D_{i})^{−1} ∘ κ̃(D_{i})]

\underline{\underline{Proof}} : Because of Lemma 2.11 it suffices to prove

ht(id : (C, κ(C)) ⟶ (C, κ̃(C))) = ∑_{i≥0}(−1)^{i} · [κ(C_{i})^{−1} ∘ κ̃(C_{i})]

Because of Proposition 2.9 we can assume that C is concentrated in dimension zero. If ϕ : (C_{0}, κ(C_{0})) ⟶ (C_{0}, κ̃(C_{0})) is an isometric RG-isomorphism, we compute

ht(id : (C, κ(C)) ⟶ (C, κ̃(C))) = [ϕ] + [ϕ^{∗}] = [ϕ] + [κ̃(C_{0}) ∘ ϕ^{−1} ∘ κ(C_{0})^{−1}]

= [ϕ ∘ κ̃(C_{0}) ∘ ϕ^{−1} ∘ κ(C_{0})^{−1}] = [κ(C_{0})^{−1} ∘ κ̃(C_{0})]

Next we compare K_{1}(RG)^{Z/2} and R ⊗_{Z} Rep_{R}(G). Let I be a complete set of representatives for the isomorphism classes of irreducible RG-representations. For any finitely generated RG-module W we have the natural RG-isomorphism

2.21 ϕ : ⊕_{V ∈I} Hom_{RG}(V, P) ⊗_{EndRG(V)} V ⟶ P f ⊗ v ↦ f(v)

By Schur’s Lemma End_{RG}(V) is a skew field. Hence the canonical homomorphism End_{RG}(V)^{∗} ⟶ K_{1}(End_{RG}(V)) induces an isomorphism from the abelianization of the group of units End_{RG}(V)_{ab}^{∗} ⟶ K_{1}(End_{RG}(V)) (see Silvester [33] page 133). We obtain from 2.21 an isomorphism

2.22 Φ : K_{1}(A) ⟶ ⊕_{V ∈I}End_{RG}(V)_{ab}^{∗} [g : P ⟶ P] ↦ {[Hom_{RG}(id_{V} , g)] | V ∈ I}

There is an involution ∗ : End_{RG}(V)_{ab}^{∗} ⟶ End_{RG}(V)_{ab}^{∗} sending [f] to [k^{−1} ∘ f^{∗} ∘ k] for any RG-isomorphism k : V ⟶ V^{∗}satisfying k = k^{∗}. Then ϕ is compatible with the involution ∗ on K_{1}(A) and the direct sum of the involutions ∗ on End_{RG}(V)_{ab}^{∗}. Because End_{RG}(V) is a skew field over R, it is isomorphic to R,C or H and we accordingly call V of real, complex or quaternionic type. Under these isomorphisms the involution on End_{RG}(V) corresponds to the trivial, complex or quaternionic involution. The map R^{+} × S^{3} ⟶ H^{∗} sending (λ, z) to λ · z is an isomorphism of Lie groups, if S^{3} inherits the Lie group structure from S^{3} ⊂ H. The Lie group S^{3} = SU(2) is its own commutator group. The inclusion R^{+} ↪ H^{∗} and the norm map H^{∗} ⟶ R sending a ∈ H to \sqrt{aā} induce to another inverse isomorphisms R^{+} ⟶ H_{ab}^{∗} and H_{ab}^{∗} ⟶ R^{+}. Now the inclusion i : R^{∗} ⟶ (End_{RG}(V)_{ab}^{∗})^{Z/2} mapping λ ∈ R^{∗} to λ · id : V ⟶ V induces an isomorphism

2.23 i : R^{∗} ⟶ (End_{RG}(V)_{ab}^{∗})^{Z/2} , if V is of real or complex type

i : R^{+} ⟶ (End_{RG}(V)_{ab}^{∗})^{Z/2} , if V is of quaternionic type

Let the isomorphisms γ_{1} : R^{+} ⟶ R resp. γ_{1} ⊕ γ_{2} : R^{∗} ⟶ R ⊕ Z/2 send r to ln(r) resp. (ln(| r |), r/ | r |). Denote by Î the subset of I consisting of V ’s of real or complex type and by R̂ep_{R}(G) the subgroup of Rep_{R}(G) generated by Î. For an abelian group A we get identifications

2.24 ⊕_{V ∈I}A ⟶ A ⊗_{Z} Rep_{R}(G)

⊕_{V ∈Î}A ⟶ A ⊗_{Z} R̂ep_{R}(G) from the Z-bases I resp. Î Define

2.25 Γ_{1} ⊕ Γ_{2} : K_{1}(RG)^{Z/2} ⟶ (R ⊗_{Z} Rep_{R}(G)) ⊕ (Z/2 ⊕_{Z} R̂ep_{R}(G))

to be the composition ((⊕_{V ∈Î} γ_{1} ⊕ γ_{2}) ⊕ (⊕_{V ∈I−Î} γ_{1})) ∘ (⊕_{V ∈I} i^{−1}) ∘ Φ^{Z/2}.

Proposition 2.26 The map Γ_{1} ⊕ Γ_{2} is an isomorphism and is natural with respect to induction and restriction and the operations of K_{0}(RG) = Rep_{R}(G). If we denote by m(V) = dimR (End_{RG}(V)) the Schur index of V, we get

Γ_{1} ([f : P ⟶ P]) =

∑_{V ∈I} \frac{1}{2·m(V)} · ln((det(Hom_{RG}(id_{V} , f) : Hom_{RG}(V, P) ⟶ Hom_{RG}(V, P))^{2})

where the determinant is taken for a linear map over R. The inverse of Γ_{1} ⊕ Γ_{2} maps (λ ⊗_{R} [V], ±[W]) to [exp(λ) · id : V ⟶ V] + [±id : W ⟶ W].

2.27 Until now we have dealt with homology and chain complexes. There is also a cohomology and cochain version. If C^{∗} is a finite RG-Hilbert cochain complex, let co(C^{∗}) be the finite RG-Hilbert complex with co(C^{∗})_{n} := C^{−n} and c^{−n} as n^{th}-differential. A RG-cochain equivalence f^{∗} : C^{∗} ⟶ D^{∗} induces a RG-chain equivalence co(f^{∗}) : co(C^{∗}) ⟶ co(D^{∗}) We define

ht(f^{∗}) := ht(co(f^{∗})) hr(C^{∗}) := hr(co(C^{∗}))

All the results of this section have cochain analogoues.

Let C be a finite RG-Hilbert chain complex with differential c_{n} : C_{n} ⟶ C_{n−1}. The adjoint of c_{n+1} is denoted by γ_{n} : C_{n} ⟶ C_{n+1}. Define a symmetric and non-negative definite RG-- homomorphism Δ_{n} : C_{n} ⟶ C_{n} by c_{n+1} ∘ γ_{n} + γ_{n−1} ∘ c_{n}. Let E_{λ}(Δ_{n}) be the eigenspace for λ ≥ 0. Define the holomorphic Zeta-function

2.28 ζ_{n} : C ⟶ C ⊗_{Z} Rep_{R}(G) s ↦ ∑_{λ>0} λ^{−s} · [E_{λ}(Δ_{n})]

By the finite-dimensional analogue of the Hodge Decomposition Theorem 1.10 we have a canonical isomorphism i : E_{0}(Δ_{n}) ⟶ H_{n}(C). Equip E_{0}(Δ_{n}) ⊂ C_{n} and H_{n}(C) with the induced RG-Hilbert structures. The following result motivates the definition of analytic torsion and Hilbert-Reidemeister torsion (cf. Ray-Singer [29], Prop. 1.7)

Proposition 2.29

a.) Γ_{1}(hr(C)) = ∑_{n≥0}(−1)^{n} · n · \frac{d}{ds}ζ_{n} |_{s=0}

b.) Γ_{2}(hr(C)) = 1

\underline{\underline{Proof}} : First we treat the case where C is acyclic. Then Δ_{n} is bijective. Let f_{n} : C_{n} ⟶ C_{n} be Δ_{n}^{n} = Δ_{n} ∘ Δ_{n} ∘ … ∘ Δ_{n}. Then the following diagram commutes

[figure: C_{odd}; C_{ev}; c + c^{∗}Δ^{−1}; f_{odd}; f_{ev}; C_{odd}; C_{ev}; cΔ^{−1} + c^{∗}]

As the lower horizontal arrow in the square above is the inverse of the adjoint of the upper horizontal arrow and c^{∗} ∘ Δ^{−1} is a chain contraction for C, hr(C) = − ∑_{n≥0}(−1)^{n} · n · [Δ_{n}] is true. One easily checks

Γ_{1}([Δ_{n}]) = Γ_{1}(∑_{λ>0}[λ · id : E_{λ}(Δ_{n}) ⟶ E_{λ}(Δ_{n})]) =

∑_{λ>0} ln(λ) · [E_{λ}(Δ_{n})] = −\frac{d}{ds}ζ_{n} |_{s=0}

This proves the claim for acyclic C. The general case follows from the obvious exact sequence of finite RG-Hilbert complex 0 ⟶ H(C) ⟶^{i} C ⟶ cok(i) ⟶ 0 and Proposition 2.10. We leave b.) to the reader.

3. PL-Torsion

Let G be a finite group. A finite G-CW -complex X is a CW -complex X together with a G-action such that for any g ∈ G and any open cell e ⊂ X with e ∩ l(g)(e) ≠ ∅ we have l(g)(e) = e and l(g) |_{e}= id_{e} where l(g) : X ⟶ X is left-multiplication with g. We want to define PL-torsion for finite G-CW -complexes with a RG-Hilbert structure on the homology resp. cohomology groups. We prefer the name PL-torsion instead of topological torsion, as in the equivariant setting torsion invariants are not invariants under G-homeomorphisms in general (see Cappell-Shaneson (1981) [7]). We will apply this to Riemannian G-manifolds using the equivariant triangulation theorem. Having these invariants also for G-CW -complex gives some useful flexibility.

Consider a pair (X, Y) of finite G-CW -complexes. Suppose for simplicity that X is connected. The following definitions are easily extended to the non-connected case. Let p : X̃ ⟶ X be the universal covering and Ỹ := p^{−1}(Y). Then (X̃, Ỹ) is a finite D^{G}(X)- CW -pair. The group D^{G}(X) was introduced in 1.1. Its cellular Z[D^{G}(X)]-chain complex C_{∗}(X̃, Ỹ) is free over Z[π_{1}(X)]. Fix an orthogonal D^{G}(X)-representation V. As we think of π = π_{1}(X) as group of deck transformations, D^{G}(X) and π act from the left on C_{∗}(X̃, Ỹ). Let¯ : Z[D^{G}(X)] ⟶ Z[D^{G}(X)] be the involution sending ∑ λ_{d} · d to ∑ λ_{d} · d^{−1} and similar for Rπ. There is an induced right module structure given by u · s := s̄ · u for s ∈ R[D^{G}(X)], u ∈ C_{∗}(X̃, Ỹ). Define the RG-Hilbert complex

3.1 C_{∗}(X, Y; V) := C_{∗}(X̃, Ỹ) ⊗ RπV

The RG-module structure comes from g · (u ⊗ Rπv) := g̃u ⊗ Rπg̃v for any lift g̃ ∈ D^{G}(X) of g ∈ G and u ∈ C_{∗}(X̃, Ỹ) and v ∈ V. We obtain a RG-Hilbert structure by requiring that for one (and hence all) cellular Zπ-base B of C_{∗}(X̃, Ỹ) the following R-isomorphism is an isometry with respect to the orthogonal structure on V

C_{∗}(X, Y; V) ⟶ ⊕_{B}V  (∑_{b∈B} a_{b} · b) ⊗ Rπv ↦ {ā_{b} · v | b ∈ B}

Define the RG-Hilbert cochain complex

3.2 C^{∗}(X, Y; V) := Hom_{Rπ}(C_{∗}(X̃, Ỹ), V)

The RG-structure is induced from the D^{G}(X)-action on Hom_{R}(C_{∗}(X̃, Ỹ); V) given by d · f := l(d) ∘ f ∘ l(d^{−1}) and the fact that the π-fixed point set is Hom_{Rπ}(C_{∗}(X̃, Ỹ); V). The RG-Hilbert structure is determined by the property that for one (and hence all) cellular Rπbase B of C_{∗}(X̃, Ỹ) the following R-isomorphism is isometric

C^{∗}(X, Y; V) ⟶ ⊕_{B}V ϕ ↦ {ϕ(b) | b ∈ B}

We have explained before how P^{∗} inherits a RG-Hilbert structure from P if P is a finitely generated RG-module with RG-Hilbert structure. Let H_{∗}(X, Y; V) resp. H^{∗}(X, Y; V) be the homology resp. cohomology of C_{∗}(X, Y; V) resp. C^{∗}(X, Y; V). There is an isometric RG-chain isomorphism

3.3 C^{∗}(X, Y; V) ⟶ Hom_{R}(C_{∗}(X, Y; V^{∗}), R)

sending ϕ ∈ Hom_{Rπ}(C_{∗}(X̃, Ỹ), V) to C_{∗}(X̃, Ỹ) ⊗ RπV^{∗} ⟶ R, u ⊗ Rπψ ↦ ψ ∘ ϕ(u). It induces a RG-isomorphism

3.4 H^{∗}(X, Y; V) ⟶ Hom_{R}(H_{∗}(X, Y; V^{∗}); R)

Definition 3.5 Let (X,Y) be a finite G-CW -pair and V be an orthogonal D^{G}(X)-represen-tation. Let κ_{∗} be a RG-Hilbert structure on H_{∗}(X, Y; V). Define the Hilbert -PL-torsion or briefly PL-torsion

ρ_{pl}^{G}(X, Y; V, κ_{∗}) ∈ K_{1}(RG)^{Z/2}

by ρ_{pl}^{G}(X, Y; V, κ_{∗}) := hr(C_{∗}(X, Y; V), κ_{∗}) (see 2.5). Similiarly, if κ^{∗} is a RG-Hilbert structure on H^{∗}(X, Y; V), define

ρ_{pl}^{G}(X, Y; V, κ^{∗}) ∈ K_{1}(RG)^{Z/2}

by ρ_{pl}^{G}(X, Y; V, κ^{∗}) := hr(C^{∗}(X, Y; V), κ^{∗}) (see 2.27)

We derive from 2.18

Proposition 3.6 If κ_{∗} and κ^{∗} are compatible with 3.4, we have

ρ_{pl}^{G}(X, Y; V, κ_{∗}) = −ρ_{pl}^{G}(X, Y; V, κ^{∗})

Remark 3.7 It is convenient to have both the homological and the cohomological definition. The first one is more convenient for computations as other related torsion invariants like Whitehead torsion are given by chain complexes and the cellular chain complex is easier to compute than the cochain complex. The second one fits better into the context of analytic torsion and deRham cohomology.

Let (f, f_{1}) : (X, X_{1}) ⟶ (Y, Y_{1}) be a G-homotopy equivalence of pairs of finite G-CW - complexes. In Dovermann-Rothenberg (1986) [11], Illman (1974) [17] and in Lück (1989)

[21] equivariant Whitehead torsion τ^{G}(f, f_{1}) ∈ W h^{G}(Y) is defined. Let V be an orthogonal D^{G}(Y)-representation. We consider RG-Hilbert structures κ_{i}(X) on H_{i}(X, X_{1}; f^{∗}V) and κ_{i}(Y) on H_{i}(Y, Y_{1}; f^{∗}V). Let the element u_{i} ∈ K_{1}(RG)^{Z/2} be given by the composition

H_{i}(X, X_{1}; f^{∗}V) ⟶^{H_{i}(f,f_{1})} H_{i}(Y, Y_{1}; V) ⟶^{κ_{i}(Y)} H_{i}(Y, Y_{1}; V)^{∗} ⟶^{H_{i}(f,f_{1})^{∗}}

H_{i}(X, X_{1}; f^{∗}V)^{∗} ⟶^{κ_{i}(X)^{−1}} H_{i}(X, X_{1}; f^{∗}V)

Proposition 3.8 There is a natural homomorphism

ω = ω^{G}(Y; V) : W h^{G}(Y) ⟶ K_{1}(RG)^{Z/2}

such that

ρ_{pl}^{G}(Y, Y_{1}; V, κ_{∗}(Y)) − ρ_{pl}^{G}(X, X_{1}; f^{∗}V, κ_{∗}(X)) = ω(τ^{G}(f, f_{1})) − ∑_{i≥0}(−1)^{i} · u_{i}

\underline{\underline{Proof}} : We describe ω in the language developed in Lück (1989) [21]. A class [k] ∈ W h^{G}(Y) is represented by an automorphism k : P ⟶ P of a finitely generated projective ZΠ(G, Y)- module P. Let x ∈ Ob(Π(G, Y)) be represented by the G-map x : G ⟶ Y with G as domain. Let v ∈ K_{1}(A) be the class of the RG-automorphism k(x) ⊗ Zπid_{V} of P (x) ⊗ ZπV where k(x) is given by evaluating k at x. Define ω([g]) := v + ∗v. We get from the definitions ω(τ^{G}(f, f_{1})) = ht(C_{∗}(f, f_{1}; V)). Now apply Proposition 2.12.

Remark 3.9 A G-homotopy equivalence (f, f_{1}) : (X, X_{1}) ⟶ (Y, Y_{1}) is simple if τ^{G}(f, f_{1}) vanishes. Hence ρ_{pl}^{G} depends only on the simple G-homotopy type, provided that H_{∗}(Y, Y_{1}; V) vanishes.

Consider the cellular G-push out of pairs of finite G-CW -complexes,where i_{1} is an inclusion of such pairs

3.10 [figure: (X_{0}, A_{0}); (X_{2}, A_{2}); i_{2}; i_{1}; j_{0}; j_{2}; (X_{1}, A_{1}); (X, A); j_{1}]

If V is an orthogonal D^{G}(X)-representation, we get an exact sequence of RG-chain complexes

{0} ⟶ C_{∗}(X_{0}, A_{0}, j_{0}^{∗}V) ⟶^{i_{1∗}⊕i_{2∗}} C_{∗}(X_{1}, A_{1}; j_{1}^{∗}V) ⊕ C_{∗}(X_{2}, A_{2}, j_{2}^{∗}V)

⟶^{j_{1∗}−j_{2∗}} C_{∗}(X, A; V) ⟶ {0}

Denote by M_{∗} the long homology sequence of the sequence above. Suppose that we have RG-Hilbert structures κ_{i} on H_{∗}(X_{i}, A_{i}, j_{i}^{∗}V) for i =, 0, 1, 2 and κ on H_{∗}(X, A, V). Then M_{∗} inherits the structure of an acyclic finite RG-Hilbert complex. We derive from Proposition 2.10.

Proposition 3.11 (Sum formula for PL-torsion) We have ρ_{pl}^{G}(X, A; V, κ) = ρ_{pl}^{G}(X_{1}, A_{1}; j_{1}^{∗}V, κ_{1}) + ρ_{pl}^{G}(X_{2}, A_{2}; j_{2}^{∗}V, κ_{2}) − ρ_{pl}^{G}(X_{0}, A_{0}; j_{1}^{∗}V, κ_{0}) + hr(M_{∗})

Let (X, A) resp. (Y, B) be a pair of finite G- resp. H-CW -complexes and V resp. W be orthogonal D^{G}(X)- resp D^{H}(Y)-representations. Equip H_{∗}(X, A, V) resp. H_{∗}(Y, B; W) with RG- resp. RH-Hilbert structures κ(X) resp. κ(Y). Then (X, A) × (Y, B) is a pair of finite G × H-CW -complexes and V ⊗_{R} W an orthogonal representation of D^{G×H}(X × Y) = D^{G}(X) × D^{H}(Y). Put on H_{∗}((X, A) × (Y, B); V ⊗_{R} W) the RG × H-Hilbert structure κ(X ×Y) induced by the RG×H-Künneth isomorphism from H_{∗}(X, A; V) ⊗_{R} H_{∗}(Y, B; W) to H_{∗}((X, A) × (Y, B); V ⊗_{R} W). We define the equivariant Euler characteristic

3.12 χ^{G}(X, A; V) ∈ K_{0}(RG) = Rep_{R}(G)

by χ^{G}(X, A; V) := ∑_{n≥0}(−1)^{n} · [C_{n}(X, A; V)] = ∑_{n≥0}(−1)^{n} · [H_{n}(X, A; V)] as we have already done for manifolds in 1.31. We derive from Proposition 2.16

Proposition 3.13 (Product Formula for PL-Torsion) ρ_{pl}^{G×H}(X, A) × (Y, B); V ⊗_{R} W, κ(X × Y)) =

ρ_{pl}^{G}(X, A; V, κ(X)) ⊗_{R} χ^{H}(Y, B; W) + χ^{G}(X, A; V) ⊗_{R} ρ_{pl}^{H}(Y, B; W, κ(Y))

Remark 3.14 The PL-torsion is compatible with induction and restriction by 2.17.

Next we deal with manifolds. A Riemannian G-manifold triad (M; M_{1}, M_{2}) consists of a Riemannian G-manifold M together with G-invariant codimension zero submanifolds M_{1} and M_{2} of the boundary ∂M satisfying ∂M = M_{1} ∪ M_{2} and ∂M_{1} = M_{1} ∩ M_{2} = ∂M_{2}. We do not require in this section M_{1} ∩ M_{2} = ∅ what we did in previous sections. Regard an equivariant triangulation (f; f_{1}, f_{2}) : (K; K_{1}, K_{2}) ⟶ (M; M_{1}, M_{2}) (see Illman (1978) [18]. Roughly speaking, this is an ordinary triangulation together with a regular simplicial action of G on K for which f is G-equivariant. In particular (K, K_{1}) is a pair of finite G-CW - complexes. Let V be an orthogonal D^{G}(M)-representation. The Riemannian metric on M gives an inner product << ω, η >>= ∫_{M} < ω, η > dM = ∫_{M} ω ∧ ∗η on Λ^{p}(M). Equip the subspace of harmonic forms H_{harm}^{p}(M, M_{1}; V) with the induced innner product. There is a natural RG-isomorphism j : H^{p}(K, K_{1}; f^{∗}V) ⟶ H_{s}^{p}(K, K_{1}, f^{∗}V)between singular and cellular homology. Denote by κ_{harm}^{∗} the RG-Hilbert structure on H^{∗}(K, K_{1}; f^{∗}V) for which the following RG-isomorphism becomes an isometry where ī is the Hodge isomorphism (see Proposition 1.10)

H^{p}(K, K_{1}; f^{∗}V) ⟶^{j} H_{s}^{p}(K, K_{1}; f^{∗}V) ⟶^{(f^{∗})^{−1}} H_{s}^{p}(M, M_{1}; V) ⟶^{ī^{−1}} H_{harm}^{p}(M, M_{1}; V)

Let κ_{∗}^{harm} be the RG-Hilbert structure on H_{∗}(K, K_{1}; f^{∗}V) given by κ_{harm}^{∗} and 3.4

Definition 3.15 Define the PL-torsion

ρ_{pl}^{G}(M, M_{1}; V) ∈ K_{1}(RG)^{Z/2}

by ρ_{pl}^{G}(K, K_{1}; f^{∗}V, κ_{∗}^{harm})

If (g; g_{1}, g_{2}) : (L; L_{1}, L_{2}) ⟶ (M; M_{1}, M_{2}) is a second triangulation, g ∘ f^{−1} is simple, i.e. τ^{G}(g^{−1} ∘ f) = 0 in W h^{G}(L). Now Proposition 3.8 shows that the choice of equivariant triangulation does not matter. In the sequel we will identify M with a triangulation. We emphasize that our notion on PL-torsion is based on homology (cf. Proposition 3.6)

Definition 3.16 We call the RG-Hilbert structures κ_{harm}^{∗} and κ_{∗}^{harm} constructed above the harmonic Hilbert structure.

In the equivariant setting there is a new invariant involved which does not occur in the non-equivariant case. For a generator [M] ∈ H_{m}(C_{∗}(M̃, ˜ ∂M) ⊗ Zπ^{w}Z) ≅ Z we obtain by Poincaré duality a RG-chain equivalence unique up to homotopy

3.17 ∩[M] : C^{n−∗}(M, M_{2};^{w} V) ⟶ C_{∗}(M, M_{1}; V)

It induces an RG-isomorphism

3.18 H(∩[M]) : H^{n−∗}(M, M_{2};^{w} V) ⟶ H_{∗}(M, M_{1}; V)

The construction of ∩[M] as a RG-chain map uses the existence of equivariant approximations of the diagonal G-map M ⟶ M × M. Recall that C_{∗}(M, M_{1}; V) and C^{n−∗}(M, M_{1}; V) have preferred RG-Hilbert structures coming from cellular R-bases and the inner product on V (see 3.1 and 3.2).

Definition 3.19 Define the Hilbert-Poincaré torsion or briefly Poincaré torsion

ρ_{pd}^{G}(M, M_{1}; V) ∈ K_{1}(RG)^{Z/2}

to be the Hilbert torsion of ∩[M] : C^{n−∗}(M, M_{2};^{w} V) ⟶ C_{∗}(M, M_{1}; V)

This definition is independent of the choice of [M] as the Hilbert torsion of −id is always zero.

Proposition 3.20 If G acts freely, then ρ_{pd}^{G}(M, M_{1}; V) vanishes.

\underline{\underline{Proof}} : If G acts freely on M, then also D^{G}(M) on M̃. The proof of Poincaré duality by the dual cell decomposition shows that the Z[D^{G}(M)]-chain equivalence of finitely generated based free Z[D^{G}(M)]-chain complexes ∩[M] : C^{n−∗}(M, M_{2};^{w} V) ⟶ C_{∗}(M, M_{1}; V) is base preserving and hence has vanishing Hilbert torsion.

Remark 3.21 For non-free actions the dual CW -complex structure obtained from an equivariant triangulation is not a G-CW -complex structure. For example, consider S^{1} with the Z/2-action given by complex conjugation. The upper and lower hemispheres give obvious equivariant triangulations. The dual cell decomposition is obtained from it by rotation about 90^{0}. It is not an equivariant cell decomposition as Z/2 acts non-trivially on each of the 1cells. Therefore it can happen that the Poincaré torsion is not trivial, if the group acts non-freely.

On the homology level nothing happens.

Lemma 3.22 The RG-map H(∩[M]) : H^{n−∗}(M, M_{2};^{w} V) ⟶ H_{∗}(M, M_{1}; V) is isometric with respect to the harmonic RG-Hilbert structures. In particular the Hilbert torsion of it is zero

\underline{\underline{Proof}} : The claim follows from the commutativity of the following diagram, where A is the deRham isomorphism and <, > denotes the Kronecker pairing resp. is given by the Hilbert structure.

[figure: H_{harm}^{n−p}(M, M_{2};^{w} V); H^{n−p}(M, M_{2};^{w} V); A; ∗; H(∩[M]); H_{harm}^{p}(M, M_{1}; V); H_{p}(M, M_{1}; V); <, >; <, >; H_{harm}^{p}(M, M_{1}; V)^{∗}; H^{p}(M, M_{1}; V)^{∗}; A^{∗}]

Proposition 3.23 (Poincaré Duality for Poincaré and PL-Torsion) Let (M; M_{1}, M_{2}) be a m-dimensional Riemannian G-manifold triad with orientation homomorphism w : D^{G}(M) ⟶ {±1}. Let V be an orthogonal D^{G}(M)-representation. Then we have :

a.) ρ_{pl}^{G}(M, M_{1}; V) + (−1)^{m} · ρ_{pl}^{G}(M, M_{2};^{w} V) = ρ_{pd}^{G}(M, M_{1}; V)

b.) ρ_{pd}^{G}(M, M_{1}; V) = (−1)^{m} · ρ_{pd}^{G}(M, M_{2};^{w} V)

c.) χ^{G}(M, M_{1}; V) = (−1)^{m} · χ^{G}(M, M_{2};^{w} V)

\underline{\underline{Proof}} : a.) Because of Proposition 3.6 and Lemma 3.22 the claim follows from the Comparision formula 2.12 applied to ∩[M] : C^{m−∗}(M, M_{2}; V) ⟶ C_{∗}(M, M_{1}; V).

b.) The RG-chain maps ∩[M] and (∩[M])^{m−∗} from C^{m−∗}(M, M_{2}; V) to C_{∗}(M, M_{1}; V) are chain homotopic. Now apply Proposition 2.4 and Proposition 2.18. c.) follows from 3.4.

Remark 3.24 If one compares the statements about Poincaré duality for analytic torsion 1.20 and for PL-torsion 3.23, it becomes obvious that the Poincaré torsion has to enter in a formula relating analytic and PL-torsion

Example 3.25 Let Z/2 act on S^{1} by complex conjugation. Equip S^{1} with the standard metric scaled by a factor such that the volume of S^{1} is the positive real number μ. We use the Z/2-CW -structure with the upper and lower hemispheres S_{+}^{1} and S_{−}^{1} as 1-cells and the points −1, +1 ∈ C as 0-cells. The cellular RZ/2-chain complex C_{∗}(S^{1}; R) is

R[Z/2] ⟶^{(−ϵ,ϵ)} R ⊕ R

where ϵ is the augmentation a + b · t ↦ a + b for t ∈ Z/2 the generator, S_{+}^{1} resp. S_{−}^{1} corresponds to 1 resp. t ∈ R[Z/2] and the points −1, 1 to (1, 0), (0, 1) ∈ R ⊕ R. The cellular R[Z/2]-Hilbert structure is given by the orthonormal R-bases {1, t} ⊂ R[Z/2] and {(1, 0), (0, 1)} ⊂ R ⊕ R. The 1-form μ^{−1/2} · dvol for dvol the volume form has norm 1 because

<< μ^{−1/2} · dvol, μ^{−1/2} · dvol >>= μ^{−1}· << dvol, dvol >>= μ^{−1} · ∫_{S^{1}} dvol ∧ ∗dvol

= μ^{−1} · ∫_{s^{1}} dvol = μ^{−1} · μ = 1

Notice that the generator (1 − t) of H_{1}(S^{1}; R) is represented by id : S^{1} ⟶ S^{1} and we have

∫_{S^{1}} μ^{−1/2} · dvol = μ^{1/2}

The harmonic 0-form S^{1} ⟶ R, z ↦ μ^{−1/2} has norm 1 and evaluating it at the generator (1, 1) of H_{0}(S^{1}; R) yields 2 · μ^{−1/2}. Hence the following maps are R[Z/2]-isometries, if R^{−} and R carry the standard and H_{∗}(S^{1}; R) the harmonic R[Z/2]-Hilbert structures.

R^{−} ⟶ H_{1}(S^{1}; R) 1 ↦ μ^{−1/2} · (1 − t) R ⟶ H_{0}(S^{1}; R) 1 ↦ 1/2 · (μ^{1/2}, μ^{1/2})

We use them as identifications of R[Z/2]-Hilbert spaces. The following R[Z/2]-chain map i : H_{∗}(C^{∗}(S^{1}; R)) ⟶ C^{∗}(S^{1}, R) satisfies H(i) = id

[figure: R^{−}; R; 0; μ^{−1/2} · (1 − t); (\frac{μ^{1/2}}{2} , \frac{μ^{1/2}}{2} ); R[Z/2]; R ⊕ R; (−ϵ, ϵ)]

Its mapping cone is concentrated in dimension 0, 1, 2 and has the following differential c and chain contraction γ,where δ : R[Z/2] ⟶ R^{−} maps a + bt to a − b

[figure: R; ( \begin{matrix}0 \\ μ^{−1/2} · (1 − t) \end{matrix} ); ( 0 μ^{1/2}/2 · δ ); R ⊕ R[Z/2]; ( \begin{matrix}μ^{1/2}/2 & ϵ \\ μ^{1/2}/2 & −ϵ \end{matrix} ); ( \begin{matrix}μ^{−1/2} & μ^{−1/2} \\ (t + 1)/4 & −(t + 1)/4 \end{matrix} ); R ⊕ R]

Hence (c + γ) : Cone(i)_{odd} ⟶ Cone(i)_{ev} is the R[Z/2]-isomorphism

( \begin{matrix}μ^{−1/2}/2 & ϵ \\ μ^{−1/2}/2 & −ϵ \\ 0 & μ^{1/2}/2 · δ \end{matrix} ) : R ⊕ R[Z/2] ⟶ R ⊕ R ⊕ R^{−}

We have the R[Z/2]-isometry

ϕ = ( \begin{matrix}1 & 0 & 0 \\ 0 & \frac{\sqrt{2}}{2} · (t + 1) & \frac{\sqrt{2}}{2} · (t − 1) \end{matrix} ) : R ⊕ R ⊕ R^{−} ⟶ R ⊕ R[Z/2]

Hence [ϕ ∘ (c + γ)] = [(c + γ) ∘ ϕ] is represented by

( \begin{matrix}μ^{1/2}/2 & \sqrt{2} & 0 \\ μ^{1/2}/2 & −\sqrt{2} & 0 \\ 0 & 0 & −\sqrt{2}/2 · μ^{1/2} \end{matrix} ) : R ⊕ R ⊕ R^{−} ⟶ R ⊕ R ⊕ R^{−}

This shows

3.26 ρRZ/2_{pl}(S^{1}; R) = [2μ · id : R ⟶ R] + [μ/2 · id : R^{−} ⟶ R^{−}]

We get by restriction to the trivial subgroup 3.27 ρ_{pl}(S^{1}; R) = μ^{2} ∈ R^{∗} We conclude from 3.26 and Lemma 3.32

3.28 ρRZ/2_{pl}(S^{1}; R^{−}) = [μ/2 · id : R ⟶ R] + [2μ · id : R^{−} ⟶ R^{−}]

We derive from 3.26, 3.28 and Proposition 3.23

3.29 ρRZ/2_{pd}(S^{1}; R) = [4 · id : R ⟶ R] + [\frac{1}{4} · id : R^{−} ⟶ R^{−}]

Example 3.30 Equip D^{1} = [0, 1] with the standard metric scaled my μ > 0. Then D^{1} has volume μ. The cellular R-chain complex is

R ⟶^{(1,−1)} R ⊕ R

The element (μ^{1/2}/2, μ^{1/2}/2) ∈ H_{0}(D^{1}; R) has norm 1 with respect to the harmonic Hilbert structure. Now one easily checks 3.31 ρ_{pl}(D^{1}; R) = μ ∈ R^{∗}

Recall the map q : D^{G}(M) ⟶ G of 1.2 and the operation of K_{0}(RG) = Rep_{R}(G) on K_{1}(RG)^{Z/2} and R ⊗_{Z} Rep_{R}(G) defined in 2.14 and 1.25. Lemma 3.32 Let W be an orthogonal G-representation. We have ρ_{an}^{G}(M, M_{1}; V ⊗_{R} q^{∗}W) = ρ_{an}^{G}(M, M_{1}; V) ⊗_{R} [W] ρ_{pl}^{G}(M, M_{1}; V ⊗_{R} q^{∗}W) = ρ_{pl}^{G}(M, M_{1}; V) ⊗_{R} [W] ρ_{pd}^{G}(M, M_{1}; V ⊗_{R} q^{∗}W) = ρ_{pd}^{G}(M, M_{1}; V) ⊗_{R} [W]

\underline{\underline{Proof}} : There are natural isometric RG-isomorphisms

Λ^{∗}(M; V) ⊗_{R} W ⟶ Λ^{∗}(M; V ⊗_{R} W) and C_{∗}(M, M_{1}; V) ⊗_{R} W ⟶ C_{∗}(M, M_{1}; V ⊗_{R} q^{∗}W)

Let (M; M_{1}, M_{2}) and (N; N_{1}, N_{2}) be m-dimensional Riemannian G-manifold triads and (f; f_{1}, f_{2}) : (M; M_{1}, M_{2}) ⟶ (N; N_{1}, N_{2}) be a G-homotopy equivalence of such triads. Let V be an orthogonal D^{G}(N)-representation and κ_{∗}^{harm}(M) and κ_{∗}^{harm} be the harmonic Hilbert structures on H_{∗}(M, M_{1}; f^{∗}V) and H_{∗}(N, N_{1}; V). The following composition represents an element u_{i} ∈ K_{1}(RG)^{Z/2}

H_{i}(M, M_{1}; f^{∗}V) ⟶^{H_{i}(f,f_{1})} H_{i}(N, N_{1}; V) ⟶^{κ_{i}^{harm}(N)} H_{i}(N, N_{1}; V)^{∗} ⟶^{H_{i}(f,f_{1})^{∗}}

H_{i}(M, M_{1}; f^{∗}V)^{∗} ⟶^{κ_{i}^{harm}(M)^{−1}} H_{i}(M, M_{1}; f^{∗}V)

We have introduced the map ω^{G}(N; V) : W h^{G}(N) ⟶ K_{1}(RG)^{Z/2} in Proposition 3.8. We derive from Proposition 3.8, Proposition 2.12 and Proposition 2.18

Proposition 3.33 We have

a.) ρ_{pl}^{G}(N, N_{1}; V) − ρ_{pl}^{G}(M, M_{1}; f^{∗}V) = ω^{G}(N; V)(τ^{G}(f, f_{1})) − ∑_{i≥0}(−1)^{i} · u_{i}

b.) ρ_{pd}^{G}(N, N_{1}; V) − ρ_{pd}^{G}(M, M_{1}; f^{∗}V) = ω^{G}(N; V)(τ^{G}(f, f_{1})) + (−1)^{m} · ω^{G}(N;^{w} V)(τ^{G}(f, f_{2}))

Proposition 3.33 tells in particular how the PL-torsion varies under change of Riemannian metric. Notice that the Poincaré torsion depends only on the simple G-homotopy type. This is also true for the PL-torsion, if H_{i}(N, N_{1}; V) vanishes.

The following pairings are special cases of 2.14

⊗_{R} : K_{1}(RG)^{Z/2} ⊗_{Z} Rep_{R}(Z/2) ⟶ K_{1}(R[G × Z/2])^{Z/2}

⊗_{R} : Rep_{R}(G) ⊗_{Z} K_{1}(R[Z/2])^{Z/2} ⟶ K_{1}(R[G × Z/2])^{Z/2}

Proposition 3.34 (Double Formula for Poincaré and PL-Torsion) Let M be a Riemannian G-manifold and V be an orthogonal D^{G}(M)-represention. Suppose that ∂M is the disjoint union M_{1} ∐ M_{2} and that the metric is a product near the boundary. Then a.) ρ_{pl}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) =

ρ_{pl}^{G}(M; V) ⊗_{R} [R] + ρ_{pl}^{G}(M, M_{1}; V) ⊗_{R} [R^{−}] + χ^{G}(M_{1}; V) ⊗_{R} [2 · id : R ⟶ R] b.) ρ_{pl}^{G×Z/2}(M ∪_{M_{1}} M, ∂(M ∪_{M_{1}} M); V ∪_{M_{1}} V) =

ρ_{pl}^{G}(M, M_{2}; V) ⊗_{R} [R] + ρ_{pl}^{G}(M, ∂M; V) ⊗_{R} [R^{−}] + χ^{G}(M_{1}; V) ⊗_{R} [2 · id : R ⟶ R] c.) ρ_{pd}^{G×Z/2}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) =

ρ_{pd}^{G}(M; V) ⊗_{R} [R] + ρ_{pd}^{G}(M, M_{1}; V) ⊗_{R} [R^{−}] + χ^{G}(M_{1}; V) ⊗_{R} [2 · id : R ⟶ R]−

χ^{G}(M_{1}; V) ⊗_{R} [2 · id : R^{−} ⟶ R^{−}]

\underline{\underline{Proof}} : Let i : M ⟶ M ∪_{M_{1}} M be the inclusion onto the first summand. We obtain from this map a R[D^{G}(M ∪_{M_{1}} M]-chain map C_{∗}(i) : i_{∗}C_{∗}(M̃) ⟶ C_{∗} ˜ (M ∪_{M_{1}} M) for i_{∗} the induction with the homomorphism D^{G}(i) : D^{G}(M) ⟶ D^{G}(M ∪_{M_{1}} M). As i^{∗}(V ∪_{M_{1}} V) is V, there is a RG-chain isomorphism j from i_{∗}C_{∗}(M̃) ⊗_{R[D^{G}(M∪_{M_{1}}M)]} V ∪_{M_{1}} V to i_{∗}C_{∗}(M; V). The composition of C_{∗}(i) ⊗_{R[D^{G}(M∪_{M_{1}}M)]} V ∪_{M_{1}} V and j^{−1} is denoted by 3.35 i_{∗} : i_{∗}C(M; V) ⟶ C(M ∪_{M_{1}} M; V ∪_{M_{1}} V) Let τ : M ∪_{M_{1}} M ⟶ M ∪_{M_{1}} M be the flip map. As τ^{∗}V ∪_{M_{1}} V = V ∪_{M_{1}} V holds, we also obtain a RG-chain map 3.36 τ_{∗} : C_{∗}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) ⟶ C_{∗}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) Define a R[G × Z/2]-chain map 3.37 f_{∗} : (C_{∗}(M; V) ⊗_{R} R) ⊕ (C_{∗}(M; M_{1}; V) ⊗_{R} R^{−}) ⟶ C_{∗}(M ∪_{M_{1}} M; V ∪_{M_{1}} V) by

f_{∗}(a ⊗_{R} 1 ⊕ b ⊗_{R} 1) = \frac{\sqrt{2}}{2} · (i_{∗} + τ_{∗} ∘ i_{∗})(a) + \frac{\sqrt{2}}{2} · (i_{∗} − τ_{∗} ∘ i_{∗})(b)

Counting the cellular bases we obtain isometric RG-isomorphisms for p ∈ Z 3.38 ϕ : C_{p}(M, M_{1}; V) ⊕ C_{p}(M_{1}; V) ⟶ C_{p}(M; V)

Ψ : C_{p}(M, M_{1}; V) ⊕ C_{p}(M_{1}; V) ⊕ C_{p}(M, M_{1}; V) ⟶ C_{p}(M ∪_{M_{1}} M; V ∪_{M_{1}} V)

The map ψ is a R[G × Z/2]-map if Z/2 acts trivially on C_{p}(M_{1}; V) and switches the two summands C_{p}(M, M_{1}; V). We get from ϕ an isometric R[G × Z/2]-isomorphism

(ϕ ⊗_{R} id) ⊕ id : (C_{p}(M, M_{1}; V)) ⊗_{R} R) ⊕ (C_{p}(M_{1}; V) ⊗_{R} R) ⊕ (C_{p}(M, M_{1}; V) ⊗_{R} R^{−})

⟶ C_{∗}(M; V) ⊗_{R} R ⊕ C_{∗}(M; M_{1}; V) ⊗_{R} R^{−}

Now the composition ψ^{−1} ∘ f_{p} ∘ ((ϕ ⊗_{R} id) ⊕ id) is given by

( \begin{matrix}\sqrt{2}/2 · id & 0 & \sqrt{2}/2 · id \\ 0 & \sqrt{2} · id & 0 \\ \sqrt{2}/2 · id & 0 & −\sqrt{2}/2 · id \end{matrix} ) : C_{p}(M, M_{1}; V) ⊕ C_{p}(M_{1}; V) ⊕ C_{p}(M, M_{1}; V) ⟶

(C_{p}(M, M_{1}; V)) ⊗_{R} R) ⊕ (C_{p}(M_{1}; V) ⊗_{R} R) ⊕ (C_{p}(M, M_{1}; V) ⊗_{R} R^{−})

The Hilbert torsion of this R[G × Z/2]-isomorphism of R[G × Z/2]-Hilbert modules is easily computed as [C_{p}(M_{1}; V)] ⊗_{R} [2 · id : R ⟶ R]. We get for the Hilbert torsion of the R[G × Z/2]-chain map f_{∗} of R[G × Z/2]-Hilbert chain complexes

3.39 ht^{G×Z/2}(f_{∗}) = χ^{G}(M_{1}; V) ⊗_{R} [2 · id : R ⟶ R]

Next we compute on the homology level

3.40 ht^{G×Z/2}(H(f_{∗})) = 0

The following diagram commutes where A is given by the deRham map and the isomorphism 3.4 and l is the composition of 1.29 and 1.30.

[figure: H_{p}(M ∪_{M_{1}} M, V ∪_{M_{1}} V)^{∗}; (H_{p}(M; V) ⊗_{R} R)^{∗} ⊕ (H_{p}(M, M_{1}; V) ⊗_{R} R^{−})^{∗}; H(f_{∗})^{∗}; A; (A ⊗_{R} id) ⊕ (A ⊗_{R} id); E_{0}(M ∪_{M_{1}} M; V ∪_{M_{1}} V); (E_{0}(M; V) ⊗_{R} R) ⊕ (E_{0}(M, M_{1}; V) ⊗_{R} R^{−}); \sqrt{2} · l]

The lower vertical map is an isometry because of the following computation for ω, η ∈ E_{0}(M ∪_{M_{1}} M, V ∪_{M_{1}} V)

< \sqrt{2} · l(ω), \sqrt{2} · l(η) >

=< \sqrt{2}/2 · i^{∗}(ω + τ^{∗}ω), \sqrt{2}/2 · i^{∗}(η + τ^{∗}η) >_{M} + < \sqrt{2}/2 · i^{∗}(ω − τ^{∗}ω), \sqrt{2}/2 · i^{∗}(η − τ^{∗}η) >_{M}

=< i^{∗}ω, i^{∗}η >_{M} + < i^{∗}τ^{∗}ω, i^{∗}τ^{∗}η >_{M}

=< ω, η >_{M∪_{M_{1}}M}

This implies 3.40. Now the claim a. follows from the Comparision formula 2.12 applied to the R[G × Z/2]-chain isomorphism f_{∗} defined in 3.37 if we take 3.39 and 3.40 into account. One proves b.) analogously. Then c.) follows from a.) and b.) and Poincaré duality 3.23 since the following R[G × Z/2]-representations are isomorphic

(^{w^{G}(M)}V ∪_{M_{1}} ^{w^{G}(M)}V) ⊗ R^{−} =^{w^{G×Z/2}(M∪_{M_{1}}M)} (V ∪_{M_{1}} V)

This finishes the proof of the Double Formula.

Remark 3.41 Notice that the double formulas for analytic torsion 1.27 and PL-torsion 3.34 differ by a Euler characteristic term depending only on the boundary. It appears in the PLcase as the cells in the boundary do contribute to the Hilbert structures. This is not true in the analytic situation where the Hilbert structures come from certain integrals and the boundary does not contribute to them as it is a zero set. These observations indicate that the Euler characteristic is the correction term in a formula relating analytic and PL-torsion for manifolds with boundary.

Example 3.42 In this example we show how the general results above can be used to compute the torsion invariants for S^{1} and D^{1}. We will see that we get the same answers as in examples 1.15 1.18 3.25 3.30 where we computed these invaraints directly. Equip D^{1} with the standard metric scaled my μ > 0. Then D^{1} has volume μ. The double D^{1} ∪_{∂D^{1}} D^{1} is S^{1} with Z/2-action given by complex conjugation and the S^{1}-invariant Riemannian metric for which the volume of S^{1} is 2 · μ. We get from Poincaré duality 1.20 and 3.23 and Proposition 3.20 ρ_{pl}(D^{1}) = ρ_{pl}(D^{1}, ∂D^{1}) ρ_{an}(D^{1}) = ρ_{an}(D^{1}, ∂D^{1}) We derive from the double formulas 1.27 and 3.34 3.43 ρ_{an}^{Z/2}(S^{1}; R) = ρ_{an}(D^{1}) · ([R] + [R^{−}])

ρ_{pl}^{Z/2}(S^{1}; R) = ρ_{pl}(D^{1}) · ([R] + [R^{−}]) + 2 · [2 · id : R ⟶ R]

ρ_{pd}^{Z/2}(S^{1}; R) = ρ_{pd}(D^{1}) · ([R] + [R^{−}]) + 2 · [2 · id : R ⟶ R] − 2 · [2 · id : R^{−} ⟶ R^{−}] We know ρ_{an}(S^{1}; R) = ln(ρ_{pl}(S^{1}; R)) from Cheeger (1979) [8] and Müller (1978) [26]. Hence the restriction of 3.43 to the trivial subgroup shows 3.44 ρ_{an}(D^{1}) = ln(ρ_{pl}(D^{1})) + ln(2)

One easily checks that the generator of H_{0}(D^{1}; Z) has norm μ^{−1/2} with respect to the harmonic Hilbert structure when considered in H_{0}(D^{1}; R). The projection D^{1} ⟶ {pt.} is a simple homotopy equivalence. Hence we get from 3.8 3.45 ρ_{pl}(D^{1}) = μ We get from 3.44 3.46 ρ_{an}(D^{1}) = ln(2 · μ) We conclude from 3.43 if ν = 2 · μ is the volume of S^{1} 3.47 ρ_{an}^{Z/2}(S^{1}; R) = ln(ν) · [R] + ln(ν) · [R^{−}])

ρ_{pl}^{Z/2}(S^{1}; R) = [2ν · id : R ⟶ R] + [ν/2 · id : R^{−} ⟶ R^{−}]

ρ_{pd}^{Z/2}(S^{1}; R) = [4 · id : R ⟶ R] − [1/4 · id : R^{−} ⟶ R^{−}]

Let M resp. N be Riemannian G- resp. H-manifold with boundary. Some care is necessary to put a G×H-manifold structure on M×N, one has to straighten the angle. There seems to be no canonical Riemannian metric on M × N. If f : K ⟶ M, g : L ⟶ N and h : X ⟶ M × N are equivariant triangulations, h^{−1} ∘ (f × g) : K × L ⟶ X is a simple G × H-homotopy equivalence. Hence we get from 3.8 and 3.13

Proposition 3.48 (Product Formula for Poincaré and PL-torsion) Let (M; M_{1}, M_{2}) resp. (N; N_{1}, N_{2}) be a Riemannian G- resp. H-manifold triad and V resp. W be an orthogonal D^{G}(M)- resp. D^{H}(N)-representation. Put on the G × Hmanifold (M × N; M × N_{1} ∪ M_{1} × N, M × N_{2} ∪ M_{2} × N) an invariant Riemannian metric such that the Künneth R[G × H]-isomorphism from H_{∗}(M, M_{1}; V) ⊗_{R} H_{∗}(N, N_{1}; W) to H_{∗}(M × N; M_{1} × N ∪ M × N_{1}; V ⊗_{R} W) becomes an isometry with respect to the harmonic structures. Then we get

a.) ρ_{pl}^{G×H}(M × N, M_{1} × N ∪ M × N_{1}; V ⊗_{R} W) = χ^{G}(M, M_{1}; V) ⊗_{R} ρ_{pl}^{H}(N, N_{1}; W) + ρ_{pl}^{G}(M, M_{1}; V) ⊗_{R} χ^{H}(N, N_{1}; W)

b.) ρ_{pd}^{G×H}(M × N, M_{1} × N ∪ M × N_{1}; V ⊗_{R} W) = χ^{G}(M, M_{1}; V) ⊗_{R} ρ_{pd}^{H}(N, N_{1}; W) + ρ_{pd}^{G}(M, M_{1}; V) ⊗_{R} χ^{H}(N, N_{1}; W)

c.) χ^{G×H}(M × N, M_{1} × N ∪ M × N_{1}; V ⊗_{R} W) = χ^{G}(M, M_{1}; V) ⊗_{R} χ^{H}(N, N_{1}; W)

Next we examine how the Poincaré and PL-torsion behave under glueing.

Proposition 3.49 (Sum Formula for Poincaré and PL-Torsion) Let (M; M_{1}, M_{2}) and (N; N_{1}, N_{2}) be G-manifold triads with invariant Riemannian metric. Let V and W be orthogonal D^{G}(M)- and D^{G}(N)-representations. Let f : M_{2} ⟶ N_{1} be a G-diffeomorphism and f̄: V | M_{2} ⟶ f^{∗}W | N_{1} an isometric R[D^{G}(M_{2})]-isomorphism. Put an invariant Riemannian metric on M ∪_{f} N. Let M_{∗} be the acyclic finite RG-Hilbert complex given by the long Mayer-Vietoris homology sequence and the harmonic RG-Hilbert structures. Define P_{∗} analogously by the long homology sequence of the pair (M, M_{1}). Then we have:

a. ) ρ_{pl}^{G}(M ∪_{f} N, V ∪f̄ W) = ρ_{pl}^{G}(M; V) + ρ_{pl}^{G}(N; W) − ρ_{pl}^{G}(M_{2}; V) − hr^{G}(M_{∗}) b.) ρ_{pd}^{G}(M ∪_{f} N, V ∪f̄ W) = ρ_{pd}^{G}(M; V) + ρ_{pd}^{G}(N; W) − ρ_{pd}^{G}(M_{2}; V) c.) χ^{G}(M ∪_{f} N, V ∪f̄ W) = χ^{G}(M; V) + χ^{G}(N; W) − χ^{G}(M_{2}; V) d.) ρ_{pl}^{G}(M, M_{1}; V) = ρ_{pl}^{G}(M; V) − ρ_{pl}^{G}(M_{1}; V) − hr^{G}(P_{∗}) e.) ρ_{pd}^{G}(M, M_{1}; V) = ρ_{pd}^{G}(M; V) − ρ_{pd}^{G}(M_{1}; V) f.) χ^{G}(M, M_{1}; V) = χ^{G}(M; V) − χ^{G}(M_{1}; V)

\underline{\underline{Proof}} : a.) and d.) follow directly from Proposition 3.11. e.) We have exact sequences of RG-chain complexes

3.50 0 ⟶ C_{∗}(M_{1}; V) ⟶ C_{∗}(M; V) ⟶ C_{∗}(M, M_{1}; V) ⟶ 0

3.51 0 ⟶ C_{∗}(M_{1}, ∂M_{1};^{w} V) ⟶ C_{∗}(M, M_{2};^{w} V) ⟶^{p} C_{∗}(M, ∂M;^{w} V) ⟶ 0

We have the canonical exact sequences if Cyl(p) and Cone(p) are the mapping cylinder and cone of p and P_{∗} and Q_{∗} are the mapping cones of C_{∗}(M, ∂M;^{w} V) and C_{∗}(M, M_{2};^{w} V)

3.52 0 ⟶ C_{∗}(M, M_{2};^{w} V) ⟶ Cyl(p) ⟶ Cone(p) ⟶ 0
0 ⟶ C_{∗}(M, ∂M;^{w} V) ⟶ Cyl(p) ⟶ Q_{∗} ⟶ 0
0 ⟶ ∑ C_{∗}(M_{1}, ∂M_{1};^{w} V) ⟶ Cone(p) ⟶ P_{∗} ⟶ 0

As P_{∗} and Q_{∗} are contractible, the last two sequences in 3.52 split as exact sequences of RG-chain complexes (see Cohen (1973) [9]). Choose such a splitting. Then we get from 3.52 by applying C_{∗} ↦ C^{m−∗} the exact sequence

3.53 {0} ⟶ C^{m−1−∗}(M_{1}, ∂M_{1};^{w} V) ⊕ P^{m−∗} ⟶ C^{m−∗}(M, ∂M;^{w} V) ⊕ Q^{m−∗} ⟶ C^{m−∗}(M, M_{2};^{w} V) ⟶ {0}

Now one constructs a commutative diagram of RG-Hilbert complexes with 3.50 and 3.53 as exact rows such that f resp. g resp. h represents ∩[M] ∘ pr resp. ∩[M] ∘ pr resp. ∩[M] for pr the projection.

[figure: C^{m−1−∗}(M_{1}, ∂M_{1};^{w} V) ⊕ P^{m−∗}; ⟶; C^{m−∗}(M, ∂M;^{w} V) ⊕ Q^{m−∗}; ⟶; C^{m−∗}(M, M_{2}; wV); C_{∗}(M_{1}; V); ⟶; C_{∗}(M; V); ⟶; C_{∗}(M, M_{1}; V); f; g; h]

Since pr^{m−∗} : P^{m−∗} ⟶ {0} and pr^{m−∗} : Q^{m−∗} ⟶ {0} have vanishing Hilbert torsion, we get from Proposition 2.4 and Proposition 2.11 ρ_{pd}^{G}(M_{1}; V) = ht^{G}(f) ρ_{pd}^{G}(M; V) = ht^{G}(g) ρ_{pd}^{G}(M, M_{1}; V) = ht^{G}(h) Now the claim follows from 2.9. b.) is proven analogously starting with 0 ⟶ C_{∗}(M_{1}; V) ⟶ C_{∗}(M; V) ⊕ C_{∗}(N; V) ⟶ C_{∗}(M ∪_{f} N; V ∪f̄ W) ⟶ 0 {0} ⟶ C_{∗}(M_{1}; ∂M_{1}; V) ⟶ C_{∗}(M, ∂M; V) ⊕ C_{∗}(N, ∂N; V)

⟶ C_{∗}(M ∪_{f} N, ∂(M ∪_{f} N; V ∪f̄ W) ⟶ {0}

c.) and f.) are obvious. This finishes the proof of the Sum formula

4. Comparision of Analytic and PL-Torsion

Let G be a finite group. Consider a Riemannian G-manifold M whose boundary ∂M is M_{1} ∐ M_{2}. We have introduced an extension 0 ⟶ π_{1}(M) ⟶^{i} D^{G}(p) ⟶^{q} G ⟶ 0 in 1.2 and an operation of D^{G}(M) on the universal covering M̃ of M extending the π_{1}(M)-action and covering the G-action in 1.3 if π_{1}(M) is identified with the group of deck transformations of the universal covering. Let V be an orthogonal D^{G}(M)-representation. We have introduced analytic torsion

ρ_{an}^{G}(M, M_{1}; V) ∈ R ⊗_{Z} Rep_{R}(G)

in 1.14, PL-torsion

ρ_{pl}^{G}(M, M_{1}; V) ∈ K_{1}(RG)^{Z/2}

in 3.15 and Poincaré torsion

ρ_{pd}^{G}(M, M_{1}; V) ∈ K_{1}(RG)^{Z/2}

in 3.19. In 2.25 we have defined an isomorphism

Γ_{1} ⊕ Γ_{2} : K_{1}(RG)^{Z/2} ⟶ (R ⊗_{Z} Rep_{R}(G)) ⊕ (Z/2 ⊕_{Z} R̂ep_{R}(G))

We want to relate these invariants by this isomorphism. One easily checks

Proposition 4.1 Γ_{2}(ρ_{pl}^{G}(M, M_{1}; V)) = Γ_{2}(ρ_{pd}^{G}(M, M_{1}; V)) = 0

Hence only the images of PL-torsion and Poincaré torsion under Γ_{1} are interesting. We need the following technical condition. We call D^{G}(M)-representations V and W coherent if for

any H ⊂ G, x ∈ M^{H} their restrictions to H by H = D^{H}({pt.}) ⟶^{D^{H}(j(x))} D^{H}(M) ⟶^{k} D^{G}(M)

are RH-isomorphic, where j(x) : {pt.} ⟶ M has {x} as image and k is the obvious inclusion. This is equivalent to the assumption that the G-vector bundles M ×_{π_{1}(M)} V and M ×_{π_{1}(M)} W are locally isometrically isomorphic (cf. proof of Lemma 4.14).

Definition 4.2 We call a D^{G}(M)-representation V coherent to a G-representation if V and q^{∗}W are coherent for an appropiate G-representation W.

Example 4.3 If M^{H} is non-empty and connected for all H ⊂ G, then any G-representation is coherent to a G-representation W. Namely, for a fixed element x ∈ M^{G} the homomorphism

s : G = D^{G}({pt.}) ⟶^{D^{G}(j(x))} D^{G}(M) splits q : D^{G}(M) ⟶ G. Put W = s^{∗}V. For any y ∈ M^{H}

the H-maps j(x) and j(y) are H-homotopic. Hence the composition H ↪ G ⟶^{s} D^{G}(M)

agrees with the composition H = D^{H}({pt.}) ⟶^{D^{H}(j(y))} D^{H}(M) ⟶^{k} D^{G}(M).

Example 4.4 Let Z/2 act on S^{1} by complex conjugation. For −1, 1 ∈ S^{1} we get different sections s^{+}, s^{−} : Z/2 ⟶ D^{Z/2}(S^{1}) of q. Identify D^{Z/2}(S^{1}) with the semi direct product Z ×_{s} Z/2 using s^{+}. Then s^{−} : Z/2 ⟶ Z ×_{s} Z/2 sends m̄ to ((−1)^{m}, m̄). Consider the 1dimensional D^{Z/2}(S^{1})-representation given by Z ×_{s} Z/2 ⟶ {±1} , (n, m̄) ↦ (−1)^{n+m}. It cannot be coherent to Z/2-representation since it restrictions with s^{+} and s^{−} are not the same.

Our main result is :

Theorem 4.5 (Torsion Formula for Manifolds with Boundary and Symmetry) Let M be a Riemannian G-manifold whose boundary is the disjoint union M_{1} ∐ M_{2}. Let V be an equivariant coefficient system which is coherent to a G-representation. Assume that the metric is a product near the boundary. Then

ρ_{an}^{G}(M, M_{1}; V) = Γ_{1}(ρ_{pl}^{G}(M, M_{1}; V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M, M_{1}; V)) + \frac{ln(2)}{2} · χ(∂M; V)

Recall that ^{w}V is the w-twisted D^{G}(M)-representation given by V and the orientation homomorphism w : D^{G}(M) ⟶ {±1} defined in 1.4. We will give examples, computations and applications in the next section. The remainder of this section is devoted to the proof of Theorem 4.5.

We verify Theorem 4.5 under certain additional assumptions and remove these one after the other. We will use the work of Lott-Rothenberg [19] where Theorem 4.5 is proved following Müller (1978) [26], provided that ∂M is empty, M is orientable and V is the trivial D^{G}(M)-representation R. Their definitions still make sense when we remove the last condition that V is trivial. Fix an orthogonal G-representation W and an orthogonal D^{G}(M)-representation V. Let σ : G ⟶ Aut_{R}(W) be the G-structure on W. We extend the basic definitions of Lott-Rothenberg [19]. One gets their definitions back if one puts V = R. Notice that V plays the role of an equivariant coefficient system for M whereas W is a base element for the representation ring when W is irreducible.

Let Δ^{p} : Λ^{p}(M; V) ⟶ Λ^{p}(M; V) be the Laplacian and the harmonic projection onto H_{harm}^{p}(M; V) be pr_{harm}^{p} : Λ^{p}(M; V) ⟶ Λ^{p}(M; V). Let \underline{Δ}^{p} : Λ^{p}(M; V) ⟶ Λ^{p}(M; V) be the sum Δ^{p} + pr_{harm}^{p}. Denote by pr : Λ^{p}(M; V) ⊗_{R} W ⟶ (Λ^{p}(M; V) ⊗_{R} W)^{G} the projection operator onto the fixed point set sending (ω ⊗_{R} w to \frac{1}{|G|} · ∑_{g∈G} l(g^{−1})^{∗}ω ⊗_{R} σ(g)(w), where l(g^{−1})^{∗} is induced from left multiplication with g^{−1}. In Lott-Rothenberg [19] a meromorphic function μ_{W}^{p} (s), analytic in 0, is constructed which is for s ∈ C with Real(s) > m/2 given by 4.6 μ_{W}^{p} (s) = \frac{1}{Γ(s)} · ∫_{0}^{∞} t^{s−1} · trace (pr ∘ exp(−t(\underline{Δ}^{p} ⊗_{R} id_{W}))) dt −

dim_{R} ((H_{harm}^{p}(M; V) ⊗_{R} W)^{G})

where the trace is taken for operators Λ^{p}(M; V) ⊗_{R} W ⟶ Λ^{p}(M; V) ⊗_{R} W. The existence of the meromorphic extension is based on standard approximations of the heat kernel. We recall the definition of analytic torsion in Lott-Rothenberg [19]

4.7 T_{W} (M; V) = \frac{1}{2} · ∑_{p≥0}(−1)^{p} · p · \frac{d}{ds}μ_{W}^{p} (s) |_{s=0} ∈ R

The PL-torsion τ_{W} (M; V) ∈ R^{∗} in Lott-Rothenberg [19] is defined in the following way. They use on C^{p}(M; V) and H^{p}(M; V) the same RG-Hilbert structures as we and equip (C^{p}(M; V) ⊗_{R} W)^{G} and H((C^{p}(M; V) ⊗_{R} W)^{G}) = (H^{p}(M; V) ⊗_{R} W)^{G}) with the RG-Hil-bert structure induced by restriction from the product structures. They choose orthonormal bases and define τ_{W} (M; V) by Milnor’s definition of torsion for the finitely generated based free R-cochain complex (C^{p}(M; V) ⊗_{R} W)^{G} with based free cohomology. One easily checks (cf. Example 2.6)

4.8 τ_{W} (M; V)^{2} = hr((C_{∗}(M; V) ⊗_{R} W)^{G}) ∈ R^{∗}

Let I be a complete set of representatives of the isomorphism classes of irreducible Grepresentations. Let m(W) = dim_{R} (W ⊗_{R} W)^{G}) = dim_{R} (Hom_{RG}(W, W)) be the Schur index of W ∈ I.

Lemma 4.9

a.) For s ∈ C with Real(s) > m/2 we have in C ⊗_{Z} Rep_{R}(G) for p > 0

ζ_{p}(M; V)(s) = 2 · ∑_{W ∈I} m(W)^{−1} · μ_{W}^{p} (M; V)(s) · [W]

b.) We have in R ⊗_{Z} Rep_{R}(G)

ρ_{an}^{G}(M; V) = 2 · ∑_{W ∈I} m(W)^{−1} · T_{W} (M; V) · [W]

c.) We get in K_{1}(RG)^{Z/2}

ρ_{pl}^{G} = ∑_{W ∈I} m(W)^{−1} · [τ_{W} (M; V)^{2} · id : W ⟶ W]

\underline{\underline{Proof}} : We obtain in Rep_{R}(G) for any G-representation P from 2.21 4.10 [P] = ∑_{W ∈I} m(W)^{−1} · dim_{R}((P ⊗_{R} W)^{G}) · [W]

As pr is the projection operator, we have

4.11 trace (pr ∘ exp(−t(\underline{Δ}^{p} ⊗_{R} id_{W})) : Λ^{p}(M; V) ⊗_{R} W ⟶ Λ^{p}(M; V) ⊗_{R} W) =

trace (exp(−t(\underline{Δ}^{p} ⊗_{R} id_{W})^{G}) : (Λ^{p}(M; V) ⊗_{R} W)^{G} ⟶ (Λ^{p}(M; V) ⊗_{R} W)^{G})

This implies

4.12 μ_{W}^{p} (s) =

\frac{1}{Γ(s)} · ∫_{0}^{∞} t^{−s} · trace (exp(−t(\underline{Δ}^{p} ⊗_{R} id_{W})^{G})) dt − dim_{R} ((H_{harm}^{p}(M; V) ⊗_{R} W)^{G}) =

∑_{λ>0} λ^{−s} · dim_{R} (E_{λ}((Δ^{p} ⊗_{R} id_{W})^{G})) =
∑_{λ>0} λ^{−s} · dim_{R} ((E_{λ}(Δ^{p}) ⊗_{R} W)^{G})

Now the claim a.) follows from 4.10 and 4.12. We derive b.) from a.) and c.) is proven similiarly using 4.8

The main result in Lott-Rothenberg [19] is the following Theorem

Theorem 4.13 (Lott-Rothenberg) Let M be an odd-dimensional orientable Riemannian G-manifold. Suppose that G acts orientation preserving and ∂M is empty, If W is an orthogonal G-representation,we get

T_{W} (M; R) = ln(τ_{W} (M; R))

Their proof is modelled upon the proof of Müller (1978) [26]. Notice that Müller allows arbitrary coefficients. Lott and Rothenberg define an equivariant version of the combinatorial torsion τ_{W}^{c} (M; V) based on Whitney’s map. They show that the estimates of the sections 1 to 5 in Müller (1978) [26] still hold in the equivariant setting. Then they define an equivariant parametrix and generalize the estimates of section 8 in Müller (1978) [26]. Since Lott and Rothenberg work with trivial coefficients V, they can leave out the first step in Müller (1978) [26], where the difference between analytic and PL-torsion is shown to be independent of V. Lott and Rothenberg carry out the second step equivariantly, where the difference is examined under surgery, and thus get their result. We want to deal with Müller’s first step.

Lemma 4.14 Let M be a closed odd-dimensional G-manifold with trivial orientation behaviour w^{G}(M). Let U and V be coherent orthogonal D^{G}(M)-representations and W an orthogonal G-representation. Then

T_{W} (M; U) − ln(τ_{W} (M; U)) = T_{W} (M; V)) − ln(τ_{W} (M; V))

\underline{\underline{Proof}} : For any x ∈ M with isotropy group H = \{g ∈ G | gx = x\} there is an open neighbourhood of the shape G ×_{H} S for a H-representation S (see Bredon (1972) [6], VI.2.4 Corollary). The key observation is that the restriction of the G-vector bundles with Riemannian metric M̃ ×_{π} U and M̃ ×_{π} V to G ×_{H} S are isometrically isomorphic since then the argument in Müller (1978) [26], section 9 goes through following the equivariant pattern of Lott and Rothenberg. The inclusion G/H ⟶ G ×_{H} S sending gH to (g, 0) is a G-homotopy equivalence so that it suffices to regard the restrictions to G/H. We must show for the inclu-

sion j : x ⟶ M that the restrictions with H = D^{H}(x) ⟶^{D^{H}(i)} D^{H}(M) ⟶^{k} D^{G}(M) of U and V for k the obvious inclusion agree. As RH-isomorphic implies isometically RH-isomorphic, this follows from the condition that U and V are coherent.

Lemma 4.15 Theorem 4.5 is true if M is closed and w^{G}(M) = 0.

\underline{\underline{Proof}} : If dim(M) is even, we get ρ_{an}^{G}(M; V) = 0 from Poincaré duality 1.20. Analogously we obtain 2 · ρ_{pl}^{G}(M; V) = ρ_{pd}(M; V) from 3.23 and the claim follows. Suppose that M is odd-dimensional. We may assume V = q^{∗}U for some G-representation U by 4.14 and the assumption that V is coherent to a G-representation. Because of 3.32 we may suppose U = R. Now apply Lemma 4.9 and Theorem 4.13.

Next we want to drop the condition that G acts orientation preserving.

Lemma 4.16 Theorem 4.5 is true if M is closed and M is orientable

\underline{\underline{Proof}} : As M is orientable w^{G}(M) : D^{G}(M) ⟶ \{±1\} factorizes over q : D^{G}(M) ⟶ G into w̄ : G ⟶ {±1}. Let K be the kernel of G. Since K operates orientation preserving, Lemma 4.15 applies to res_{K}^{G}M. It suffices to treat the case K ≠ G. Because the maps ind_{K}^{G} ∘ res_{G}^{K} on K_{1}(RG)^{Z/2} and on R ⊗_{Z} Rep_{R}(G) are given by ? ⊗_{R} ([R] + [w̄R]) and Γ_{1} is compatible with restriction, we obtain

4.17 (ρ_{an}^{G}(M; V) − Γ_{1}(ρ_{pl}^{G}(M; V)) + \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M; V))) ⊗_{R} ([R] + [w̄R]) = 0

Let Z/2 act on S^{1} by complex conjugation. We derive from the poduct formulas 1.32 and 3.13

4.18 ρ_{an}^{G×Z/2}(M × S^{1}, V ⊗_{R} R) − Γ_{1}(ρ_{pl}^{G×Z/2}(M × S^{1}; V ⊗_{R} R))

+ \frac{1}{2} · Γ_{1}(ρ_{pd}^{G×Z/2}(M × S^{1}; V ⊗_{R} R)) =

χ^{G}(M; V) ⊗_{R} (ρ_{an}^{Z/2}(S^{1}; R) − Γ_{1}(ρ_{pl}^{Z/2}(S^{1}; V)) + \frac{1}{2} · Γ_{1}(ρ_{pd}^{Z/2}(S^{1}; V))) +
(ρ_{an}^{G}(M; V) − Γ_{1}(ρ_{pl}^{G}(M; V)) + \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M; V))) ⊗_{R} χ^{Z/2}(S^{1}; R)

We conclude from 1.16 , 3.26 and 3.29

4.19 ρ_{an}^{Z/2}(S^{1}; R) − Γ_{1}(ρ_{pl}^{Z/2}(S^{1}; V)) + \frac{1}{2} · Γ_{1}(ρ_{pd}^{Z/2}(S^{1}; V)) = 0

Obviously we have

4.20 χ^{Z/2}(S^{1}; R) = [R] − [R^{−}]

If we restrict the G × Z/2 on M × S^{1} to G by id × w̄ : G ⟶ G × Z/2, we obtain an orientation preserving action. Hence we can apply Lemma 4.15 and obtain from 4.18, 4.19 and 4.20

4.21 (ρ_{an}^{G}(M; V) − Γ_{1}(ρ_{pl}^{G}(M; V)) + \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M; V))) ⊗_{R} ([R] − [w̄R]) = 0

Now add 4.17 and 4.21, divide by 2 and the claim follows.

Lemma 4.22 Theorem 4.5 is true if M is closed.

\underline{\underline{Proof}} : Assume that M is not orientable. Let p : M̂ ⟶ M be the orientation covering. There is a group extension 0 ⟶ Z/2 ⟶ Ĝ ⟶^{q̂} G ⟶ 0 and a Ĝ-operation on M̂ extending the Z/2-action on M̂ and covering the G-action on M. Since M̂ is orientable, the claim for the Ĝ-manifold M̂ follows from Lemma 4.16. Applying induction with q̂ gives the assertion also for M.

Now we are ready to prove Theorem 4.5. Assume that M_{1} = ∅. Let the orthogonal D^{G}(M)-representation V be coherent to the G-representation W. Then V ∪_{∂M} V is coherent to W ∪_{M_{1}} W as a G × Z/2-representation. Because of Lemma 4.22 we get for the closed G × Z/2-manifold M ∪_{∂M} M

4.23 ρ_{an}^{G×Z/2}(M ∪_{∂M} M, V ∪_{∂M} V) =

Γ_{1}(ρ_{pl}^{G×Z/2}(M ∪_{∂M} M; V ∪_{∂M} V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G×Z/2}(M ∪_{∂M} M; V ∪_{∂M} V))

Put the double formulas 1.27 and 3.34 into 4.23

4.24 ρ_{an}^{G}(M; V) ⊗_{R} [R] + ρ_{an}^{G}(M, ∂M; V) ⊗_{R} [R^{−}] =

Γ_{1}(ρ_{pl}^{G}(M; V)) ⊗_{R} [R] + Γ_{1}(ρ_{pl}^{G}(M, ∂M; V)) ⊗_{R} [R^{−}] + ln(2) · χ^{G}(∂M; V) ⊗_{R} [R] −

\frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M; V)) ⊗_{R} [R] − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M, ∂M; V)) ⊗_{R} [R^{−}] −
\frac{1}{2} · ln(2) · χ^{G}(∂M; V) ⊗_{R} [R] + \frac{1}{2} · ln(2) · χ^{G}(∂M; V) ⊗_{R} [R^{−}] =
(Γ_{1}(ρ_{pl}^{G}(M; V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M; V)) + \frac{1}{2} · ln(2) · χ^{G}(∂M; V)) ⊗_{R} [R] +
(Γ_{1}(ρ_{pl}^{G}(M, ∂M; V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M, ∂M; V)) + \frac{1}{2} · ln(2) · χ^{G}(∂M; V)) ⊗_{R} [R^{−}]

Now the claim in the case M_{1} = ∅ follows from comparing the coefficients of [R] in 4.24. In the general case we repeat this argument, but now we glue along M_{1} instead of ∂M. Then the equation corresponding to 4.24 looks like

4.25 ρ_{an}^{G}(M; V) ⊗_{R} [R] + ρ_{an}^{G}(M, M_{1}; V) ⊗_{R} [R^{−}] =

Γ_{1}(ρ_{pl}^{G}(M; V)) ⊗_{R} [R] + Γ_{1}(ρ_{pl}^{G}(M, M_{1}; V)) ⊗_{R} [R^{−}] + ln(2) · χ^{G}(M_{1}; V) ⊗_{R} [R] −

\frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M; V)) ⊗_{R} [R] − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M, M_{1}; V)) ⊗_{R} [R^{−}] − \frac{ln(2)}{2} · χ^{G}(M_{1}; V) ⊗_{R} [R] +

\frac{1}{2} · ln(2) · χ^{G}(M_{1}; V) ⊗_{R} [R^{−}] + \frac{ln(2)}{2} · χ^{G×Z/2}(∂(M ∪_{M_{1}} M), V ∪_{M_{1}} V) =
(Γ_{1}(ρ_{pl}^{G}(M; V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M; V)) + \frac{1}{2} · ln(2) · χ^{G}(∂M; V)) ⊗_{R} [R] +
(Γ_{1}(ρ_{pl}^{G}(M, M_{1}; V)) − \frac{1}{2} · Γ_{1}(ρ_{pd}^{G}(M, M_{1}; V)) + \frac{1}{2} · ln(2) · χ^{G}(∂M; V)) ⊗_{R} [R^{−}]

Now Theorem 4.5 follows from comparing the coefficients of R^{−}.

We end this section by indicating the proof of Lemma 1.13 about the meromorphic extension of the equivariant Zeta-function. As μ_{W}^{p} (s) defined in 4.6 has a meromorphic extension to the complex plane, analytic in 0, the same is true for ζ(M; V)(s) by Lemma 4.9, povided that M is closed and w^{G}(M) = 0. Since the product formula 1.32 and double formula 1.27 give explicit identities of Zeta-functions for Real(s) > m/2, the arguments in the proof of Theorem 4.5 can also be used to verify Lemma 1.13

5. Some Computations

In this section we treat some special cases as an illustration. Firstly, suppose that G is trivial. Then Γ_{1} ⊕ Γ_{2} : K_{1}(RG)^{Z/2} ⟶ (R ⊗_{Z} Rep_{R}(G)) ⊕ (Z/2 ⊕_{Z} R̂ep_{R}(G)) reduces to R^{∗} ⟶ R, r ↦ ln(r/ | r |) and ρ_{pl}^{G}(M, M_{1}; V) is just a positive real number. Moreover, χ^{G}(∂M; V) is just dim_{R}(V) · χ(∂M),where χ(∂M) is the ordinary Euler characteristic. By Lemma 3.20 ρ_{pd}^{G}(M, M_{1}; V) vanishes. We get from Theorem 4.5 :

Corollary 5.1 Let M be a Riemannian manifold whose boundary is the disjoint union of M_{1} and M_{2}. Suppose that the metric is a product near the boundary. Let V be an orthogonal π_{1}(M)-representation. Then :

ρ_{an}(M, M_{1}; V) = ln(ρ_{pl}(M, M_{1}; V)) + \frac{ln(2)}{2} · χ(∂M) · dim_{R}(V)

Notice that our definition of analytic torsion differs from those in Ray-Singer [29] by a factor of 2 and our PL-torsion is the square of theirs. Corollary 5.1 for closed manifolds was independently proved by Cheeger [8] and Müller [26].

For trivial G and V the PL-torsion can be computed as follows. Fix a dimension p ≥ 0. Choose an orthonormal bases {ω_{1}, ω_{2}, …, ω_{β_{p}}} for the space of harmonic p-forms H_{harm}^{p}(M, M_{1}; V). Let {σ_{1}, σ_{2}, …, σ_{β_{p}}} be a set of cocycles in the singular (or cellular) chain complex with integral coefficients of (M, M_{1}) such that the set of their classes in H(M, M_{1}; Z)/T ors(H(M, M_{1}; Z)) is an integral bases. Let r_{p} be the determinant of the following matrix

( \begin{matrix}∫_{σ_{1}} ω_{1} & ∫_{σ_{1}} ω_{2} & ... & ∫_{σ_{1}} ω_{β_{p}} \\ ∫_{σ_{2}} ω_{1} & ∫_{σ_{2}} ω_{2} & ... & ∫_{σ_{2}} ω_{β_{p}} \\ {. \atop .} & {. \atop .} & ... & {. \atop .} \\ ∫_{σ_{βp}} ω_{1} & ∫_{σ_{βp}} ω_{2} & ... & ∫_{σ_{βp}} ω_{β_{p}} \end{matrix} )

Define

5.2 r(M, M_{1}) = ∏_{p=0}^{m} r_{p}^{2·(−1)^{p+1}}

Define the multiplicative Euler characteristic to be

5.3 mχ(M, M_{1}) = ∏_{p=0}^{m} | T ors(H_{i}(M, M_{1}; Z)) |^{(−1)^{p}} The following result was proved in Cheeger [8] (0.1) and (1.4)

Proposition 5.4 Let (M, M_{1}, M_{2}) be a triad of Riemannian manifolds. Then

ρ_{pl}(M, M_{1}; V) = mχ(M, M_{1})^{2} · r(M, M_{1})

\underline{\underline{Proof}} : Let κ_{i}^{int} be the Hilbert structure on H_{i}(M, M_{1}; R) for which the bases given by {σ_{1}, σ_{2}, ..., σ_{β_{p}}} is an orthonormal bases.We derive from Proposition 3.8

ρ_{pl}(M, M_{1}; R) = ρ_{pl}(M, M_{1}, κ_{harm}) = ρ_{pl}(M, M_{1}, κ_{∗}^{int}) · r(M, M_{1}).

Hence it suffices to show

ρ_{pl}(M, M_{1}, κ_{int}) = mχ(M, M_{1})

Its elementary proof can be found for example in Cheeger [8] (1.4) and in Lück [21] Lemma 18.34

Corollary 5.5 Let M be a m-dimensional Riemannian manifold. a.) If M is a rational homology sphere, i.e. H_{∗}(M; Q) = H_{∗}(S^{m}; Q),then :

ρ_{an}(M) = ln(ρ_{pl}(M)) = (1 + (−1)^{m+1}) · ln(V ol(M)) + 2 · ∑_{p=1}^{m−1}(−1)^{p} · ln(| H_{p}(M, Z) |)

b.) If M is a rational homology point, i.e. H_{∗}(M; Q) = H_{∗}({point}; Q), then : ρ_{an}(M; R) = ln(V ol(M)) + (1 + (−1)^{m+1}) · \frac{ln(2)}{2} + 2 · ∑_{p=1}^{m−1}(−1)^{p} · ln(| H_{p}(M, Z) |)

Next we treat the case where M is orientable and G acts orientation preserving on M, or equivalently, where w^{G}(M) is trivial. We derive from Theorem 4.5 and Poincaré duality 1.20 and 3.23 :

Corollary 5.6 Let M be a Riemannian G-manifold with invariant Riemannian metric which is a product near the boundary. Suppose M is closed and w^{G}(M) = 0. Let V be an orthogonal D^{G}(M)-representation coherent to a G-representation.

a.) If dim(M) is odd , we have :

ρ_{an}^{G}(M; V) = Γ_{1}(ρ_{pl}(M; V))

ρ_{pd}(M; V) = 0

b.) If dim(M) is even, we get :

ρ_{an}(M; V) = 0

ρ_{pl}^{G}(M; V) = \frac{1}{2}ρ_{pd}^{G}(M; V)

Remark 5.7 The assumption w^{G}(M) = 0 is necessary in Corollary 5.6. We have already shown ρ_{pd}^{Z/2}(S^{1}; R) ≠ 0 for Z/2 acting by complex conjugation in 3.29. Let Z/2 act on S^{2} by sending (x, y, z) to (−x, y, z). Then S^{2} is the double of D^{2} for appropiate Riemannian metrics on S^{2}. We derive from the Double formula 1.27, Poincaré duality 1.20 ,Proposition 3.8 and Corollary 5.1 that

ρ_{an}^{Z/2}(S^{2}; R) = ρ_{an}(D^{2}; R) · ([R] − [R^{−}]) = ln(V ol(S^{2})/2) · ([R] − [R^{−}])

Hence ρ_{an}^{Z/2}(S^{2}) is not necessarily zero. Let RP^{n} be the n-dimensional real projective space. It is rationally a point if n is even. We get from Corollary 5.5

ρ_{an}(RP^{2n}; R) = ln(V ol(RP^{2n})) − 2n · ln(2)

Again this may be non-zero.

Next we analyse how the analytic torsion changes under variation of the metric and under G-homotopy equivalence. We have just to combine Theorem 4.5 and Proposition 3.33. Suppose that (f; f_{1}, f_{2}) : (M; M_{1}, M_{2}) ⟶ (N; N_{1}, N_{2}) and V and W satisfy the same hypothesis as in Proposition 3.33 and assume additionally, that M_{1} ∩ M_{2} and N_{1} ∩ N_{2} are empty. We get using the same notation as in Proposition 3.33

Proposition 5.8 ρ_{an}^{G}(N, N_{1}; V) − ρ_{an}^{G}(M, M_{1}; f^{∗}V) = \frac{1}{2} · ω^{G}(N; V)(τ^{G}(f, f_{1})) + (−1)^{m−1} · \frac{1}{2} · ω^{G}(N;^{w} V)(τ^{G}(f, f_{2})) − ∑_{i≥0}(−1)^{i} · u_{i}

Consider an isometric G-diffeomorphism f : M_{2} ⟶ N_{1} and an isometric RD^{G}(M_{2})- isomorphism f̄: V |_{M_{2}}⟶ W |_{N_{1}}. Denote by D^{∗}(M ∪_{f} N, M, N; V, W) resp. D^{∗}(M, M_{1}; V) the acyclic finite RG-Hilbert chain complexes given by the long Mayer-Vietoris sequence resp. the long homology sequence of the pair and the harmonic RG-Hilbert structures. We derive from the Sum Formula 3.49 and Theorem 4.5 :

Theorem 5.9 (Sum Formula for Analytic Torsion)

a.) ρ_{an}^{G}(M ∪_{f} N; V ∪f̄ W) = ρ_{an}^{G}(M; V)+ρ_{an}^{G}(N; W)−ρ_{an}^{G}(M_{2}; V)−Γ_{1}(hr(D^{∗}(M ∪_{f} N, M, N; V, W)))−ln(2)·χ^{G}(M_{2}; V)

b.) ρ_{an}^{G}(M, M_{1}; V) = ρ_{an}^{G}(M; V) − ρ_{an}^{G}(M_{1}; V) − Γ_{1}(hr(D^{∗}(M, M_{1}; V)))

Remark 5.10 The sum formula is very useful for computations. One may chop a manifold into elementary peaces, compute the analytic torsion for each piece and use the Sum Formula to get the analytic torsion for the manifold itself. The existence of such a formula is remarkable from the analytic point of view since it is very hard to get information about the spectrum of the Laplace operator on M ∪_{f} N from the spectrum of the Laplace operator on M, N and M_{1}. We do not have a direct analytic proof of the sum formula, but such a proof may be hidden in the paper of Müller [26] section 10. If one has a proof for the Sum Formula not using Theorem 4.5 one can prove Theorem 4.5 by using induction over the number of handles.

Next we compute the various torsion invariants for G-representations (cf. Ray [28]). Define for a Riemannian G-manifold

5.11 ρ̂_{an}^{G}(M) := ρ_{an}^{G}(M) + ln(V ol(M)) · (χ^{G}(M) − 2 · [R])

ρ̂_{pl}^{G}(M) := ρ_{pl}^{G}(M) + ln(V ol(M)) · (χ^{G}(M) − 2 · [R])

Let V be a G-representation. Choose any orthogonal structure and any invariant Riemannian metric on DV which is a product near the boundary SV. Then ρ̂_{an}^{G} and ρ̂_{pl}^{G} are defined for SV and DV and depend only on the G-diffeomorphism type, but not on the other choices by Proposition 5.8 since the equivariant Whitehead torsion of a G-diffeomorphism is zero.

Lemma 5.12

a.) ρ_{pd}^{G}(D(V ⊕ W)) = ρ_{pd}^{G}(DV) + ρ_{pd}^{G}(DW)

b.) ρ̂_{pl}^{G}(DV) = 0

c.) ρ̂_{an}^{G}(DV) = −\frac{1}{2} · Γ_{1} (ρ_{pd}^{G}(DV)) + \frac{ln(2)}{2} · χ^{G}(SV)

d.) ρ_{pd}^{G}(SV) = χ^{G}(SV) ⊗_{R} ρ_{pd}^{G}(DV)

e.) ρ̂_{pl}^{G}(SV) = ρ_{pd}^{G}(DV) ⊗_{R} (χ^{G}(SV) − [R])

f.) ρ̂_{an}^{G}(SV) = Γ_{1} (ρ_{pd}^{G}(DV)) ⊗_{R} (\frac{1}{2} · χ^{G}(SV) − [R])

\underline{\underline{Proof}} : a.) is a consequence of the product formula for Poincaré torsion 3.48. b.) As the projection from DV to a point is a simple G-homotopy equivalence, the claim follows from Proposition 3.8. c.) follows from Theorem 4.5 and b. ) d.) and e.) follow from Poincaré duality 3.23 and the Sum Formula 3.49 f.) is a consequence of Theorem 4.5 and d.) and e.)

Because of the Lemma 5.12 above we obtain a homomorphism 5.13 Rep_{R}(G) ⟶ R ⊗_{Z} Rep_{R}(G) [V] ↦ ρ_{pd}^{G}(DV)

Hence it suffices to compute the torsion invariants for irreducible G-representations in order to know it for all G-representations. Elements in R ⊗_{Z} Rep_{R}(G) and hence by Proposition 2.26 also elements in K_{1}(RG)^{Z/2} are detected by restriction to cyclic subgroups so that it is enough to compute the torsion elements of irreducible G-representations for cyclic G.

Let G be the cyclic group Z/n. Let d > 0 be a divisor of n. If d ≥ 3, let V (j, d) for 1 ≤ j < d/2 be the (real) 2-dimensional G-representation given by multiplication with primitive d^{th}-root of unity exp(2πi\frac{j}{d}) on C. Let V (d/2, d) be the unique 1-dimensional nontrivial G-representation, if d is even, and R the trivial G-representation. Denote by I(d) the set {j ∈ Z | 1 ≤ j ≤ d/2, (d, j) = (1)}. Then the set

{V (j, d) | d ≥ 2, d | n, 1 ≤ j ≤ d/2, (j, d) = (1)} ∪ {R}

is a complete set of representatives for the irreducible G-representations. One easily computes using Lemma 5.12 in the first case and the Double Formula 3.34 in the second case

5.14 ρ_{pd}^{G}(DV (j, d)) = −ρ_{pl}^{G}(SV (j, d)) =

[d^{2} · id : R ⟶ R] − ∑_{1≤k≤d/2} [| exp(2πi\frac{jk}{d}) − 1 |^{2} ·id : V (k, d) ⟶ V (k, d)]
for d | n, d ≥ 3, j ∈ I(d)
ρ_{pd}^{G}(DV (1, 2)) = [2 · id : R ⟶ R] − [2 · id : V (1, 2) ⟶ V (1, 2)]
ρ_{pd}^{G}(D(R)) = 0

Now one can describe the homorphism 5.13 above by characters

Lemma 5.15 (Character Formula) Let G be a finite group and V a G-representation with character ch_{V}. Then ρ_{pd}^{G}(DV) ∈ R ⊗_{Z} Rep_{R}(G) can be viewed as a class function on G with values in R. Let g ∈ G be an element. Let n be the order of the cyclic subgroup < g > generated by g. Then the value of ρ_{pd}^{G}(DV) at g is 0, if n = 1 and otherwise :

ρ_{pd}^{G}(DV)(g) =

∑_{d|n,d≥2} ∑_{j∈I(d)} < ch_{V |_{<g>}}, ch_{V (j,d)} > · < ch_{V (j,d)}, ch_{V (j,d)} >^{−1} ·ρ_{pd}^{G}(DV (j, d))(g) =

∑_{1≤l≤n} \frac{1}{n} · ch_{V} (g^{l}) · (−1)^{l} · ln(4) + ∑_{d|n,d≥3} ∑_{j∈I(d)} ∑_{1≤l≤n} \frac{1}{n}ch_{V} (g^{l}) · cos(2π \frac{jl}{d}) ·

(ln(d^{2}) − ∑_{1≤k≤d/2} ln ((sin(2π \frac{kj}{d})^{2} + (cos(2π \frac{kj}{d}) − 1)^{2}) · 2 · cos(2π \frac{k}{d}))

Theorem 5.16 Let G be a finite group. There is an injective ring homomorphism

ρR^{G} : Rep_{R}(G) ⟶ Z ⊕ (⊕_{(H)}K_{1}(R[W H])^{Z/2}) [V] ↦ dim_{R}(V) ⊕ (ρ_{pd}^{WH}(DV^{H}))_{(H)}

\underline{\underline{Proof}} : This homomorphism is compatible with restriction to subgroups of G for an appropiate restriction homomorphism on the right side, we do not explain here (see Lück [21]). Hence it suffices to consider the case where G is cyclic.

Let RepR^{free}(G) be the subgroup of Rep_{R}(G) generated by all free G-representations. Let res_{G/H}^{G} : Rep_{R}(G/H) ⟶ Rep_{R}(G) be restriction with the projection G ⟶ G/H for a subgroup H ⊂ G. Then we obtain an isomorphism, provided that G is cyclic

5.17 ⊕_{H⊂G}res_{G/H}^{G} : ⊕_{H⊂G}RepR^{free}(G/H) ⟶ Rep_{R}(G)

With respect to this splitting of the representation ring ρR^{G} is given by a upper triangle matrix. Hence it suffices to show injectivity for the diagonal entrees, i.e. the injectivity of

RepR^{free}(G) ⟶ K_{1}(RG)^{Z/2} [V] ↦ ρ_{pd}^{G}(DV)

Consider the ring homomorphism

ψ_{k} : R[Z/n] ⟶ C^{∗}

sending the generator to the root of unity exp(2πi\frac{k}{n}). The set \{V (j) | j ∈ I(n)\} is a bases for RepR^{free}(G). The image of [V (j)] under ψ_{k} ∘ ρR^{G} is | exp(2πi\frac{jk}{n} − 1) |^{−2} for d ≥ 3, 1 ≤ k < d/2 and 1/2 for d = 2. Now an application of Franz’ Lemma finishes the proof (see Franz [13], Cohen [9])

We get as immediate conclusion of Theorem 5.16 the celebrated result of deRham [30] that two orthogonal G-representations V and W are isometrically RG-isomorphic if and only if their unit spheres are G-diffeomorphic. Similar proofs using PL-torsion resp. analytic torsion can be found in Rothenberg [31] and Lott-Rothenberg [19]. The result is an extension of the classification of Lens spaces which is carried out for example in Cohen [9] and Milnor [25].

The result of deRham does not hold in the topological category. Namely, there are nonlinearly isomorphic G-representations V and W whose unit spheres are G-homeomorphic (see Cappell-Shaneson [7]). However, if G has odd order, G-homeomorphic implies G-diffeomor-phic for unit spheres in G-representations as shown by Hsiang-Pardon [16] and Madsen-Rothenberg [24].

The Sum formula 3.11 implies a local formula for Poincaré torsion. For this purpose we have to recall a different notion of equivariant Euler characteristic as defined in Lück [20]. Given a G-space X, denote by {G/? ⟶ X} the set of all G-maps G/H ⟶ X for all subgroups H of G. Call x : G/H ⟶ X and y : G/K ⟶ X equivalent if there is a G-iso-morphism σ : G/H ⟶ G/K such that y ∘ σ and x are G-homotopic. Let {G/? ⟶ X}/ ∼ be the set of equivalence classes under this equivalence relation on {G/? ⟶ X}. Given a G-map x : G/H ⟶ X, define X^{H}(x) to be the component of X^{H} containing the point x(eH). There is a bijection

5.18 {G/? ⟶ X}/ ∼ ⟶ ∐_{(H)} π_{0}(X^{H})/WH [x : G/H ⟶ X] ↦ [X^{H}(x)]

where the coproduct runs over the set of conjugacy classes (H) of subgroups of G. Let U^{G}(X) be the free abelian group generated by {G/? ⟶ X}/ ∼. Let X^{>H} be the subset of those points in X, whose isotropy group G_{x} satisfies G_{x} ⊃ H, G_{x} ≠ H, and WH(x) is the isotropy group of [X^{H}(x)] ∈ π_{0}(X^{H}) under the WH-action. Given a pair (X, Y) of finite G-CW -complexes, define its universal equivariant Euler characteristic in the sense of Lück [20]

5.19 χ_{univ}^{G}(X) ∈ U^{G}(X)

by assigning to x : G/H ⟶ X the integer which is given by the ordinary Euler characteristic χ(X^{H}(x)/WH(x), (X^{H}(x) ∩ (X^{>H} ∪ Y))/WH(x)). It is connected to the equivariant Euler characteristic, we have defined in 3.12, by the following map

5.20 θ^{G}(X) : U^{G}(X) ⟶ Rep_{R}(G) [x : G/H ⟶ X] ↦ [R[G/H]]

We derive from the universal property of χ_{univ}^{G} and the Sum Formula 3.49 for χ^{G} (see Lück [20]) :

Lemma 5.21 θ^{G}(X)(χ_{univ}^{G}(X, Y)) = χ^{G}(X, Y)

Let (M, M_{1}, M_{2}) be a G-manifold triad and V be an orthogonal D^{G}(M)-representation. Given x : G/H ⟶ X, we define the H-representation x^{∗}V by restricting V to H by H =

E^{H}({point} ⟶^{D^{H}(x|_{eH})} D^{H}(res_{H}^{G}(X)) ↪ D^{G}(X). Let the homomorphism

5.22 Φ^{G}(M; V) : U^{G}(M) ⟶ K_{1}(RG)^{Z/2}

send [x : G/H ⟶ X] to ind_{H}^{G} (ρ_{pd}^{H}(D(T M_{x})) ⊗_{R} x^{∗}V ), where T M_{x} is the tangent space of M at the point x(eH) and ⊗_{R} the pairing defined in 2.14. Now we obtain a local formula for the Poincaré torsion in terms of the various tangent representations of the components of the fixed point sets.

Proposition 5.23 (Local Formula for Poincaré Torsion)

Φ^{G}(M; V) (χ_{univ}^{G}(M, M_{1})) = ρ_{pd}^{G}(M, M_{1}; V)

\underline{\underline{Proof}} : In the sequel we have not to worry about corners and straightening the angle because ρ_{pd}^{G} depends only on the simple homotopy type by Proposition 3.33. Moreover, we assume V = R for simplicity, the general case is done similiarly. Let N_{1} and N_{2} be G-manifolds and N_{0} ⊂ ∂N_{1} and N′_{0} ⊂ ∂N_{2} be submanifolds of codimension 0. Let f : N_{0} ⟶ N′_{0} be a G-diffeomorphism and put N := N_{1} ∪_{f} N_{2}. Then we get from the Sum Formula 3.49

5.24 ρ_{pd}^{G}(N) = ρ_{pd}^{G}(N_{1}) + ρ_{pd}^{G}(N_{2}) − ρ_{pd}^{G}(N_{0})

Since the equivariant Euler characteristic is additive in the sense of Lück [20], we have

5.25 Φ^{G}(N) (χ_{univ}^{G}(N)) =

Φ^{G}(N_{1}) (χ_{univ}^{G}(N_{1})) + Φ^{G}(N_{2}) (χ_{univ}^{G}(N_{2})) − Φ^{G}(N_{2}) (χ_{univ}^{G}(N_{2}))

We first verify the claim for empty M_{1}. We begin with the special case M = Dξ where ξ ↓ B is a G-vector bundle over a G-manifold X having precisely one orbit type, say G/H. Then X/G is a manifold and we use induction over the number of handles. If X/G is empty the claim is trivial. Suppose that X is obtained from Y by attaching an equivariant handle :

[figure: G/H × S^{k−1} × D^{d−k}; Y; G/H × D^{k} × D^{d−k}; X]

We may suppose that X is connected,otherwise treat each component seperately.Let Hrepresentation W be the typical fibre of ξ ↓ X.Then we obtain a G-push out :

[figure: G ×_{H} (S^{k−1} × D^{d−k} × DW); Dξ ↓ Y; G ×_{H} (D^{k} × D^{d−k} × DW); Dξ ↓ X]

By induction hypothesis the claim is true for Dξ ↓ Y.Because of 5.24 and 5.25 it suffices to verify the claim for G-manifolds of the shape G ×_{H} (Z × DW) for Z a manifold with trivial H-action.One easily computes

ρ_{pd}^{G}(G ×_{H} (Z × DW)) = ind_{H}^{G}(ρ_{pd}^{H}(Z × DW)) =

ind_{H}^{G}(ρ_{pd}^{H}(Z) ⊗_{R} χ^{H}(DW) + χ^{H}(Z) ⊗_{R} ρ_{pd}^{H}(DW)) = χ(Z) · ind_{H}^{G}(ρ_{pd}^{H}(DW)) =

Φ^{G}(G ×_{H} (Z × DW))(χ^{G}(G ×_{H} (Z × DW)))

This finishes the proof in the case M = Dξ, M_{1} = ∅. Now the claim for M_{1} = ∅ follows by induction over the orbit types of M. Namely, choose H ∈ Iso(M) such that H ⊂ K, K ∈ Iso(M) implies H = K. Let ν be the normal bundle of M^{(H)} = G · M^{H} in M. Define M̄ := M − int(Dν). The induction hypothesis applies to M̄ and Sν and the considerations above to Dν. Since M is M̄ ∪_{Sν} Dν, the assertion follows from 5.24 and 5.25, provided that M_{1} = ∅.

The claim in general follows from the relations

ρ_{pd}^{G}(M, M_{1}) = ρ_{pd}^{G}(M) − ρ_{pd}^{G}(M_{1})

Φ^{G}(M)(χ^{G}(M, M_{1})) = Φ^{G}(M)(χ^{G}(M)) − Φ^{G}(M_{1})(χ^{G}(M_{1}))

Example 5.26 Suppose that the G-manifold M is modelled upon the G-representation V, i. e. there is a G-representation V such that for any x ∈ M the G_{x}-representations resV and T M_{x} are linearly G_{x}-isomorphic. This is true for example if all fixed point sets of M are connected and non-empty. Then the Local Formula for Poincaré torsion 5.23 reduces to

ρ_{pd}^{G}(M, M_{1}; R) = χ^{G}(M, M_{1}) ⊗_{R} ρ_{pd}^{G}(DV)

Let Lef(l(g)) be the Lefschetz index of the map l(g) : (M, M_{1}) ⟶ (M, M_{1}) given by multiplication with g ∈ G. Then we obtain in terms of class functions on G with values in R :

Γ_{1} (ρ_{pd}^{G}(M, M_{1}; R)) (g) = Lef(l(g)) · Γ_{1} (ρ_{pd}^{G}(DV)) (g)

Remark 5.27 In Connolly-Lück [10] a duality formula is established for a G-homotopy equivalence (f, ∂f) : (M, ∂M) ⟶ (N, ∂N). A correction term appears which depends on the universal Euler characteristic and the G_{x}-representations T M_{x} for all x ∈ M. It is closely related to, firstly, the Local Formula for Poincaré torsion 5.23, and, secondly, to Proposition 3.33 which says that the difference of the Poincaré torsion is an obstruction for a duality formula for equivariant Whitehead torsion. The duality formula is important for the proof of the equivariant π-π-theorem in the simple category (see Dovermann-Rothenberg [12], Lück-Madsen [22] and [23])

Remark 5.28 The Euler characteristic term in our main Theorem 4.5 may also be interpreted as the index of the deRham complex. This leads to the following question.

Let P^{∗} be an elliptic complex of partial differential operators. Denote by Δ(P)^{∗} the associated Laplacian. It is an elliptic non-negative self-adjoint partial differential operator in each dimension. Hence its analytic torsion ρ_{an}(Δ(P)^{∗}) can be defined as done before for the ordinary Laplace operator. Suppose that the complex restricts on the boundary of M to an elliptic complex ∂P^{∗} in an appropiate sense. Can one find a more or less topological invariant ρ_{top}(P^{∗}) such that the following equation holds

ρ_{an}(P^{∗}) = ρ_{top}(P^{∗}) + \frac{ln(2)}{2} · index(∂P^{∗})

If we take P^{∗} to be the deRham complex and put ρ_{top} to be ρ_{pl}, the equation above becomes just Corollary 5.1.

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Address Wolfgang Lück^{∗} Fachbereich Mathematik und Informatik Westfälische Wilhelms-Universität Münster Einsteinstr. 62 48149 Münster Germany email: lueck@math.uni-muenster.de Fax: 0251 838370

* Partially supported by NSF Grant DMS - 9103327