Eta invariant and concordance classes of positive scalar curvature metrics

László Fehér Alfred Renyi Mathematical Research Institute e-mail: lfeher@math-inst.hu

Contents

1 Introduction 2 1.1 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 The assembly map and existence of positive scalar curvature metrics 5 2.1 The equivariant Dirac operator D(W) . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Definition of the assembly map and existence of positive scalar curvature metrics . . 8 2.3 Homotopy theoretic description of the assembly map . . . . . . . . . . . . . . . . . . 10 2.4 The assembly map with ℤ_{p^{∞}} coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 14

3 Concordance classes of positive scalar curvature metrics 16 3.1 Dirac operators on manifolds with boundary . . . . . . . . . . . . . . . . . . . . . . 16 3.2 Positive scalar curvature and the R_{n}-groups . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 The surjectivity of θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

3.3.1 The definition of Diagram 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.3.2 Three surjective maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Connection between η and the classical eta invariants 23 4.1 Complexification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 4.2 The eta invariants and the APS index theorem for manifolds with boundary . . . . . 24

5 Commutativity of square 2 26

It turned out that the proof of Proposition 3.26 wasn’t correct. I am working on a proof now. The consequence of this missing proof is that I can prove only a weaker version of the main Theorem 3.14. Surjectivity should be replaced by p-local surjectivity. (I inserted the necessary changes in the text in boldface.) However p-local surjectivity can be proved by different methods like the techniques developed by Peter Gilkey and Boris Botvinnik. See e.g. [BG95]. So my excuses for using deeper results are that on one hand I establish a language which is useful in the field and on the other hand this result on the topological description of the mod ℤ reduced eta invariant might be of independent interest.

Abstract

Suppose that M is a compact n-1-dimensional (n ≥ 6) spin manifold admitting positive scalar curvature metric with finite fundamental group π = π_{1}(M). We give a “lower bound” on the size of π̃_{0}(Riem^{+}(M)), the set of concordance classes of positive scalar curvature metrics on M. Concordance is a (possibly) weaker equivalence relation than homotopy through positive scalar curvature metrics. S. Stolz showed in [Sto] that there is a certain relative bordism theory such that the corresponding bordism groups R_{n}(Bπ) acts on π̃_{0}(Riem^{+}(M)) freely and transitively. The groups R_{n}(Bπ) are not known for any π. But there is an index type map: θ : R_{n}(Bπ) → KO_{n}(ℝπ) which is usually non-trivial. (This map can be defined for infinite groups, too. Then the group algebra ℝπ should be replaced by the reduced C^{∗}-algebra of π.) We show that for a finite p-group π the map θ is surjective if n is large enough. As a main ingredient of the proof we use a real version of the Atiyah-Patodi-Singer index theorem for flat bundles, which compares the η-invariant with the the assembly map with ℤ_{p^{∞}}-coefficients.

1 Introduction

A well known result of Lichnerowicz shows that the index of the Dirac operator on a spin manifold M is an obstruction for the existence of a positive scalar curvature metric on M. Nigel Hitchin [Hit74] was the first to observe that the Dirac operator method can be used for classification problems, too: An appropriate index can distinguish different components of Riem^{+}(M) – the space of positive scalar curvature metrics on M. The eta invariant of Atiyah, Patodi and Singer [APS75a] gave a powerful tool to distinguish plenty of components of Riem^{+}(M) provided that M has a non-trivial fundamental group (see e.g. [BG95]). In fact our method can be considered as a systematic way to exhaust the η-invariant method. The novelty is that instead of constructing examples, we use abstract homotopy theoretic arguments.

Instead of π_{0}(Riem^{+}(M)) we look at π̃_{0}(Riem^{+}(M)), the set of concordance classes of positive scalar curvature metrics on M. Concordance is a (possibly) weaker equivalence relation than homotopy through positive scalar curvature metrics. S. Stolz showed in [Sto] that there is a certain relative bordism theory such that the corresponding bordism groups R_{n}(Bπ) have the following properties: Suppose that M is a compact n-1-dimensional (n ≥ 6) connected spin manifold with fundamental group π = π_{1}(M). Then:

1. M admits a positive scalar curvature metric if and only if R_{n−1}(Bπ) ∋ [M] = 0.

2. If M admits a positive scalar curvature metric then R_{n}(Bπ) acts on π̃_{0}(Riem^{+}(M)) freely and transitively.

This result shows that these groups are of fundamental importance in this field. The group R_{n}(Bπ) can be defined as follows:

R_{n}(Bπ): = \{(W, f, g) | \begin{matrix}W is an n-dimensional spin manifold with boundary \\ f : W → Bπ and g ∈ Riem^{+}(∂W) \end{matrix} \}/ bordism,

Unfortunately the groups R_{n}(Bπ) (n ≥ 6) are not known for any group π. However, we can define an index type map: θ : R_{n}(Bπ) → KO_{n}(ℝπ) where ℝπ is the real group algebra of π. (This map can be defined for infinite groups, too. Then the group algebra ℝπ should be replaced by the reduced C^{∗}-algebra of π.) The groups

KO_{n}(ℝπ) = {ℤ_{2}-graded ℝπ ⊗ Cl_{n}-modules}/ ∼,

where Cl_{n} denotes the n-th real Clifford algebra, are very easy to calculate and all the previous statements about different positive scalar curvature metrics on a manifold can be translated to statements about the image of θ. We will define a certain ℝπ ⊗ Cl_{n}-equivariant Dirac operator D(W, f, g) for any triple (W, f, g) representing an element in R_{n}(Bπ). The equivariance implies that Ker(D(W)) has an ℝπ ⊗ Cl_{n}-module structure. So θ(W, f, g): = [Ker(D(W))] defines an element in KO_{n}(ℝπ). By a theorem of U. Bunke θ(W, f, g) depends only on the bordism class of (W, f, g) in R_{n}(Bπ).

Closed manifolds represent elements in KO_{n}(Bπ) (see 2.29) and the restriction of θ to these elements defines a map A : KO_{n}(Bπ) → KO_{n}(ℝπ). The map A is called the assembly map. This map is induced by a map of spectra A : KO∧Bπ_{+} → KO(ℝπ) where KO(ℝπ) is the KO-spectrum of ℝπ. This enables us to define the assembly map with any coefficients.

KO_{n}(ℝπ) – the range of θ – is 8-periodic. It has the consequence that θ can only distinguish stable concordance classes of positive scalar curvature metrics: Two positive scalar curvature metrics g_{1}, g_{2} on M are stably concordant if for sufficiently many copies of B the induced positive scalar curvature metrics on M × B × ⋯ × B are concordant. B denotes the Bott manifold, a certain 8-dimensional spin manifold which represents the Bott periodicity in KO-theory. So we can define a stable version of θ:

θ_{st} : R_{n}(Bπ)[B^{−1}] → KO_{n}(ℝπ),

where R_{n}(Bπ)[B^{−1}] denotes R_{n}(Bπ) with [B] ∈ R_{8}(pt) inverted. A recent example of T. Schick (see Remark 2.17) shows that stable concordance is different from concordance, i.e. there is a group and an n such that the map R_{n}(Bπ) → R_{n}(Bπ)[B^{−1}] is not injective. The most general conjecture was formulated by Stephan Stolz:

Conjecture 1.1 θ_{st} is an isomorphism for any finitely presented group π.

Proving the injectivity of θ_{st} is unlikely with the existing methods. For torsion free groups the Baum-Connes conjecture implies the surjectivity of θ_{st}. In this paper we prove the (p-local) surjectivity for finite p-groups:

Theorem 3.14 For every finite p-group  θ_{st} : R_{n}(Bπ)[B^{−1}] → KO_{n}(ℝπ)  is (p-locally)

surjective.

(In other words the cokernel of θ_{st} is a finite group with order coprime to p.)

To prove the Theorem we define a map η on bordism classes of spin manifolds admitting positive scalar curvature metric with finite fundamental group π = π_{1}(M), which can be considered as the collection of all the η_{α}’s where α runs through the irreducible representations of π. These η_{α}’s are the spectral invariants defined by Atiyah, Patodi and Singer [APS75a]. The target of η is KO_{n}(ℝ̃π) ⊗ ℤ[\frac{1}{p}], where ℝ̃π is the augmentation ideal of the real group algebra of π. The key step in the proof is to find a close relationship between η and A(ℤ_{p^{∞}}) – the assembly map with ℤ_{p^{∞}}-coefficients. This relation is a real version of the Atiyah-Patodi-Singer index theorem for flat bundles.

We need the finite p-group assumption since a major step in the proof is that A(ℤ_{p^{∞}}) is an isomorphism for finite p-groups and the analogous statement is false for non-p-groups. However there are some promising results in this direction. Joint work in progress with Stephan Stolz indicates that θ_{st} is surjective for finite groups which are products of an odd order group and a 2-group (in particular for nilpotent groups).

This paper is based on my PhD. thesis at the University of Notre Dame, Indiana. The help of my advisor, Stephan Stolz, was essential at every stage of this work. I also owe a lot to the faculty and fellow graduate students of the Mathematics Department for stimulating discussions. Part of this research was done while enjoying the hospitality of the Max Plack Institute in Bonn.

1.1 Outline

In Chapter 2 we introduce the assembly map A and review the history of the Existence or Gromov-Lawson-Rosenberg Conjecture. Then we describe the Stable Existence Theorem which says that a spin manifold M with finite fundamental group admits stably a positive scalar curvature metric if and only if A(M) = 0.

Also we describe the various KO-groups involved, and describe a spectrum map A which induces the assembly map. That allows us to define the assembly map with coefficients.

In Chapter 3 we introduce the notion of concordance. We define the R_{n}-groups and the map θ : R_{n}(Bπ) → KO_{n}(ℝπ). In 3.3 we define a diagram (Diagram 1) involving θ, the analytically defined map η and a topologically defined map \overline{η}. Diagram chasing shows that the commutativity of Diagram 1 together with the surjectivity of η, \overline{η} and a third map f implies the surjectivity of θ. In 3.3.1 we prove most of the commutativity of Diagram 1. In 3.3.2 we show the surjectivity of η and \overline{η}.

In Chapter 4 we explain the connection between the classical eta invariants and η. This result is needed to prove the the surjectivity of f.

In Chapter 5 – using results from Chapter 4 – we complete the proof of the commutativity of Diagram 1, in other words we compare η and \overline{η}. This proof is based on the Atiyah-Patodi-Singer index theorem for flat bundles [APS76, (5.3)] which describes the classical eta invariants in topological terms.

2 The assembly map and existence of positive scalar curvature metrics

The results of this section can be found in several place in the literature, e. g. in [RS95].

2.1 The equivariant Dirac operator D(W)

The history of this area starts with the result of Lichnerowicz [Lic63] who showed that if a spin manifold W admits a positive scalar curvature metric then the index of a certain Dirac operator vanishes. For certain manifolds with infinite fundamental group [SY79] and recently for four dimensional manifolds – the Seiberg-Witten theory – different obstructions were developed later. But for spin manifolds W with π_{1}(W) finite and dim(W) > 4 still all the known obstructions are indices of certain Dirac operators. There is a way to combine all these Dirac operators into a single one:

Let Cl_{n} denote the n-th real Clifford algebra:

Cl_{n}: = ⟨e_{1}, ⋯, e_{n} : e_{i}e_{j} = −e_{j}e_{i} for i ≠ j, e_{i}^{2} = −1⟩.

Cl_{n} comes with a ℤ_{2}-grading: Cl_{n} = Cl_{n}^{0} ⊕ Cl_{n}^{1} where Cl_{n}^{0} = ⟨e_{i}e_{j} : 1 ≤ i, j ≤ n⟩.

There is an invariant way to define the Clifford algebras [ML89, I.§1]:

Definition 2.1 For any vector space V equipped with a quadratic form q

Cl(V, q) = T (V)/I_{q}(V)

where T (V) = ∑_{i=0}^{∞} ⊗^{i}V is the tensor algebra of V and the ideal I_{q}(V) is generated by the elements of the form

v ⊗ v + q(v)1.

In this terminology Cl_{n} = Cl(ℝ^{n}, q) where q(v) = ⟨v, v⟩ is the standard positive definite quadratic form on ℝ^{n}.

The group Spin(n) – generated by products of even number of unit vectors in ℝ^{n} ⊂ Cl_{n} – is a subgroup of Cl_{n}^{×} where Cl_{n}^{×} is the group of invertible elements in Cl_{n} [ML89, p.14]. Let W be an n-dimensional Riemannian spin manifold with metric G. The group Spin(n) is a double cover of SO(n) and a spin structure of the oriented manifold W defines a principal Spin(n)-bundle Spin(W) which is a double cover of SO(W), the oriented frame bundle of W. The group Spin(n) acts on Cl_{n} by left multiplication so we can define an associated bundle:

Definition 2.2 For any spin manifold W let Cl_{Spin}(W) = Spin(W) ×_{Spin(n)} Cl_{n}.

As a rule of thumb, we choose the notation to be consistent with [ML89] since this is the best general reference on the subject.

Suppose that π = π_{1}(W). Then W̃ – the universal cover of W – is a principal π-bundle over W. Suppose that π is finite and let ℝπ denote the real group-algebra of π, i. e. ℝπ = {f : π → ℝ}. Multiplication is defined by “convolution”: f_{1} · f_{2}(h): = ∑_{g∈π} f_{1}(g)f_{2}(g^{−1}h). We have π ⊂ ℝπ^{×} if we identify g ∈ π with ǧ : π → ℝ, ǧ(h) = δ_{h}^{g}. This way we can define an action of π on ℝπ by left multiplication.

Definition 2.3 For a manifold W with π = π_{1}(W) finite, let ℝπ(W) = W̃ ×_{π} ℝπ

Using that right multiplication commutes with left multiplication, Cl_{Spin}(W) is a right Cl_{n}- module bundle and ℝπ(W) is a right ℝπ-module bundle.

We will frequently use ℤ_{2}-graded objects:

Definition 2.4 A ℤ_{2}-graded vector space is an ordered pair (V^{0}, V^{1}) of vector spaces. Let V = V^{0} ⊕ V^{1}. The operator T : V → V defined by the property that the 1-eigenspace of T is V^{0} and the -1-eigenspace of T is V^{1}. T is called the grading operator.

Remark 2.5 Equivalently, we could define a ℤ_{2}-graded vector space as a pair (V, T), where V is a vector space and T : V → V is a linear operator with T^{2} = 1.

Cl_{Spin}(W) admits a connection ∇^{W} : Γ(Cl_{Spin}(W) ⊗ T W) → Γ(Cl_{Spin}(W)) (or ∇ for short) coming from the Levi-Civita connection ∇^{LC} (think ∇^{LC} as a connection on SO(W) – the orthogonal frame bundle of W. It has a unique lift to the double cover Spin(W)). The pair (Cl_{Spin}(W), ∇) is a so called Dirac bundle (see [ML89, p.114]), so we can define a Dirac operator on it ([ML89, §5]):

Let Cl(T W) denote the ”Cliffordization” of the Euclidean vector bundle T W:

Cl_{x}(T W): = Cl(T_{x}W, q)

where q(v) = ⟨v, v⟩ and ⟨·, ·⟩ denotes the scalar product defined by the Riemannian metric of W. Cl(T W) is an algebra bundle, the fibers are equipped with an algebra structure.

Definition 2.6 [ML89, II.5.2] A Dirac bundle over a Riemannian manifold W is a bundle S of left modules over Cl(T W) together with a metric ⟨·, ·⟩ and a metric connection ∇^{S} having the following properties:

1. ⟨eσ_{1}, eσ_{2}⟩ = ⟨σ_{1}, σ_{2}⟩ for all σ_{1}, σ_{2} ∈ S_{x} and for all e ∈ T_{x}W with ||e|| = 1, 2. ∇_{X}^{S}(φσ) = ∇_{X}(φ)σ + φ(∇_{X}^{S}σ) for all smooth sections X ∈ Γ(T W), φ ∈ Γ(Cl(T W)) and

σ ∈ Γ(S).

These properties allow us to define a Dirac operator D(S) : Γ(S) → Γ(S) by the local formula:

Definition 2.7 (D(S)σ) (x): = (∑_{i}^{n} e_{i} · ∇_{e_{i}}) (x)σ

where {e_{i}} is an orthonormal basis of T_{x}W ⊂ Cl_{x}(T W) and · denotes the Clifford multiplication.

For example every vector bundle associated to Spin(W) admits a natural Dirac bundle structure ([ML89, II.5.9]). Also, if S is a Dirac bundle and E is a vector bundle with a metric and a compatible connection then S ⊗ E admits a natural Dirac bundle structure.

ℝπ(W) – having a discrete structure group – admits a flat connection ∇^{π}. Therefore

(Cl_{Spin}(W) ⊗ ℝπ(W), ∇ ⊗ ∇^{π})

is a Dirac bundle, too. Here we follow the notation of [ML89]:

(∇ ⊗ ∇′)_{X}(s ⊗ s′): = ∇_{X}s ⊗ 1 + 1 ⊗ ∇′_{X}s′

for any X ∈ Γ(T W).

Definition 2.8 D(W): = D(Cl_{Spin}(W) ⊗ ℝπ(W), ∇ ⊗ ∇^{π})

This operator is our main interest. D(W) is a first order elliptic differential operator, it acts on the smooth sections of Cl_{Spin}(W) ⊗ ℝπ(W). By standard methods we can extend it to the Hilbert space of L^{2}-sections. This is an unbounded self-adjoint operator. D(W) has degree 1: It interchanges the 0- and the 1-part of the space of L^{2}-sections [ML89]. Degree 1 operators are called odd operators as well.

Proposition 2.9 [ML89, p.115] For a compact Riemannian spin manifold W the vector space Ker(D(W)) is finite dimensional.

Since D(W) commutes with the Cl_{n} ⊗ ℝπ-action, Ker(D(W)) is a ℤ_{2}-graded Cl_{n} ⊗ ℝπ-module.

Let’s compare D(W) with the classical Dirac operator: if n ≡ 0 (mod 2) then ℂl_{n} = Cl_{n} ⊗ ℂ, the complex Clifford algebra has two irreducible ℤ_{2}-graded modules:

Definition 2.10 Let 𝕊 = 𝕊_{n} be the unique irreducible ℤ_{2}-graded left ℂl_{n}-module for which the action of ωn^{ℂ} ∈ ℂl_{n} coincides with the grading operator T, where ωn^{ℂ} is the complex volume form

(see [ML89, p. 34]): ωn^{ℂ} = i^{[ \frac{n+1}{2} ]}e_{1} ⋯ e_{n}.

Then Spin(W) ×_{Spin(n)} 𝕊 is the so-called spinor bundle. The corresponding Dirac operator D_{AS} is the Atiyah-Singer Dirac operator. Traditionally the zero-part of the spinor bundle was called positive half spinors and they considered D_{AS}^{0} – the restriction of D_{AS} to the positive half spinors. (Similarly D_{AS}^{1} denotes the restriction to the one-part.) So we have:

Ker(D_{AS}) = (Ker(D(W)) ⊗_{ℂl_{n}} 𝕊)^{π}

where the superscript π means the fixpoint set of π, and

Index_{ℂ}D_{AS}^{0} = dim_{ℂ} Ker(D_{AS}^{0}) − dim_{ℂ} Coker(D_{AS}^{0}) = dim_{ℂ} ((KerD_{AS})^{0}) − dim_{ℂ} ((KerD_{AS})^{1}).

2.2 Definition of the assembly map and existence of positive scalar curvature metrics

Though we suppressed it in the notation, Ker(D(W)) depends on the metric G. But just slightly:

Definition 2.11 Let C be a ℤ_{2}-graded ℝ-algebra:

𝔐(C): = Gr{finitely generated projective ℤ_{2}-graded right C-modules},

where Gr stands for Grothendieck group. Let i^{∗} : 𝔐(C ⊗ Cl_{n+1}) → 𝔐(C ⊗ Cl_{n}) be the map which forgets the e_{n+1}-action and

KO_{n}(C): = 𝔐(C ⊗ Cl_{n})/i^{∗}𝔐(C ⊗ Cl_{n+1}).

Remark 2.12 For finite dimensional semisimple algebras the finitely generated projective ℤ_{2}- graded right C-modules are exactly the finite dimensional ones.

Definition 2.13 For a compact spin manifold W with π = π_{1}(W) finite let

A^{π}(W) = [Ker(D(W))] ∈ 𝔐(ℝπ ⊗ Cl_{n})/i^{∗}𝔐(ℝπ ⊗ Cl_{n+1}) = KO_{n}(ℝπ).

If π is clear from the context we suppress it from the notation. For simply connected manifolds we get an invariant in KO_{n}(ℝ) ≅ KO^{−n}(pt). This map is the “improved” version of the Â-genus defined by Hitchin ([Hit74] see also in [ML89, p.149]).

We recall now an appropriate form of the Lichnerowicz theorem:

Theorem 2.14 ([Ros86]) If G is a positive scalar curvature metric on the spin manifold W then Ker(D(W)) = 0.

As we will see (Theorem 2.30), A^{π}(W) is independent of the metric. It implies the following:

Corollary 2.15 If [Ker(D(W))] ∈ KO_{n}(ℝπ) is non-trivial then W doesn’t admit a positive scalar curvature metric.

A conjecture of Gromov and Lawson – later modified by Rosenberg – states that this is the only obstruction:

Conjecture 2.16 (Existence Conjecture [RS95]) Suppose W^{n} is a spin manifold, π = π_{1}(W) and n ≥ 5. Then W admits a positive scalar curvature metric if and only if A^{π}(W) = 0.

Remark 2.17 We defined A^{π}(W) only for finite π, but it can be defined for the general case. If |π| = ∞ then A^{π}(W) ∈ KO_{n}(C_{r}^{∗}π) where C_{r}^{∗}π is the real reduced group C^{∗}-algebra of π (see [Ros91, §2]). Recently Thomas Schick found a counter example to the conjecture: a 5-dimensional spin manifold with fundamental group ℤ^{4} × ℤ_{3}. The proof is based on the hypersurface method of Schoen and Yau [SY79].

The conjecture is known to be true only in a few cases. Even for the case π trivial the proof is quite involved [Sto92]. But a weaker version of the conjecture was proved for a wide class of groups, including all finite groups. It is easy to see that A is multiplicative in the following sense:

Proposition 2.18 A(M × N) = A(M) ∘ A(N) where

∘ : KO_{n}(ℝπ_{1}) ⊗ KO_{m}(ℝπ_{2}) → KO_{n+m}(ℝ(π_{1} × π_{2}))

is induced by the ℤ_{2}-graded tensor product of modules.

Proof [ML89, p.150]. We implicitly assumed that the spin structure on M × N is the product of the spin structures on M and N. It implies that

Cl_{Spin}(M × N) ⊗ ℝ(π_{1} × π_{2})(M × N) ≅ pr_{M}^{∗}(Cl_{Spin}(M) ⊗ ℝπ_{1}(M)) ⊗ pr_{N}^{∗}(Cl_{Spin}(N) ⊗ ℝπ_{2}(N))

where pr_{M} and pr_{N} denotes the projections. This implies that D(M × N, g_{M} × g_{N}) = D(M, g_{M}) ⊗ 1 + T ⊗ D(N, g_{N}) where T is the grading operator and D^{2}(M × N, g_{M} × g_{N}) = D^{2}(M, g_{M}) ⊗ 1 + 1 ⊗ D^{2}(N, g_{N}). The operator D^{2} is a non-negative with Ker(D^{2}) = Ker(D), therefore

KerD(M, g_{M}) ⊗ KerD(N, g_{N}) ≅ KerD(M × N, g_{M} × g_{N}).

If the modules are ℤ_{2}-graded, then ⊗ always denotes the ℤ_{2}-graded tensor product. We are interested in the KO-theory of the group algebra ℝπ. This is an example of a C^{∗}-algebra:

Definition 2.19 [Goo82, §8] A real ∗-algebra is a real algebra C equipped with a linear involution ∗ such that (xy)^{∗} = y^{∗}x^{∗}. A real Banach ∗-algebra is a real ∗-algebra equipped with a norm making it a real Banach space, such that ||xy|| ≤ ||x|| · ||y|| for all x, y in C. (If C is unital we also require that ||1|| = 1.)

Definition 2.20 [Sch93, p.1] A real C^{∗}-algebra C is a Banach ∗-algebra over the reals which is *-isometrically isomorphic to a norm closed subalgebra of the algebra of continuous linear operators on a real Hilbert space.

Equivalently ([Goo82, §8]) a real C^{∗}-algebra C is a real Banach ∗-algebra such that ||x^{∗}x|| = ||x||^{2} and 1 + x^{∗}x is invertible for every x ∈ C.

For finite dimensional algebras this condition is equivalent to semisimplicity [Sch93, p. 1], what is clear for ℝπ. For completeness we mention that f^{∗}(g): = f(g^{−1}) defines a ∗-operation on ℝπ. An other example for a real C^{∗}-algebra is C(X) – the space of continuous functions on a compact space X – with the trivial ∗-operation.

Proposition 2.21 For any real C^{∗}-algebra C we have KO_{n}(C) ≅ KO_{n+8}(C) where the isomorphism is induced by ⊗S for the irreducible ℤ_{2}-graded Cl_{8}-module S, which represents a generator of KO_{8}(ℝ) ≅ ℤ [Sch93].

We denote by S the unique irreducible ℤ_{2}-graded Cl_{8}-module which has the property that S⊗ℂ is isomorphic to 𝕊_{8} as a ℂl_{8}-module.

Proposition 2.22 [ML89, p.151] There is an 8-dimensional simply connected spin manifold B such that A(B) = [S].

B is usually called a Bott manifold since it represents the Bott periodicity. Such a manifold is not unique, but the choice doesn’t effect the following. Proposition 2.18 implies that A(W) = 0 if and only if A(W × B) = 0.

Definition 2.23 W admits stably a positive scalar curvature metric if there is a k such that W × \underbrace{B × ⋯ × B}_{k times} has positive scalar curvature metric.

Corollary 2.24 If W admits stably a positive scalar curvature metric then A(W) = 0.

Theorem 2.25 (Stable Existence Theorem – Rosenberg, Stolz [RS95]) If W is a spin manifold and π_{1}(W) is finite then A(W) = 0 if and only if W admits stably a positive scalar curvature metric.

2.3 Homotopy theoretic description of the assembly map

The assembly map A has strong invariance properties. To see this we first extend the domain of A:

Definition 2.26 Let W be a spin manifold, let f : W → Bπ be a continuous map and let ℝπ(Bπ): = Eπ ×_{π} ℝπ be the universal ℝπ-bundle over Bπ. Then

A(W, f): = [KerD(W, f)] ∈ KO_{n}(ℝπ)

where D(W, f): = D(Cl_{Spin}(W) ⊗ ℝπ(W, f)) and ℝπ(W, f): = f^{∗}ℝπ(Bπ).

Definition 2.27 Ω_{n}^{Spin}(X): = {(W, f) : W^{n} spin, f : W → X}/ ∼

where (W_{1}, f_{1}) ∼ (W_{2}, f_{2}) if there are spin manifolds with boundary U_{1}, U_{2} and maps f_{i} : U_{i} → X such that ∂U_{1} ∐ W_{1} ≅ ∂U_{2} ∐ W_{2}. Here ≅ means a diffeomorphism compatible with the spin structure and the maps to X. We assume that the spin structure on ∂U_{i} is induced by the spin structure of U_{i}.

Ω_{n}^{Spin}(X) is a group with addition induced by the operation ∐ ([Swi75, p.226]).

Theorem 2.28 A(W, f) depends only on [W, f] ∈ Ω_{n}^{Spin}(Bπ).

We will prove a much stronger statement. The KO-orientation u : MSpin → KO induces a map U : Ω_{n}^{Spin}(Bπ) → KO_{n}(Bπ). There is a geometric way to describe this map [KS93, Thm C.]:

Theorem 2.29 There is a canonical isomorphism U:

KO_{n}(X) ≅ (⨁_{k} Ω_{n+8k}^{Spin}(X)) / ∼,

where the equivalence is generated by

1. [E, f ∘ p] ∼ 0 if p : E → M is an ℍℙ^{2}-bundle over a spin manifold M and f is a map from M to X.

2. [M, f] ∼ [M × B, f ∘ pr_{1}], where B is the Bott manifold and pr_{1} denotes the projection to the first factor.

Theorem 2.30 A(W, f) depends only on U[W, f] ∈ KO_{n}(Bπ).

Theorem 2.30 is a consequence of the homotopy theoretic description of A in Theorem 2.37.

This is an important strengthening of Theorem 2.28 since it is much easier to describe KO_{n}(Bπ) (see Proposition 2.50) than the group Ω_{n}^{Spin}(Bπ). The proof of this theorem is based on a version of the Atiyah-Singer index theorem which gives a topological formula for A(W, f) in terms of [W, f] ∈ KO_{n}(Bπ). This formula plays a fundamental role in this work so we present it in details.

Definition 2.31 We will call A : KO_{n}(Bπ) → KO_{n}(ℝπ) the Assembly map.

At this point the reader might be interested in what these groups look like:

Proposition 2.32 [AS69] Let Irr(ℝπ) denote the set of irreducible ℝπ-modules in other words the real representations of π. Then

KO_{n}(ℝπ) ≅ ∑_{α∈Irr(ℝπ)} KO_{n}(F (α))

where F (α) = Hom_{ℝπ}(α, α) is the commuting field of α – can be ℝ, ℂ or ℍ ([AS69]) and

0 1 2 3 4 5 6 7
KO_{n}(ℝ) ℤ ℤ_{2} ℤ_{2} 0 ℤ 0 0 0
KO_{n}(ℂ) ℤ 0 ℤ 0 ℤ 0 ℤ 0
KO_{n}(ℍ) ℤ 0 0 0 ℤ ℤ_{2} ℤ_{2} 0

Remark 2.33 If π is a group of odd order then F (α) = ℂ for every non-trivial irreducible ℝπmodule α [FH91, p.34].

We want to show that A is induced by a map A between spectra. KO_{n}(Bπ) is the n-th homotopy group of the spectrum KO ∧ Bπ_{+}. But it is less clear that KO_{n}(ℝπ) is the n-th homotopy group of a spectrum as well.

Theorem 2.34 [Ros95, theorem 2.1] For every real (ungraded) C^{∗}-algebra C there is a spectrum KO(C) such that:

1. π_{n}(KO(C)) = KO_{n}(C)

2. (KO(C))_{0} = KO_{0}(C) × BGL(C). GL(C) is the limit of GL_{n}(C) = Aut_{C}(C^{n}) where C^{n} considered as a C-module.

3. KO(C) is 8-periodic i.e. we can assume that KO(C) is an Ω-spectrum and (KO(C))_{8} homotopy equivalent to (KO(C))_{0}.

To describe A, we need one more property of KO(C):

Theorem 2.35 [Ros95, p. 353] KO(C) is naturally a module spectrum over KO(ℝ) via tensor product.

Definition 2.36  A : KO ∧ Bπ_{+} → KO(ℝπ),  A: = μ ∘ (Id_{KO} ∧ Bi_{+})

where i : π → GL_{1}(ℝπ) ⊂ GL(ℝπ), so Bi : Bπ → BGL(ℝπ) ⊂ (KO(ℝπ))_{0} and Bi_{+} is the extension of Bi sending + to the basepoint of BGL(ℝπ), and μ : KO ∧ KO(ℝπ) → KO(ℝπ) is the module spectrum map.

Theorem 2.37 [Ros91, p. 175] A induces A on the homotopy groups.

This theorem shows that A is indeed an assembly map in the sense of [WW95]. We show now how to express Theorem 2.37 in terms of the cohomological slant product:

Bi : Bπ → BGL_{1}(ℝπ) is the classifying map of the ℝπ-module bundle

Bi^{∗}(EGL_{1}(ℝπ) ×_{GL_{1}(ℝπ)} ℝπ).

But pulling back with Bi – where i is a homomorphism of groups – corresponds to the reduction of the structure group (see [Swi75, p. 202]) so we have

Bi^{∗}(EGL_{1}(ℝπ) ×_{GL_{1}(ℝπ)} ℝπ) ≅ Eπ ×_{π} ℝπ = ℝπ(Bπ)

For non-compact spaces vector bundles don’t always define an element in the corresponding Ktheory but in our case ℝπ(Bπ) admits a classifying map – this is how we constructed it – which defines an element [Bi_{+}] in KO(ℝπ)^{0}(Bπ) = [Bπ_{+}, KO(ℝπ)]. So let us use the notation [ℝπ(Bπ)] as well for this element.

Let E be a ring spectrum and F a module spectrum over E with module structure μ : E∧F → F:

Definition 2.38 [Swi75, p. 270-271] The (cohomological) slant product is a map

/ : F^{p}(Y) ⊗ E_{q}(Y) → F^{p−q}(pt).

Suppose f : Y_{+} → Σ^{p}(F) represents an element [f] ∈ F^{p}(Y) and g : S^{q} → E ∧ Y_{+} represent an element [g] ∈ E_{q}(Y). Then:

[f]/[g]: = [h]

where

\xymatrix{pt_{+} = S^{0} \ar[r]^{Σ^{−q}g} \ar[rrr]_{h} & Σ^{−q}E ∧ Y_{+} \ar[r]^{1∧f} & Σ^{−q}E ∧ Σ^{p}F \ar[r]^{Σ^{p−q}μ} & Σ^{p−q}F}

From the definition we can see that:

Corollary 2.39 For a spin Bπ-manifold (W, f)

A_{∗}[W, f] = [ℝπ(Bπ)]/[W, f]

where [ℝπ(Bπ)] ∈ KO(ℝπ)^{0}(Bπ) and [W, f] ∈ KO_{n}(Bπ).

2.4 The assembly map with ℤ_{p^{∞}} coefficients

Since A is induced by a map of spectra, we can define the assembly map with any coefficients. In this section we explain how to do this and we describe the map A(ℤ_{p^{∞}}). The fact that this map is an isomorphism (Theorem 2.45) plays a crucial role in the proof of the surjectivity of θ. Also, as a corollary of Theorem 2.45, we get a description of the image of the assembly map A.

Let us recall now the notion of generalized homology theories with coefficients:

Proposition 2.40 [Ada70, p. 200] For every Abelian group G there is a spectrum MG – called the Moore spectrum of G – with the following properties:

1. π_{n}(MG) = 0 if n < 0

2. π_{0}(MG) ≅ G ≅ H_{0}(MG)

3. H_{n}(MG) = 0 if n > 0.

It is easy to see that such a spectrum is unique up to homotopy. Let E be a spectrum and G be an Abelian group:

Definition 2.41 E_{n}(X; G): = (E ∧ MG)_{n}(X) = π_{n}(X_{+} ∧ E ∧ MG)

where MG is the Moore spectrum of G.

For a map of spectra f : E → F and a group G the maps f ∧ Id_{MG} : E ∧ MG → F ∧ MG induce maps f(G) : E_{n}(pt; G) → F_{n}(pt; G). In particular we can define A(ℤ_{p^{∞}}) : KO_{n}(Bπ; ℤ_{p^{∞}}) → KO_{n}(ℝπ; ℤ_{p^{∞}}). First we describe the range of A(ℤ_{p^{∞}}).

For the spectrum KO(C) we want to use the following conventions:

Definition 2.42

\begin{matrix}KO_{n}^{C}(X): = KO(C)_{n}(X) = π_{n}(X_{+} ∧ KO(C)), & KO_{n}^{C}(X; G): = (KO(C) ∧ MG)_{n}(X), \\ KO_{C}^{n}(X): = KO(C)^{n}(X) = [X_{+}, Σ^{n}KO(C)], & KO_{C}^{n}(X; G): = (KO(C) ∧ MG)^{n}(X). \end{matrix}

For X = pt we also want to use: KO_{n}(C): = KO_{n}^{C}(pt) and KO_{n}(C; G): = KO_{n}^{C}(pt; G).

We are mostly interested in the algebra ℝπ. ℝπ can be written as a direct sum of algebras: ℝπ = ℝ̃π ⊕ ℝ where ℝ is generated by the constant 1 function and ℝ̃π is the augmentation ideal i.e. the kernel of the augmentation map ε : ℝπ → ℝ, ε(f) = ∑_{g∈π} f(g). The algebra ℝ̃π plays a fundamental role in our considerations. As it can be easily checked, the K-theory of ℝ̃π is very similar to the K-theory of ℝπ:

KO_{n}(ℝ̃π) ≅ ∑_{α∈_{Irr}(ℝπ)} KO_{n}(F (α))

where Ĩrr(ℝπ) denotes the set of non-trivial irreducible ℝπ-modules. In other words KO_{n}(ℝπ) = KO_{n}(ℝ̃π) ⊕ KO_{n}(ℝ), so we can think of ℝ̃π as a “reduced” version of ℝπ.

Proposition 2.43 For any spectrum E a short exact sequence of Abelian groups

0 ⟶ A ⟶^{α} B ⟶^{β} C ⟶ 0

induces a cofibration MA ⟶ MB ⟶ MC which induces a long exact sequence:

⟶^{δ} E_{n}(X; A) ⟶^{α^{∗}} E_{n}(X; B) ⟶^{β^{∗}} E_{n}(X; C) ⟶^{δ} E_{n−1}(X; A) ⟶

e.g. for 0 ⟶ ℤ ⟶^{α} ℤ[1/p] ⟶^{β} ℤ_{p^{∞}} ⟶ 0 we have:

Coker(α^{∗}) = E_{n}(X) ⊗ ℤ_{p^{∞}} and Ker(α^{∗}) = Tor_{p}E_{n−1}(X).

The proof of this proposition is essentially the same as of the exactness of the universal coefficient sequence in [Ada70, p. 200].

Corollary 2.44 For any prime p there is a short exact sequence:

0 ⟶ KO_{n}(ℝ̃π) ⊗ ℤ_{p^{∞}} ⟶^{i} KO_{n}(ℝ̃π; ℤ_{p^{∞}}) ⟶ Tor_{p}KO_{n−1}(ℝ̃π) ⟶ 0

where Tor_{p} denotes the p-torsion.

The torsion term is of course 0 if p ≠ 2. With the help of this corollary we can calculate KO_{n}(ℝ̃π; ℤ_{p^{∞}}): Since we have

KO_{n}(ℝ̃π; ℤ_{p^{∞}}) ≅ ∑_{α∈_{Irr}(ℝπ)} KO_{n}(F (α); ℤ_{p^{∞}}),

it is enough to look at the following table:

\begin{aligned}& 0 & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ KO_{n}(ℝ; ℤ_{p^{∞}}) & ℤ_{p^{∞}} & 0 & (ℤ_{2}) & (ℤ_{2}) & ℤ_{p^{∞}} & 0 & 0 & 0 \\ KO_{n}(ℂ; ℤ_{p^{∞}}) & ℤ_{p^{∞}} & 0 & ℤ_{p^{∞}} & 0 & ℤ_{p^{∞}} & 0 & ℤ_{p^{∞}} & 0 \\ KO_{n}(ℍ; ℤ_{p^{∞}}) & ℤ_{p^{∞}} & 0 & 0 & 0 & ℤ_{p^{∞}} & 0 & (ℤ_{2}) & (ℤ_{2}) \end{aligned} (ℤ_{2}) = \{ \begin{matrix}ℤ_{2} & if p = 2 \\ 0 & otherwise \end{matrix}

Theorem 2.45 (J. Rosenberg, S. Stolz [RS]) For every finite p-group π the map

A(ℤ_{p^{∞}}) : K̃O_{n}(Bπ; ℤ_{p^{∞}}) → KO_{n}(ℝ̃π; ℤ_{p^{∞}})

is an isomorphism.

Remark 2.46 This theorem obviously implies that A(ℤ_{p^{∞}})(K̃O_{n}(Bπ; ℤ_{p^{∞}})) ⊂ KO_{n}(ℝ̃π; ℤ_{p^{∞}}),

and it is easy to see that A(K̃O_{n}(Bπ)) ⊂ KO_{n}(ℝ̃π) holds, too.

Proposition 2.47 ∂ : K̃O_{n}(Bπ; ℤ_{p^{∞}}) → K̃O_{n−1}(Bπ)

is an isomorphism where ∂ is the boundary homomorphism in the long exact sequence induced by the short exact coefficient sequence 0 ⟶ ℤ ⟶ ℤ[1/p] ⟶ ℤ_{p^{∞}} ⟶ 0.

This is a simple consequence of the following fact :

Proposition 2.48 K̃O_{n−1}(Bπ) is a p-torsion group.

Proof. Bπ = K(π, 1) so all homotopy groups of Bπ are p-torsion. By the mod-C Hurewitz theorem [Spa66, p. 512] this implies that all the reduced homology groups of Bπ are p-torsion. The Atiyah-Hirzebruch spectral sequence E_{p,q}^{r} ⇒ K̃O_{n−1}(Bπ) in this case has E^{2}-term:

E_{p,q}^{2} ≅ H̃_{p}(Bπ; KO_{q}(pt)).

By the universal coefficient theorem H̃_{p}(Bπ; KO_{q}(pt)) is also p-torsion, so K̃O_{n−1}(Bπ) is a p-torsion group.

Remark 2.49 The same argument shows that Ω̃_{n}^{Spin}(Bπ) is p-torsion. But Ω_{n}^{Spin} is zero for n < 0, which implies a stronger result: Ω̃_{n}^{Spin}(Bπ) is a finite p-group.

Corollary 2.50 K̃O_{n−1}(Bπ) ≅ KO_{n}(ℝ̃π; ℤ_{p^{∞}}), so it is a sum of ℤ_{p^{∞}}-s and perhaps some ℤ_{2}-s.

Corollary 2.51 ([RS95]) For A : K̃O_{n−1}(Bπ) → KO_{n−1}(ℝ̃π) we have:

(i) Ker(A) = ∂ ∘ (A(ℤ_{p^{∞}}))^{−1}i(KO_{n}(ℝ̃π) ⊗ ℤ_{p^{∞}})

(ii) Im(A) = Tor_{p}KO_{n−1}(ℝ̃π).

Proof. Standard diagram chasing in the diagram:

\xymatrix{0 \ar[r] \ar[d] & K̃O_{n}(Bπ; ℤ_{p^{∞}}) \ar[r]^{∂}_{≅} \ar[d]^{A(ℤ_{p^{∞}})}_{≅} & K̃O_{n−1}(Bπ) \ar[r] \ar[d]^{A} & 0 \ar[d] \\ KO_{n}(ℝ̃π) ⊗ ℤ_{p^{∞}} \ar[r]^{i} & KO_{n}(ℝ̃π; ℤ_{p^{∞}}) \ar[r] & Tor_{p}KO_{n−1}(ℝ̃π) \ar[r] & 0}

3 Concordance classes of positive scalar curvature metrics

3.1 Dirac operators on manifolds with boundary

The amazing fact is that the Dirac operator method will give results for the classification problem if we apply it to manifolds with boundary. Suppose that W is an n-dimensional Riemannian spin manifold, ∂W = M and f : W → Bπ. Assume moreover that the metric G is product near the boundary, i.e. it has the form g + dt^{2} in a collar neighbourhood of the boundary, where g is a metric on M and dt^{2} is the standard metric on the line. The same way as for closed spin manifolds in the previous chapter, we can define Cl_{Spin}(W) and ℝπ(W) and D(W), but the kernel will be infinite dimensional. There are several ways to solve this problem. We want to use the so called L^{2}-index, which in our case – positive scalar curvature metric on the boundary – coincides with the the Atiyah-Patodi-Singer index. The θ-map – which is in our main interest – however was defined in terms of a third index [Bun95]. The fact that these indices coincide was proved in [Feh98]. In the meanwhile new developments in the spectral theory of operators on Hilbert-A-modules made it possible to use a new index definition for which both the well definedness of θ and the equality with the classical index are easier to prove (see [Sto]).

Suppose that W is a spin manifold with boundary ∂W = M, f : W → Bπ and G is a metric on W which has product metric near the boundary and g = G|_{M} is a positive scalar curvature metric. Attach an infinite cylinder M × [0, ∞) to it:

Ŵ: = W ∪_{M} M × [0, ∞).

We extend G to a metric Ĝ on Ŵ via the the product metric on M ×[0, ∞) so (Ŵ , Ĝ) is a complete Riemannian manifold. The map f extends to a map f̂: Ŵ → Bπ. The metric Ĝ defines a measure on Ŵ so it makes sense to speak about L^{2}-sections. By [APS75a] we can define D(Ŵ) acting on the Hilbert space L^{2}(Cl_{Spin}(Ŵ) ⊗ ℝπ(Ŵ)) and KerD(Ŵ) will be finite dimensional.

Definition 3.1 For a spin manifold W with boundary KerD(W): = KerD(Ŵ).

As we will see this index depends – and only depends – on the concordance class of g. This way it can distinguish different positive scalar curvature metrics. Let us explain this in a more systematic way:

3.2 Positive scalar curvature and the R_{n}-groups

The best introduction is [Sto]. Here we give a brief summary in the cases we are interested in.

Definition 3.2 Let W be an n-dimensional spin manifold with boundary and let g be a positive scalar curvature metric on ∂W:

Riem^{+}(W rel g): = \{G | \begin{matrix}G is a product metric near the boundary, \\ G|_{M} = g and has positive scalar curvature \end{matrix} \} .

If W is closed then Riem^{+}(W) denotes the space of positive scalar curvature metrics on W.

Definition 3.3 Let X be an n-dimensional spin manifold, let f be a continuous map to B and let g be a positive scalar curvature metric on X:

Pos_{n}(B): = {(X, f, g)|f : X → B, g ∈ Riem^{+}(X)}/ ∼

where (X, f, g) is bordant to (X_{1}, f_{1}, g_{1}) if there is a spin manifold W, a continuous map F : W → B and a metric G ∈ Riem^{+}(W rel g ⨿ g_{1}) such that W is a spin bordism between X and X_{1}, and F extends f and f_{1}.

There is a forgetful map f : Pos_{n}(B) → Ω_{n}^{Spin}(B). In such situations we always have a relative bordism group which fits into a long exact sequence:

… ⟶ R_{n+1}(B) ⟶ Pos_{n}(B) ⟶ Ω_{n}^{Spin}(B) ⟶ R_{n}(B) ⟶ …  (1)

Definition 3.4

R_{n}(B): = \{(W, f, g) | \begin{matrix}W is an n-dimensional spin manifold with boundary \\ f : W → B and g ∈ Riem^{+}(∂W) \end{matrix} \}/ ∼,

(W, f, g) ∼ (W_{1}, f_{1}, g_{1}) if there is an n+1-dimensional spin manifold U, a map F : U → B, an n-dimensional spin manifold V with boundary ∂W ⨿ −∂W_{1} and a metric G ∈ Riem^{+}(V rel g ⨿ g_{1}) such that

∂U = (W ∐ −W_{1}) ∪_{∂W ⨿−∂W_{1}} V

and F extends f and f_{1}. It is straightforward to check that R_{n}(B) is a group and fits into the long exact sequence (1). These groups turned out to be crucial in the existence and classification problem of positive scalar curvature metrics:

Theorem 3.5 (Existence [Haj91]) Given a connected n-dimensional spin manifold W, n≥5, and a metric g ∈ Riem^{+}(∂W), g extends to a positive scalar curvature metric on W (product near the boundary) if and only if

[(W, c, g)] = 0 ∈ R_{n}(Bπ)

where π_{1}(W) = π and c : W → Bπ is the classifying map of the universal cover of W.

Definition 3.6 Two metrics g, g_{1} on a closed manifold M are concordant if there is a metric G in Riem^{+}(M × [0, 1] rel g ⨿ g_{1}).

Remark 3.7 If two metrics are in the same (path)component of Riem^{+}(M) then they are concordant (as an application of the shrinking lemma shows, see Definition 3.11 below). Nothing is known in the other direction.

Theorem 3.8 (Classification [Sto]) If W is a connected n-dimensional spin manifold, n≥5, such that Riem^{+}(W rel g) is not empty, then R_{n+1}(Bπ) acts freely and transitively on the concordance classes of metrics in Riem^{+}(W rel g), where π = π_{1}(W).

Now we define the map θ : R_{n}(Bπ) → KO_{n}(ℝπ):

Definition 3.9 Let (W, f, g) be a triple representing an element [W, f, g] in R_{n}(Bπ):

θ([W, f, g]): = [KerD(Cl_{Spin}(W) ⊗ ℝπ(W, f))].

Theorem 3.10 [Bun95, 1.17] θ is a well defined homomorphism.

Similarly to the existence problem, it is easier to handle the stable case, when we don’t distinguish between M and M × B, where B is the Bott manifold: Definition 3.11

R_{n}(Bπ)[B^{−1}]: = lim_{→} R_{n+8k}(Bπ)

where the connecting maps of the inverse system are the following:

j_{k} : R_{n+8k}(Bπ) → R_{n+8(k+1)}(Bπ), j_{k}[W, f, g] = [W × B, f × ∗, g̃]

where g̃ is a positive scalar curvature metric on ∂W × B constructed the following way:

Take any metric g_{B} on B and start to rescale it. By [GL80, p. 430] there is a t such that tg_{B} ×g has positive scalar curvature. This trick is usually called the shrinking lemma. Proposition 3.12 The map θ factors through R_{n}(Bπ)[B^{−1}]. Proof. We have to show that θ(W) = θ(W ×B) under Bott periodicity. This is a slight generalization of Proposition 2.18. Choose a metric g_{B} on B such that g × g_{B} has positive scalar curvature. Then the argument of Proposition 2.18 shows that Ker(D(W × B)) ≅ Ker(D(W)) ⊗ Ker(D(B)). Conjecture 3.13 (Stephan Stolz) θ : R_{n}(Bπ)[B^{−1}] → KO_{n}(C_{r}^{∗}π) is an isomorphism for any finitely presented group π.

Here C_{r}^{∗}π denotes the reduced C^{∗}-algebra of π. For a finite group π we have C_{r}^{∗}π ≅ ℝπ. Now we are in the position to state our main result: Theorem 3.14 For every finite p-group θ : R_{n}(Bπ)[B^{−1}] → KO_{n}(ℝπ) is (p-locally) surjective.

Since KO_{n}(ℝπ) is the sum of finitely many copies of ℤ and ℤ_{2}, Theorem 3.14 implies that: Proposition 3.15 θ : R_{n}(Bπ) → KO_{n}(ℝπ) is (p-locally) surjective for sufficiently large n.

Remark 3.16 For θ : R_{n}(pt) → KO_{n}(pt) surjectivity is implied by Carr [Car88] if n ≡ 0 and by Hitchin [Hit74] if n ≢ 0 modulo 4. They show that A : Ω_{n}^{Spin} → KO_{n}(pt) is surjective in those dimensions. Since θ(W) = A(W) for closed manifolds, it is enough to prove the following:

Theorem 3.17 For every finite p-group θ : R̃_{n}(Bπ)[B^{−1}] → KO_{n}(ℝ̃π) is a surjection where R̃_{n}(Bπ) is the kernel of the forgetful map f : R_{n}(Bπ) → R_{n}(pt).

Remark 3.18 For n odd KO_{n}(ℝπ) = Tor_{p}KO_{n}(ℝπ) so Corollary 2.51 already proves the surjectivity in odd dimensions.

It would be interesting to find a method to construct elements in R_{n}(Bπ) with a given value of θ. For cyclic groups such a construction – using lens spaces and bundles of lens spaces – is known [BG95, p.512]. This construction – and the fact that the rational representation ring R(π) ⊗ ℚ is generated by representations induced from cyclic subgroups for any finite group π – can be used to construct examples such that their image is a subgroup of finite index in KO_{n}(ℝπ).

3.3 The surjectivity of θ

In this section π denotes a finite p-group. Our main tool for the proof is the following diagram:

[figure: R̃_{n}(Bπ)[B^{−1}]; θ; ∂; 1.; P̃os_{n−1}(Bπ)[B^{−1}]; η; f; 2.; Ω̃_{n−1}^{Spin}(Bπ)[B^{−1}]; \overline{η}; 3.; R̃_{n−1}(Bπ)[B^{−1}]; θ; KO_{n}(ℝ̃π); q; KO_{n}(ℝ̃π; ℤ[1/p]); r; KO_{n}(ℝ̃π; ℤ_{p^{∞}}); δ; KO_{n−1}(ℝ̃π)]

Diagram 1

In section 3.3.1 we define the various groups and maps involved in Diagram 1 and we show the exactness of the top row. Then we prove the commutativity of Diagram 1, except square 2, which will be postponed to section 5. In section 3.3.2 we use this diagram to prove the surjectivity of θ. The argument is a slight generalization of the 5-lemma, which can be proved by standard diagram chasing:

Lemma 3.19 Suppose that in the following commutative diagram all the rows and columns are exact:

\xymatrix{ &  & • \ar[d] \ar@{->>}[r] & • \ar[d] \\ • \ar@{->>}[d] \ar[r] & • \ar[d]^{θ} \ar[r] & • \ar@{->>}[d] \ar[r] & • \ar[d] \\ • \ar[r] & • \ar[r] & • \ar[r] & •}

Then the map θ is surjective.

Comparison of Diagram 1 with the lemma shows that we have to prove the surjectivity of three maps:

1. \overline{η} : Ω̃_{n}^{Spin}(Bπ)[B^{−1}] → KO_{n+1}(ℝ̃π; ℤ_{p^{∞}}) – Proposition 3.25,

2. η : P̃os_{n−1}(Bπ)[B^{−1}] → KO_{n}(ℝ̃π; ℤ[1/p]) – Proposition 3.26,

3. f restricted to Ker(η) gives a surjection: Ker(η) → Ker(\overline{η}) – Proposition 3.27. 3.3.1 The definition of Diagram 1 Ω̃_{n}^{Spin}(Bπ)[B^{−1}] and P̃os_{n}(Bπ)[B^{−1}] have definitions similar to the definition of R̃_{n}(Bπ)[B^{−1}]: Definition 3.20 Ω̃_{n}^{Spin}(X): = Ker(p^{∗} : Ω_{n}^{Spin}(X) → Ω_{n}^{Spin}(pt)) and P̃os_{n}(X): = Ker(p^{∗} : Pos_{n}(X) → Pos_{n}(pt)).

Ω̃_{n}^{Spin}(Bπ)[B^{−1}]: = lim_{→} Ω̃_{n+8k}^{Spin}(Bπ),

where the connecting maps are the following:

j_{k} : Ω̃_{n+8k}^{Spin}(Bπ) → Ω̃_{n+8(k+1)}^{Spin}(Bπ), j_{k}[M, f] = [M × B, f × ∗]

and

P̃os_{n}(Bπ)[B^{−1}]: = lim_{→} P̃os_{n+8k}(Bπ)

where the connecting maps are the following:

j_{k} : P̃os_{n+8k}(Bπ) → P̃os_{n+8(k+1)}(Bπ), j_{k}[M, g] = [M × B, g̃]

where g̃ is defined by shrinking as in Definition 3.11. Remark 3.21 The top row of Diagram 1 is exact since it is the direct limit of exact sequences.

The bottom row is the long exact sequence induced by the coefficient sequence:

0 ⟶ ℤ ⟶ ℤ[1/p] ⟶ ℤ_{p^{∞}} ⟶ 0.

Definition 3.22 Let U : Ω̃_{n−1}^{Spin}(Bπ)[B^{−1}] → K̃O_{n−1}(Bπ) denote the map induced by the Atiyah orientation and let ∂ : K̃O_{n}(Bπ; ℤ_{p^{∞}}) → K̃O_{n−1}(Bπ) denote the boundary map. Then

\overline{η}: = A(ℤ_{p^{∞}})∂^{−1}U.

In order to define η first we show the commutativity of the already defined square 3.

Proposition 3.23 Square 3. is commutative.

Proof. The commutativity of the diagram below implies the commutativity of square 3.

\xymatrix{Ω̃_{n−1}^{Spin}(Bπ)[B^{−1}] \ar[d]^{U} \ar[r] & R̃_{n−1}(Bπ)[B^{−1}] \ar[ddd]^{θ} \\ K̃O_{n−1}(Bπ) \ar[d]^{∂^{−1}} \ar[ddr]^{A} \\ K̃O_{n}(Bπ; ℤ_{p^{∞}}) \ar[d]^{A(ℤ_{p^{∞}})} \\ KO_{n}(ℝ̃π; ℤ_{p^{∞}}) \ar[r]^{δ} & KO_{n−1}(ℝ̃π)}

The maps A and θ are defined as the index of the Dirac operator of a manifold representing an element in K̃O_{n−1}(Bπ) and R̃_{n−1}(Bπ)[B^{−1}] respectively. For a closed manifold the two indices coincide which implies the commutativity of the upper square. The commutativity of the lower – triangle shaped – square is the consequence of the fact that A and A(ℤ_{p^{∞}}) are induced by the same spectrum map and the two other sides are the connecting homomorphisms in the long exact sequences induced by the coefficient sequence:

0 ⟶ ℤ ⟶ ℤ[1/p] ⟶ ℤ_{p^{∞}} ⟶ 0.

Strictly speaking it shows the commutativity of the diagram in which ∂^{−1} is replaced with ∂ and the corresponding arrow is reversed, but these two statements are equivalent.

Corollary 3.24 There exists a unique map η which makes square 1. commutative.

Proof. Ω̃_{n−1}^{Spin}(Bπ) is p-torsion by Remark 2.49, so for any x ∈ P̃os_{n−1}(Bπ) there is a p-power k such that f(kx) = 0. By the exactness of the top row of Diagram 1 there exists a y ∈ R̃_{n}(Bπ) such that ∂y = kx. The desired commutativity of square 1. then forces us to define

η(x): = \frac{1}{k} qθ(y) ∈ KO_{n}(ℝ̃π) ⊗ ℤ[1/p].

(Note that k – being a power of p – is invertible in ℤ[1/p].) The commutativity of square 3. implies that η is well defined.

The commutativity of square 2. will be proved in the next chapter.

3.3.2 Three surjective maps

Proposition 3.25 The homomorphism \overline{η} : Ω̃_{n}^{Spin}(Bπ)[B^{−1}] → KO_{n+1}(ℝ̃π; ℤ_{p^{∞}}) is surjective.

Proof. A(ℤ_{p^{∞}}) and ∂^{−1} are isomorphisms and U is surjective by Theorem 2.29.

Proposition 3.26 The homomorphism η is surjective.

Proof. This proof wasn’t correct. I am still working on a correct one.

Proposition 3.27 f maps Ker(η) to Ker(\overline{η}) surjectively.

We postpone the proof to the next section (after Theorem 4.11), since we need the comparison between η and the classical η-invariants. It is possible to give a purely homotopy theoretical proof. As we mentioned in Theorem 2.29 the results of [KS93] imply that if a ∈ Ker(\overline{η}) = Ker(U) then a can be represented by the total space of a geometric ℍℙ^{2}-bundle P such that the Bπ-structure comes from the base. They also imply that there is an other geometric ℍℙ^{2}-bundle such that for its total space W we have ∂[W] = kP, which immediately implies Proposition 3.27, since W admits a positive scalar curvature metric extending the standard positive scalar curvature metric on P therefore θ(W) = 0. The argument is somewhat involved so we preferred the analytic proof. The existence of a homotopy theoretical proof is very interesting since it shows that it is possible to give a proof of the surjectivity of θ without any reference to the classical η-invariant.

This completes the proof of Theorem 3.14.

4 Connection between η and the classical eta invariants

4.1 Complexification

In this section we study the complexification map from the KO-groups of ℝπ to the KO-groups of ℂπ. We calculate the kernel of the complexification map for different coefficients. In section 4.2 we use these results to compare the map η and a map η_{ℂ} which connects η with the classical eta invariants. On the other hand in section 5 we show that the index theorem for flat bundles implies a complexified version of the commutativity of square 2. of Diagram 1. These calculations about the complexification map imply that this complexified version can be “lifted” to the real case.

For the spectrum KO(ℂπ) we will also use the notation K(ℂπ) (this is compatible with the usual definition of the K-theory of complex C^{∗}-algebras).

Tensoring with ℂ induces a map c : KO_{n}(ℝπ) → K_{n}(ℂπ). For a coefficient group G let c(G) denote the induced map:

c(G) : KO_{n}(ℝπ) ⊗ G → K_{n}(ℂπ) ⊗ G

Definition 4.1 Let M be a spin manifold and recall that η̂ = r ∘ η is the mod ℤ reduction of η.

\begin{matrix}A_{ℂ}: = c ∘ A, & θ_{ℂ}: = c ∘ θ, \\ η_{ℂ}: = c(ℤ[1/p]) ∘ η, & η̂_{ℂ}: = c(ℤ_{p^{∞}}) ∘ η̂. \end{matrix}

Theorem 4.2 A_{ℂ} : K(Bπ) → K(ℂπ) is induced by a map of spectra A_{ℂ}, and can be represented as slant product with [ℂπ(Bπ)] ∈ K_{ℂπ}^{0}(Bπ) where ℂπ(Bπ) = ℝπ(Bπ) ⊗ ℂ.

Proof. Completely analogous to the proof of Theorem 2.37.

At this point we can raise the question how much information is lost by complexification. We use a long exact sequence related to the cofibration of spectra S^{0} ⟶ Cη ⟶ S^{2} where η represents the generator of the stable homotopy group π_{1}^{S}(S^{0}):

Proposition 4.3 [Sch93, p.22] For any real C^{∗}-algebra C we have the long exact sequence:

⟶ KO_{n−1}(C) ⟶^{γ_{n−1}} KO_{n}(C) ⟶^{c} KO_{n}(C ⊗_{ℝ} ℂ) ⟶ KO_{n−2}(C) ⟶^{γ_{n−2}} KO_{n−1}(C) ⟶

where γ is multiplication by the generator of KO_{1}(ℝ).

Corollary 4.4

Ker(c) = Tor_{p}KO_{n}(ℝπ), Coker(c) ≅ Ker(γ_{n−2}),
Ker(c(ℤ[\frac{1}{p}])) = 0,  Ker(c(ℤ_{p^{∞}})) = Tor_{p}(Ker(γ_{n−2})).

The last statement can be derived by a snake lemma argument (see [Feh98, App.B] for details). What will be important later is that Ker(c(ℤ_{p^{∞}})) is a finite group.

4.2 The eta invariants and the APS index theorem for manifolds with boundary

Let us recall now the definition of η_{α} from [APS76]. We slightly changed the notation to be consistent with ours and of the recent literature.

Definition 4.5 For n even let 𝕊^{+} be the unique irreducible ungraded left ℂl_{n−1}-module for which ωn−1^{ℂ} ∈ ℂl_{n−1} acts as the identity, where ωn−1^{ℂ} is the complex volume form i^{[ \frac{n}{2} ]}e_{1} ⋯ e_{n−1} and let M be an n-1-dimensional spin manifold with finite fundamental group π:

D_{α}^{+}(M): = D(ℂl_{Spin}(M) ⊗_{ℂ} α(M) ⊗_{ℂl_{n−1}} 𝕊^{+}) = D(𝕊^{+}(M) ⊗_{ℂ} α(M))

where α is a ℂπ-module, α(M) = M̃ ×_{π} α is equipped with the flat connection and 𝕊^{+}(M) = Spin(M) ×_{Spin(n−1)} 𝕊^{+}.

In the construction of 𝕊^{+}(M) we used the Spin(n − 1)-action coming from the inclusion Spin(n − 1) ⊂ Cl_{n−1}^{×} ⊂ ℂl_{n−1}^{×}.

Definition 4.6 For any self-adjoint operator A :

η_{A}(s): = ∑_{λ≠0} sign(λ)|λ|^{−s}

where λ runs through the eigenvalues of A counted by multiplicity. Moreover we define:

η_{α}(s): = η_{D_{α}^{+}(M)}(s).

This sum doesn’t necessarily converge. However:

Theorem 4.7 ([APS75a, 3.10]) If A is a first order self-adjoint differential operator over the compact odd dimensional manifold M then η_{A}(s) converges absolutely for Re(s) large and extends to a meromorphic function on the whole s-plane with a finite value at s = 0.

This result validates the next definition:

Definition 4.8 Let M be an n-1-dimensional spin manifold with finite fundamental group π and α be an irreducible ℂπ-module:

η_{α}(M): = −\frac{1}{2}(η_{α}(0) − η_{N}(0))

where dim_{ℂ} α = N and N also denotes the same dimensional trivial representation.

Remark 4.9 In general η_{α}(M) = −\frac{1}{2}(η_{α}(0) + dim Ker(D_{α}^{+}(M)) − η_{N}(0) − N dim Ker(D_{1}^{+}(M))) is the definition but we always assume that the metric on M has positive scalar curvature, so dim Ker(D_{α}^{+}(M)) = dim Ker(D_{1}^{+}(M)) = 0. Here 1 denotes the trivial 1-dimensional ℂπ-module.

Let us recall from 2.10 that for n even we denoted by 𝕊_{n} the unique irreducible ℤ_{2}-graded left ℂl_{n}-module for which the action of ωn^{ℂ} ∈ ℂl_{n} coincides with the grading operator T, where ωn^{ℂ} is the complex volume form i^{[ \frac{n+1}{2} ]}e_{1} ⋯ e_{n}.

Definition 4.10 Let α ∈ Irr(ℂπ) and let t_{α} : K_{n}(ℂπ) → K_{0}(ℂ) be defined by the rule:

t_{α}[N]: = [N ⊗_{ℂl_{n}⊗ℂπ} (𝕊 ⊗ α)]

Theorem 4.11 η_{α}(M) = t_{α}η_{ℂ}(M) under the canonical isomorphism K(pt; ℤ[1/p]) ≅ ℤ[1/p].

Proof. Straightforward calculation comparing the index of D(M) and D_{α}^{+}(M) and using the Atiyah-Patodi-Singer index theorem for manifolds with boundary ([APS75a, 4.11, 4.13]).

This theorem shows that the η_{α}’s are the “coordinates” of η_{ℂ} – or of η, since Ker(c(ℤ[\frac{1}{p}])) = 0. So to prove the surjectivity of f : Ker(η) → Ker(\overline{η}) (Proposition 3.27) it is enough to show that:

Proposition 4.12 If a ∈ Ker(\overline{η}) ⊂ Ω̃_{n−1}^{Spin}(Bπ) then there is a manifold T such that [T] = a, and a positive scalar curvature metric g on T such that η_{α}(T, g) = 0 for all ℂπ-module α.

Proof. By [BGS, lemma 4.12] the total space T of a geometric ℍℙ^{2}-bundle carries a positive scalar curvature metric g, such that η_{α}(T, g) = 0 for every α. On the other hand if a ∈ Ker(\overline{η}) = Ker(U) then a can be represented by the total space of a geometric ℍℙ^{2}-bundle by Theorem 2.29.

5 Commutativity of square 2

In this section we finish the the proof of the commutativity of Diagram 1. The missing piece was the commutativity of the second square:

Theorem 5.1 The following diagram is commutative:

[figure: P̃os_{n−1}(Bπ)[B^{−1}]; η; f; Ω̃_{n−1}^{Spin}(Bπ)[B^{−1}]; \overline{η}; KO_{n}(ℝ̃π; ℤ[1/p]); r; KO_{n}(ℝ̃π; ℤ_{p^{∞}})]

or equivalently we have to show that:

η̂(M) = A(ℤ_{p^{∞}})∂^{−1}Uf(M),

where η̂ = r ∘ η.

[Feh98] contains a very detailed proof. A different proof can be obtained by developing a certain additivity theorem for families of Dirac operators on manifolds with boundary ([Feh]). Here we give a sketch of the proof in [Feh98]. First we prove a complex analogue of Theorem 5.1:

Theorem 5.2 For every n-1-dimensional (n even!) spin manifold M with π_{1}(M) = π is a finite p-group and for every α ∈ Irr(ℂπ):

η̂_{ℂ}(M) = A_{ℂ}(ℤ_{p^{∞}})∂^{−1}Uf(M).

Then we show how you can “lift” this result to prove Theorem 5.1. The starting point is the index theorem of Atiyah, Patodi and Singer for flat bundles.

Theorem 5.3 ([APS76, Thm (5.3)]) Let M be an n-1-dimensional (n even) spin manifold with π_{1}(M) = π finite and α ∈ Irr(ℂπ). Then

η̂_{α}(M) = Ind([σ(D)] · [α]).

Here [σ(D)] ∈ K^{1}(T M) is the self-adjoint symbol of D = D(𝕊^{+}(M)) used by Atiyah, Patodi and Singer in [APS76, p.81]. By [APS75b, p.430] there is an element [α] ∈ K^{−1}(M; ℝ/ℤ) such that ∂[α] = [α(M)] ∈ K^{0}(M) where ∂ denotes the boundary map induced by the coefficient sequence 0 ⟶ ℤ ⟶ ℝ ⟶ ℝ/ℤ ⟶ 0, and α(M) is the flat bundle over M corresponding to the ℂπ-module α and π = π_{1}(M). · denotes the pairing

K^{−1}(M; ℝ/ℤ) ⊗ K^{1}(T M) → K^{0}(T M; ℝ/ℤ).
Ind : K^{0}(T M; ℝ/ℤ) → K^{0}(pt; ℝ/ℤ) ≅ ℝ/ℤ

is the extension of the topological index, in other words the transfer map p_{!} where p : T M → pt. Since we assume that π is a finite p-group we can replace ℝ/ℤ with ℤ_{p^{∞}}.

One can show that σ(D) represents a K-theory Thom class for T M (similarly to [ML89, p.275]) so analogously to [ML89, 16.10]:

Ind([σ(D)] · [α]) = q_{!}([α]),

where q : M → pt and q_{!} : K^{−1}(M; ℤ_{p^{∞}}) → K^{n−2}(pt; ℤ_{p^{∞}}) ≅ K^{0}(pt; ℤ_{p^{∞}}) is the transfer map corresponding to the map q. (In [ML89] you find the real case and without coefficients but the proof is essentially the same.) Almost by definition:

q_{!}([α]) = [α]/[[M]]

where [[M]] ∈ K_{n−1}(M) is the K-theory fundamental class of M.

We can work with all the ℂπ-modules at the same time and, using the fact that

ℂπ(M) = ⨁ α(M) ⊗_{ℂ} α^{∗}

by [AS69, p.15] where α^{∗} is the right ℂπ-module dual to α and α(M) = M̃ ×_{π} ℂπ, we get:

η̂_{ℂ}(M) = [V]/[[M]]

where [V] ∈ K_{ℂπ}^{−1}(M; ℤ_{p^{∞}}) such that ∂[V] = [ℂπ(M)] ∈ K_{ℂπ}^{0}(M) and ∂ denotes the boundary map induced by the coefficient sequence 0 ⟶ ℤ ⟶ ℤ[1/p] ⟶ ℤ_{p^{∞}} ⟶ 0. Also, we used the identification K(ℂπ; ℤ_{p^{∞}}) ≅ K(ℂπ) ⊗ ℤ_{p^{∞}}.

Choose an [N] ∈ K_{n}(Bπ; ℤ_{p^{∞}}) such that ∂[N] = c_{∗}^{M}[M] ∈ K_{n−1}(Bπ). Here c^{M} : M → Bπ denotes the classifying map of the universal cover of M so c_{∗}^{M}[[M]] = [M] is the class M represents in K_{n−1}(Bπ). Such an [N] exists since we assume that [M] ∈ K̃_{n−1}(Bπ) and K̃_{n−1}(Bπ) ⊂ Im∂.

A quite straightforward calculation (using that ∂a/b = a/∂b) shows that

[V]/[[M]] = [ℂπ(M)]/[N].

But slant product with [ℂπ(M)] represents the complex assembly map (Theorem 4.2), so

[ℂπ(M)]/[N] = A_{ℂ}(ℤ_{p^{∞}})∂^{−1}[M] = A_{ℂ}(ℤ_{p^{∞}})∂^{−1}Uf(M),

which proves Theorem 5.2.

Proof of Theorem 5.1. We show that Theorem 5.2 implies Theorem 5.1. First we need the following:

Proposition 5.4 η̂ has the following factorization:

[figure: P̃os_{n−1}(Bπ)[B^{−1}]; η̂; U∘f; Im(U ∘ f); φ; KO_{n}(ℝ̃π) ⊗ ℤ_{p^{∞}}]

Proof. The proposition is equivalent with the statement Ker(U ∘ f) ⊂ Ker(η̂),which is a consequence of the following two observations:

1. Ker(f) ⊂ Ker(η̂). This is a simple consequence of the definition of η: changing the metric on M changes η(M) by an “integer” – an element in the image of q : KO_{n}(ℝ̃π) → KO_{n}(ℝ̃π) ⊗ ℤ_{p^{∞}}.

2. For every M ∈ Ker(U) there is an element (M, g) in f^{−1}(M) such that η(M, g) = 0. This is just a reformulation of Proposition 3.27.

To simplify the notation let us denote the map φ by η̂ and recall (Proposition 3.26) that Im(U ∘ f) = Ker(A). So we have the following diagram:

[figure: Ker(A); δ; η̂; A(ℤ_{p^{∞}})∂^{−1}; _{η̂_{ℂ}}; 0; Tor_{p}(Ker(γ)); KO_{n}(ℝ̃π) ⊗ ℤ_{p^{∞}}; c; K_{n}(ℂ̃π) ⊗ ℤ_{p^{∞}}]

where δ: = η̂ − A(ℤ_{p^{∞}})∂^{−1}. Since η̂ and A(ℤ_{p^{∞}})∂^{−1} are both lifts of the same map, δ defines a map to Tor_{p}(Ker(γ)). But this is a finite group. On the other hand Ker(A) is isomorphic to ⨁_{m} ℤ_{p^{∞}} (Corollary 2.51) and therefore it has no nontrivial homomorphism to a finite group. Which implies that δ = 0 and η̂ = A(ℤ_{p^{∞}})∂^{−1}. And we finished the proof of Theorem 5.1.

Remark 5.5 This last diagram shows that η̂ contains more information than η̂_{ℂ} if p=2. The possibility of refining the classical mod ℤ reduced eta invariant was first observed in [BGS].Corollary 2.51 implies that η̂ is an isomorphism, so no further refinement is possible.

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