Real secondary index theory

Ulrich Bunke and Thomas Schick ^{∗}

September 25, 2003

Contents

1 Introduction 2

2 Real local index theory 5 2.1 Chern classes of geometric families in Deligne cohomology . . . . . . . . . 5 2.2 The real index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3 The analytic invariant 14 3.1 Construction of a natural transformation d_{B,k−1}^{n} . . . . . . . . . . . . . . . 14 3.2 Some properties of d_{B,k−1}^{n} . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3 Nontrivial examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Topological universal classes 22 4.1 Transgression of the Chern classes . . . . . . . . . . . . . . . . . . . . . . . 22 4.2 A topological description of d_{B,4k+1}^{0} . . . . . . . . . . . . . . . . . . . . . . 25 4.3 The topological interpretation of d_{B,4k+1−n}^{−n} . . . . . . . . . . . . . . . . . . 28 4.4 Explicit calculation of the universal class . . . . . . . . . . . . . . . . . . . 30 4.5 The relation with ordinary characteristic classes . . . . . . . . . . . . . . . 35 4.6 Extendibility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 A Transgression 39

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

A.1 Transgression in cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . 39 A.2 Transgression and products . . . . . . . . . . . . . . . . . . . . . . . . . . 43 A.3 Transgression in ordinary cohomology and the relation with the Bockstein 45

B Cohomology of BO, BU and their loop spaces 47 B.1 The cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 B.2 Maps between loop spaces of BO . . . . . . . . . . . . . . . . . . . . . . . 50

1 Introduction

1.0.1 The index of a family of Fredholm operators parametrized by a space B is an element in the K-theory K^{∗}(B) of this parameter space. If this family is in fact a family of fiberwise generalized Dirac operators on a smooth fiber bundle over B, then after adding some further geometric structures in order to define the Bismut super connection we can do local index theory in the sense of [4]. Let us denote by E the family with this collection of geometric structures, by D(E) the family of Dirac operators, and by index(E) the index of this family. Local index theory provides a closed form Ω(E) on B (see Definition 2.2) which represents a cohomology class [Ω(E)] ∈ H^{∗}(B, ℝ). The local index theorem states that

ch^{ℝ}(index(E)) = [Ω(E)] .

1.0.2 The focus of the present paper is not a generalization of this type of results. Let us illustrate the philosophy of the present paper in the case above. We start with local index theory and produce the even form Ω(E). We then observe that this form is closed and therefore represents a cohomology class [Ω(E)] ∈ H^{even}(B, ℝ). We observe that this class in fact only depends on index(E) ∈ K^{0}(B). The classifying space of the functor K^{0} is BU × ℤ. By naturality we conclude that there must be a universal class ch_{univ}^{ℝ} ∈ H^{ev}(BU × ℤ, ℝ) such that [Ω(E)] = f^{∗}ch_{univ}^{ℝ}, if f : B → BU × ℤ classifies index(E). We know that H^{ev}(BU, ℝ) is a polynomial ring in generators c_{2}^{ℝ}, c_{4}^{ℝ}, . . .. Then we finally look for a formula which expresses ch_{univ}^{ℝ} in terms of these generators, this way obtaining the Chern character. Of course this is a well known possible way toward the family index formula.

1.0.3 In fact the real subject of the present paper is a secondary version of this approach. Let K_{k}^{∗}(B) denote the k’th step of the Atiyah-Hirzebruch filtration of K-theory (see 2.1.6). Under the assumption that index(E) ∈ K_{k−1}^{∗}(B), the Chern class c_{k}(index(E)) ∈ H^{k}(B, ℤ) (note that we use a non-standard notation where the subscript is equal to the degree) admits a natural lift to smooth Deligne cohomology ĉ_{k}(E) ∈ H_{Del}^{k}(B) (see 2.1.8 and 2.1.17). This lift is a differential-geometric (or even global-analytic) invariant which varies continuously with the geometry. In particular it has a curvature ω(ĉ_{k}(E)) ∈ A^{k}(B), which can be expressed through Ω(E).

1.0.4 We rigidify the situation be imposing additional geometric constraints. We in fact assume that the family of Dirac operators D(E) is a family of twisted Dirac operators on a family of Spin-manifolds, and that the twisting bundle is a real bundle. Let n be the fiber-dimension of this family. It follows from the presence of the real structure that, if k + n ≡ 2(4) then the class ĉ_{k}(E) is flat (see 2.1.19). This means that ω(ĉ_{k}(E)) = 0. Since any two geometric structures can be connected by a path we can now conclude that under this assumptions ĉ_{k}(E) is a differential-topological invariant. In Subsection 3.3 we give some non-trivial examples.

1.0.5 Note that the flat part of H_{Del}^{k}(B) can be identified with H^{k−1}(B, ℝ/ℤ). Thus, given a family of n-dimensional spin manifolds and a real twisting bundle such that index(E) ∈ K_{k−1}^{∗}(B) we have defined a class ĉ_{k}(E) ∈ H^{k−1}(B, ℝ/ℤ). This class is natural under pull-back.

The index of the family D(E) has a real refinement index_{ℝ}(E) ∈ KO^{−n}(B) (see Subsection 2.2). We prove that in fact ĉ_{k}(E) only depends on index_{ℝ}(E) and that 2ĉ_{k}(E) = 0 (see Propositions 3.2 and 3.5).

Let U_{4k+1−n}^{−n}(B) ⊂ KO^{−n}(B) be the subset of classes which after complexification belong to K_{4k+1−n}^{−n}(B). What we have constructed so far is a natural transformation

d_{B,4k+1−n}^{−n} : U_{4k+1−n}^{−n}(B) → H^{4k+1−n}(B, ℝ/ℤ)

such that d_{B,4k+1−n}^{−n}(index_{ℝ}(E)) = ĉ_{4k+2}(E).

1.0.6 The universal situation is given by the fiber sequence

Ω^{n}U/O →^{Ω^{n}i} Ω^{n}(BO × ℤ) → Ω^{n}(BU × ℤ) .

It is obtained by application of the functor Ω^{n} to the fibration U/O ⟶^{i} BO → BU, where we construct BO = EU/O and thus obtain the inclusion i: U/O → BO. A class x ∈ KO^{−n}(B) is represented by a map f : B → Ω^{n}(BO × ℤ). If x belongs to U_{∞}^{−n}(B), then this map factors up to homotopy through a map g : B → Ω^{n}U/O. Thus there must be a universal class

d̄_{4k+1−n} ∈ H^{4k+1−n}(Ω^{n}U/O, ℝ/ℤ)

such that g^{∗}d̄_{4k+1−n} = d_{B,4k+1−n}^{−n}(x). Note that this universal class has the special property that g^{∗}d̄_{4k+1−n} only depends on the homotopy class of the composition Ω^{n}i ∘ g.

1.0.7 The main purpose of the present paper is the calculation of the universal class d̄_{4k+1−n} in terms of the classically known generators of the cohomology of Ω^{n}U/O. The result is presented in Theorem 4.15.

1.0.8 If n = 2, 3, 4, 5 or n = 1, k > 0, then d̄_{4k+1−n} = 0. If n = 0 or n = 1, k = 0 then the class d_{B,4k+1−n}^{−n}(x) is a classical characteristic class of x (i.e. it can be expressed in terms of the dimension and Stiefel-Whitney classes). The dimension n = 6 is interesting since in this case the class is definitely not just a classical characteristic class of x. For n = 7 the relation with classical classes is open (see Subsection 4.5 for all that) .

1.0.9 For the convenience of the reader we have added two appendixes. In the first we recall (with proof) some material about transgression.

In the second we recall the results of Cartan [9] about the cohomology of the spaces Ω^{n}U/O and about the action of various maps and transgressions on this cohomology.

2 Real local index theory

2.1 Chern classes of geometric families in Deligne cohomology

2.1.1 We consider a fiber bundle π : E → B with closed n-dimensional fibers. We assume that the vertical T^{v}π := ker(dπ) bundle is oriented and equipped with a spin structure. We choose a vertical Riemannian metric g^{T^{v}π} and a horizontal distribution T^{h}π, i.e. a complement of T^{v}π in T E. Finally, we let W := (W, ∇^{W} , h^{W}) be an auxiliary complex vector bundle with hermitian metric and metric connection. The data described so far make up a geometric family E over B.

2.1.2 We assume that W admits a real structure Q ∈ End(W_{|ℝ}) which is compatible with the connection and the metric. Then W is the complexification of a real bundle W_{ℝ} = (W_{ℝ}, ∇^{W_{ℝ}}, h^{W_{ℝ}}). The latter can be identified with the +1-eigenbundle of Q.

2.1.3 The data which we compressed in the notion of a geometric family induces a family of elliptic operators D(E) over B. Indeed, for b ∈ B the operator D(E)(b) is the spin Dirac operator of the Riemannian spin manifold E_{b} := π^{−1}(b) twisted by the bundle W_{|E_{b}}. The family index of D(E) is the element index(E) ∈ K^{−n}(B).

2.1.4 For k ∈ ℕ_{0} we introduce a natural transformation of c_{k} : K^{n}(B) → H^{k}(B, ℤ) given by the Chern class. In order to have a uniform notation in the even and odd dimensional case we use a notation which differs from the conventional one. So if n is even, then we set c_{2k} := c_{k} and c_{2k+1} := 0, where c_{k} : K^{0}(B) → H^{2k}(B, ℤ) is the Chern class in the usual notation. If n is odd, then we set c_{2k} = 0 and define c_{2k+1} such that the following diagram is commutative

\begin{matrix}K^{n}(B) & →^{c_{2k+1}} & H^{2k+1}(B, ℤ) \\ ∥ & & ∥ \\ K̃^{n+1}(ΣB) & →^{c_{2k+2}} & H^{2k+2}(ΣB, ℤ) \end{matrix} ,

where K̃ denotes the reduced K-theory and the vertical isomorphisms are the natural suspension isomorphisms.

2.1.5 The Chern character is a natural transformation

ch : K^{n}(B) → ⨁_{k≡n(2)} H^{k}(B, ℚ) .

Here again the even part ch : K^{2n}(B) → ⨁_{k≡0(2)} H^{k}(B, ℚ) adopts the usual convention, while the odd part is defined such that the following diagram is commutative

K^{2n−1}(B) →^{ch}  ⨁_{k≡1(2)} H^{k}(B, ℚ) ∥  ∥ K̃^{2n}(ΣB) →^{ch} ⨁_{k≡0 (mod 2)} H̃^{k}(ΣB, ℚ) .

For k ∈ ℕ_{0} let ch_{k} : K^{∗}(B) → H^{k}(B, ℚ) denote the corresponding component.

2.1.6 The ring K^{∗}(B) has a natural decreasing filtration, the Atiyah-Hirzebruch filtration [1],

⋯ ⊂ K_{k+1}^{∗}(B) ⊂ K_{k}^{∗}(B) ⊂ ⋯ ⊂ K_{0}^{∗}(B) = K^{∗}(B) .

Recall that x ∈ K_{k}^{∗}(B) iff f^{∗}x = 0 for any k − 1-dimensional CW -complex X and continuous map f : X → B.

2.1.7 Fix now k ∈ ℕ_{0} and define m ∈ ℕ such that k = m or k = 2m − 1. If x ∈ K_{k}^{∗}(B), then we have

c_{k}(x)_{ℚ} = (−1)^{m−1}(m − 1)!ch_{k}(x)  (2.1)

where c_{k}(x)_{ℚ} ∈ H^{k}(B, ℚ) is the natural image of c_{k}(x) in cohomology with rational coefficients.

2.1.8 Let H_{Del}^{∗}(B) denote the smooth Deligne cohomology of B. In the present paper we use its description in terms of differential characters given by Cheeger-Simons [10]. Let Z^{k−1} be the group of smooth singular chains on B. A class x̂ ∈ H_{Del}^{k}(B) is a homomorphism x̂ : Z^{k−1} → ℝ/ℤ such that there exists a differential form ω(x̂) ∈ A^{k}(B) with the property that for any smooth singular k-chain C we have x̂(∂C) = [∫_{C} ω(x̂)], where [r] ∈ ℝ/ℤ denotes that class of r ∈ ℝ. Note that ω(x̂) is uniquely determined by x̂. It is called the curvature of x̂. It is necessarily closed and has integral periods.

The association B ↦ H_{Del}^{∗}(B) is a contravariant functor from smooth manifolds and smooth maps to graded abelian groups. There is a natural exact sequence

H^{k−1}(B, ℤ) → A^{k−1}(B)/ im(d) →^{â} H_{Del}^{k}(B) →^{v} H^{k}(B, ℤ) → 0 ,

where a is given by

â(β)(Z) = [∫_{Z} α], β ∈ A^{k−1}(B) .

Note that ω(â(β)) = dβ. The map v has the following description. Let x̂ ∈ H_{Del}^{k}(B). We choose a smooth ℝ-valued (k − 1)-cochain T such that T_{|Z^{k−1}} = x̂. This is possible since ℝ is divisible. Then we have dT = ω − c for some ℤ-valued k-cochain c. It follows that c is closed, and we set v(x̂) := [c]. For details we refer to Cheeger-Simons [10].

2.1.9 A complex vector bundle W → B represents an element [W] ∈ K^{0}(B). Assume that W comes with a hermitian metric h^{W} and metric connection ∇^{W}. We set W := (W, h^{W} , ∇^{W}). For k ∈ ℕ_{0}, Cheeger-Simons [10] constructed a natural lift ĉ_{2k}(W) ∈ H_{Del}^{2k}(B) of c_{2k}([W]) such that v(ĉ_{2k}(W)) = c_{2k}([W]) and ω(ĉ_{2k}(W)) ∈ A^{2k}(B) is the Chern-Weyl representative of c_{2k}([W])_{ℝ} associated to the connection ∇^{W}.

2.1.10 The bundle W can be considered as a geometric family W over B with zerodimensional fiber in a natural way. In this case we have index(W) = [W].

Therefore we can consider the geometric family E over B as a generalization of a hermitian vector bundle with connection over B. It is now an obvious question whether one can define a natural lift ĉ_{k}(E) ∈ H_{Del}^{k}(B) of c_{k}(index(E)).

2.1.11 The geometric data associated with the geometric family E induce a connection ∇^{T^{v}π} on the vertical bundle in a natural way. In fact, if we choose for a moment a Riemannian metric g^{T B} on the base, then we can define a Riemannian metric g^{T E} on the total space E as the orthogonal sum of the vertical metric g^{T^{v}π} and the metric g^{T^{h}π} on the horizontal bundle obtained as lift of g^{T B}. Then ∇^{T^{v}π} is the projection of the Levi-Civita connection of g^{T E} to the vertical bundle. This connection does not depend on the choice of g^{T B}. We refer to [4] for details. By Â(∇^{T^{v}π}) ∈ A^{∗}(E) we denote the Chern-Weyl representative of the Â-class of T^{v}π. Furthermore, let ch(∇^{W}) ∈ A(E) be the Chern-Weyl representative of the Chern character of W.

Definition 2.2 The local index form Ω(E) ∈ A(B) of the geometric family E is defined by

Ω(E) := ∫_{E/B} Â(∇^{T^{v}π})ch(∇^{W}) .

2.1.12 The Atiyah-Singer index theorem for families states that

ch(index(E))_{ℝ} = [Ω(E)] ,

where [ω] ∈ H^{∗}(B, ℝ) denotes the class represented by the closed form ω. Here we once and for all fix the isomorphism between de Rham cohomology H_{dR}^{∗}(M) and singular cohomology H^{∗}(B, ℝ) which is induced by the integration map. This means that the value of the class [ω] on the cycle Z is given by ∫_{Z} ω.

2.1.13 The form Ω(E) plays the role of the Chern-Weyl representative of the Chern character of an index bundle with connection ∇^{index(E)} for D(E) though we are not able to define the latter object. In particular, the local index form also determines candidates for the Chern-Weyl representatives c_{k}(∇^{index(E)}) of the Chern classes c_{k}(index(E)). Unfortunately we are not able to define natural lifts ĉ_{k}(E) ∈ H_{Del}^{k}(B) of c_{k}(index(E)) with curvature ω(ĉ_{k}(E)) = c_{k}(∇^{index(E)}).

2.1.14 Assume that index(E) ∈ K_{k}^{∗}(B) and k = 2m or k = 2m − 1. By equation (2.1) we have

c_{k}(index(E))_{ℝ} = [(−1)^{m−1}(m − 1)!Ω^{k}(E)] .

In [7] we have constructed a natural class

ĉ_{k}(E) ∈ H_{Del}^{k}(B)

with curvature ω(ĉ_{k}(E)) = Ω^{k}(E) and v(ĉ_{k}(E)) = c_{k}(index(E)). Instead of repeating the rather indirect construction [7] we give here a direct description which could be taken as definition of ĉ_{k}(E) as well. Note that ĉ_{k}(E) = 0 by definition if n ≢ k(2) (recall that n is the dimension of the fiber of E).

2.1.15 Let Z ∈ Z^{k−1} be a smooth cycle. We must prescribe ĉ_{k}(E)(Z). We can find a smooth manifold X (not necessarily closed) of the homotopy type of a k − 1-dimensional CW -complex, a map f : X → B, and a smooth k − 1-cycle Z′ in X, such that f_{∗}Z′ = Z. We could e.g. take for f : X → B the inclusion of a thickening of the trace |Z| ⊂ B of Z and Z = Z′. Note that 0 = f^{∗}index(E) = index(f^{∗}E). Therefore we can find a perturbation of the family of Dirac operators D(f^{∗}E) by a family of selfadjoint smoothing operators Q (which are odd in the even-dimensional case) such that the family D(f^{∗}E)+Q is invertible. In [7] the pair (f^{∗}E, Q) was called a tamed geometric family and denoted by f^{∗}E_{t}.

2.1.16 If F_{t} is a tamed geometric family over some base B, then the super connection formalism provides an eta-form η(F_{t}) ∈ A(B) such that dη(F_{t}) = Ω(E). We refer to [7] and [4] for details. The form η(F_{t}) depends on the taming. Assume that F′_{t} is a second taming of the same underlying geometric family. Then the difference η(F_{t}) − η(F′_{t}) is a closed form. As a consequence of the index theorem for boundary tamed families [7] we know that

[η(F_{t}) − η(F′_{t})] = ch(x)_{ℝ}

for some x ∈ K^{∗}(B). In fact, we can take x = index((F × I)_{bt}), where the boundary taming is induced by F_{t} and F′_{t}.

2.1.17 We can now prescribe ĉ_{k}(E)(Z) as follows.

Definition 2.3

ĉ_{k}(E)(Z) := [(−1)^{m−1}(m − 1)! ∫_{Z′} η^{k−1}(f^{∗}E_{t})] ∈ ℝ/ℤ .  (2.4)

In order to see that ĉ_{k}(E) is well-defined note that (m − 1)!ch_{k} is an integral cohomology class. Therefore the right-hand side does not depend on the choice of the taming. One also checks independence of f, X, Z′. The relation ω(ĉ_{k}(E)) = Ω^{k}(E) follows from dη(f^{∗}E_{t}) = f^{∗}Ω(E).

2.1.18 Up to this point we have not employed the fact that the geometric bundle W = (W, ∇^{W} , h^{W}) comes with a real structure Q. Because of the existence of Q the geometric bundle W is isomorphic to its hermitian conjugate W̄. We conclude from the general equality

ch_{2k}(∇V̄) = (−1)^{k}ch_{2k}(∇^{V})

that ch_{l}(∇^{W}) = 0 if l ≢ 0(4).

2.1.19 Recall that n = dim(E) − dim(B).

Lemma 2.5 If k + n ≢ 0(4), then Ω^{k}(E) = 0.

Proof. We have

Ω(E) = ∫_{E/B} Â(∇^{T^{v}π})ch(∇^{W}) .

Since the form Â(∇^{T^{v}π}) is non-trivial only in degrees 4l, l ≥ 0, we immediately see that Ω^{k}(E) = 0 if k + n ≢ 0(4). □

2.1.20 We call a class x̂ ∈ H_{Del}^{k}(B) flat if ω(x̂) = 0. By Lemma 2.5 the class ĉ_{k}(E) is flat if k + n ≡ 2(4). The Deligne cohomology of B fits into the exact sequence (see [10])

0 → H^{k−1}(B, ℝ/ℤ) →^{b̂} H_{Del}^{k}(B) →^{ω} A^{k}(B)  (2.6)

such that v ∘ b̂ : H^{k−1}(B, ℝ/ℤ) → H^{k}(B, ℤ) is the Bockstein operator associated to the exact sequence of coefficients

0 → ℤ → ℝ → ℝ/ℤ → 0 .

By (2.6), a flat class in H_{Del}^{k}(B) can be considered as a class in H^{k−1}(B, ℝ/ℤ). From now on we consider H^{k−1}(B, ℝ/ℤ) as a subset of H_{Del}^{k}(B) and do not write b̂ anymore.

2.1.21 The first assertion of the following proposition is just the conclusion of the preceeding discussion.

Proposition 2.7 1. Let E be a geometric family over B such that the geometric twisting bundle W admits a real structure. Let k ≥ 0 and assume that index(E) ∈ K_{k}^{∗}(B). If furthermore k + n ≡ 2(4) (where n = dim(E) − dim(B) is the fiber dimension of E) then ĉ_{k}(E) ∈ H_{Del}^{k} is flat and therefore gives rise to a class in H^{k−1}(B, ℝ/ℤ).

2. The class ĉ_{k}(E) ∈ H^{k−1}(B, ℝ/ℤ) is independent of the geometric structures, i.e. it only depends on the differentiable fiber bundle E → B, the choice of spin structure and orientation of the vertical bundle T^{v}π, and on the real vector bundle W_{R}.

Proof. In order to show the independence of the geometric structures we argue using the connectedness of the space P of these structures. We can set up a universal family E_{univ} over P ×B and define ĉ_{k}(E_{univ}) ∈ H^{k−1}(P ×B, ℝ/ℤ). It follows from the homotopy invariance of the cohomology functor and the naturality of the construction of these classes with respect to pull-back, that the specializations of ĉ_{k}(E_{univ}) at different parameter points p, q are cohomologous on the one hand, and give the classes ĉ_{k}(E_{p}) and ĉ_{k}(E_{q}) associated to the families E_{p} and E_{q} equipped with geometric structures given by p and q, respectively, on the other hand. □

2.1.22 The main goal of the present paper is to understand the nature of the class ĉ_{k}(E) ∈ H^{k−1}(B, ℝ/ℤ) in terms of the topology of the geometric family.

2.2 The real index

2.2.1 The group KO^{0}(B) is defined as the group completion of the semigroup of isomorphism classes of real vector bundles over B. The functor B ↦ KO^{0}(B) extends to a 8-periodic multiplicative cohomology theory KO^{∗}. Complexification of real vector bundles induces a natural transformation c_{B} : KO^{0}(B) → K^{0}(B) which extends to a natural transformation c_{B} : KO^{∗}(B) → KO^{∗}(B) of multiplicative cohomology theories.

If k + n ≢ 0(4), then the composition

KO^{−n}(B) →^{c_{B}} K^{−n}(B) →^{ch_{k}} H^{k}(B, ℚ)

vanishes.

2.2.2 In view of this observation the desirable explanation of the fact that Ω^{k}(E) = 0 if k + n ≡ 2(4) is that index(E) ∈ K^{−n}(B) is in fact of the form c_{B}(index_{ℝ}(E)) for a real refinement of the index index_{ℝ}(E) ∈ KO^{−n}(B). In fact, the spinor bundle carries additional structures which are “preserved” upon twisting by real bundles. Using these structures we can indeed refine the index index(E) ∈ K^{−n}(B) to a class index_{ℝ}(E) ∈ KO^{−n}(B).

For the purpose of illustration we sketch the construction of index_{ℝ}(E). Although this is well known, the following exposition is designed to be a useful reference for the interested reader.

2.2.3 Depending on the class of n modulo 8 we are going to use quite different pictures of KO^{−n}(B). We make use of the real Clifford algebras C^{p,q} associated to ℝ^{p+q} with quadratic form −x_{1}^{2} − ⋯ − x_{p}^{2} + x_{p+1}^{2} + ⋯ + x_{p+q}^{2}.

In one picture an element of KO^{n} is represented as a family of selfadjoint odd Fredholm operators on a graded C^{n,0}-module. Another representation is as a family of antisymmetric Fredholm operators which anticommute with an action of C^{0,n−1}. In this case there is no grading. Finally an element of K^{1}(B) is represented by a family of selfadjoint Fredholm operators (and there is again no grading). We refer to [2] and [13] for further details.

2.2.4 n ≡ 0(8) The spinor bundle S(T^{v}π) is the complexification of a real spinor bundle S_{ℝ}(T^{v}π). Thus V = S(T^{v}π) ⊗ W is the complexification of V_{ℝ} := S_{ℝ}(T^{v}π) ⊗ W_{ℝ}. The Dirac operator D(E) comes from the Dirac operator D_{ℝ}(E) on V_{ℝ}. The refined index index_{ℝ}(E) ∈ KO^{0}(B) is just the index of the family of real Fredholm operators D_{ℝ}(E)^{+}.

2.2.5 n ≡ 1(8) The spinor bundle S(T^{v}π) admits a real structure, which anticommutes with Clifford multiplication. It induces a real structure on V which anticommutes with D(E). Let V_{ℝ} be again the real +1-eigenbundle of the real structure on V. The operator iD(E) commutes with this real structure and therefore induces an antisymmetric operator D(E)_{ℝ} on V_{ℝ}. This family represents index_{ℝ}(E) ∈ KO^{−1}(B).

2.2.6 n ≡ 2(8) The spinor bundle S(T^{v}π) has a quaternionic structure which is odd with respect to the ℤ/ℤ_{2}-grading and commutes with Clifford multiplication. Thus we obtain an induced quaternionic structure J on V. We consider D(E)_{ℝ} := JD(E) as an antisymmetric real operator on V_{ℝ}^{+}. It anticommutes with the action of C^{0,1} which is induced by multiplication by i. Therefore the family D(E)_{ℝ} together with the C^{0,1}-module structure represents index_{ℝ}(E) ∈ KO^{−2}(B).

2.2.7 n ≡ 3(8) The spinor bundle carries a quaternionic structure which commutes with Clifford multiplication. We get an induced quaternionic structure J on V commuting with D(E). The antisymmetric operator D(E)_{ℝ} := iD(E) anticommutes with the action of C^{0,2} generated by J and iJ. Therefore the family D(E)_{ℝ} on V_{ℝ} together with the C^{0,2}-module structure represents index_{ℝ}(E) ∈ KO^{−3}(B).

2.2.8 n ≡ 4(8) The spinor bundle S(T^{v}π) carries a quaternionic structure which commutes with the grading and Clifford multiplication. It induces a quaternionic structure J on V which commutes with D(E). We consider the antisymmetric operator D(E)_{ℝ} := iD(E) on the bundle V_{ℝ} which anticommutes with the Clifford algebra C^{0,3} generated by i, J, iJ. Therefore the family D(E)_{ℝ} on V_{ℝ} together with the C^{0,3}-module structure represents index_{ℝ}(E) ∈ KO^{−4}(B).

2.2.9 n ≡ 5(8) The spinor bundle S(T^{v}π) carries a quaternionic structure which anticommutes with the Clifford multiplication. It induces a quaternionic structure J on V which anticommutes with D(E). We form the real selfadjoint operator

D(E)_{ℝ} := ( \begin{matrix}D(E) & 0 \\ 0 & −D(E) \end{matrix} )

on V_{ℝ} ⊕ V_{ℝ} with its standard odd grading

( \begin{matrix}0 & 1 \\ 1 & 0 \end{matrix} ) .

This operator commutes with the Clifford algebra C^{3,0} generated by

( \begin{matrix}0 & J \\ −J & 0 \end{matrix} ) , ( \begin{matrix}0 & iJ \\ −iJ & 0 \end{matrix} ) , ( \begin{matrix}1 & 0 \\ 0 & −1 \end{matrix} ) .

The C^{3,0}-equivariant operator D(E)_{ℝ} represents index_{ℝ}(E) ∈ KO^{3}(B) ≅ KO^{−5}(B).

2.2.10 n ≡ 6(8) The spinor bundle S(T^{v}π) carries a real structure which anticommutes with the grading. In induces a real structure Q on V which is odd and commutes with D(E). We consider the selfadjoint operator D(E)_{ℝ} := D(E) on V_{ℝ}. This bundle is ℤ/2ℤgraded and admits an action of C^{2,0} generated by Q and iQ commuting with D(E)_{ℝ}. The C^{2,0}-equivariant operator D(E)_{ℝ} represents index(E)_{ℝ} ∈ KO^{2}(B) ≅ KO^{−6}(B).

2.2.11 n ≡ 7(8) The spinor bundle S(T^{v}π) admits a real structure which commutes with the Clifford multiplication. In induces a real structure Q on V which commutes with D(E). We consider the real symmetric operator D(E)_{ℝ} which is obtained as restriction of D(E) to the 1-eigenbundle of Q. The operator D(E)_{ℝ} represents index_{ℝ}(E) ∈ KO^{1}(B) ≅ KO^{−7}(B).

3 The analytic invariant

3.1 Construction of a natural transformation d_{B,k−1}^{n}

3.1.1 Recall that complexification of real vector bundles induces a natural transformation of multiplicative cohomology theories c_{B}: KO^{n}(B) → K^{n}(B). The real index index_{ℝ}(E) ∈ KO^{−n}(B) is a refinement of index(E) ∈ K^{−n}(B) in the sense that

c_{B}(index_{ℝ}(E)) = index(E) .

3.1.2 For k ≥ 0 and n ∈ ℤ we define the group U_{k}^{n}(B) by the following exact sequence

0 → U_{k}^{n}(B) → KO^{n}(B) →^{q_{B}} K^{n}(B)/K_{k}^{n}(B) ,

where q_{B} is the composition of c_{B} with the projection onto the quotient. We also define U_{∞}^{n}(B) by the exact sequence

0 → U_{∞}^{n}(B) → KO^{n}(B) → K^{n}(B).

Since q_{B} is a natural transformation the association B ↦ U_{k}^{n}(B) extends to a functor with values in abelian groups.

3.1.3 Assume that k − n ≡ 2(4).

Definition 3.1 We define the natural transformation

d_{B,k−1}^{n}: U_{k}^{n}(B) → H^{k−1}(B, ℝ/ℤ)

by the requirement that

d_{B,k−1}^{n}(x) := ĉ_{k−1}(E) ,

where E is any geometric family of dimension 8l − n, l ∈ ℤ, such that x = index_{ℝ}(E).

3.1.4 We must check that the definition of d_{B,k−1}^{n} makes sense.

Proposition 3.2 If k − n ≡ 2(4), then there is a unique homomorphism

d_{B,k−1}^{n}: U_{k}^{n}(B) → H^{k−1}(B, ℝ/ℤ)

such that d_{B,k−1}^{n}(index_{ℝ}(E)) = ĉ_{k}(E) for any geometric family over B of dimension 8l−n, l ∈ ℤ, with index_{ℝ}(E) ∈ U_{k}^{n}(B). This homomorphism is natural with respect to continuous maps.

Proof. The essential parts of the proof are given in Lemma 3.3 and Lemma 3.4 below. It immediately follows from these Lemmas that for given B there exist a unique map d_{B,k−1}^{n} satisfying the requirement. Additivity and naturality with respect to smooth maps of d_{B,k−1}^{n} follows from naturality and additivity of the class ĉ_{k}(E). But then naturality extends to continuous maps since U_{k}^{n}(·) as well as H^{k−1}(·, ℝ/ℤ) are weak homotopy functors. □

3.1.5

Lemma 3.3 If index_{ℝ}(E) = 0, then ĉ_{k}(E) = 0.

Proof. Assume that index_{ℝ}(E) = 0. In this case we can find a smoothing perturbation of the real operator D(E)_{ℝ} which is invertible. We call this perturbation a real taming. By complexification a real taming induces a taming E_{t} which is compatible with the additional symmetries determining the real structure.

These additional symmetries imply that the Chern form of the Bismut super connection associated to D(E) and its tamed perturbation vanishes. Since the η-form is derived from this Chern form we conclude that η(E_{t}) = 0 if the taming is induced from a real taming. The assertion of the Lemma now follows from the description of ĉ_{k}(E) in terms of the η-form (see Subsection 2.1.17). □

3.1.6

Lemma 3.4 If x ∈ KO^{n}(B), then there exists a geometric family E as above such that index_{ℝ}(E) = x.

Proof. By the periodicity of KO^{∗}(B) we can assume that n < 0. By definition,

KO^{n}(B) = K̃O^{n}(B_{+}) = K̃O^{0}(Σ^{n}B_{+}) ,

where B_{+} is obtained from B by adjoining an additional base point, and K̃O^{n}(B_{+}) denotes the reduced KO-theory.

Let now x ∈ KO^{n}(B) correspond to x̃ ∈ K̃O^{0}(Σ^{n}B_{+}). Let p: S^{n} × B_{+} → Σ^{n}B_{+} be the natural projection and W_{ℝ} = W_{ℝ}^{+} ⊕ W_{ℝ}^{−} be the real ℤ/2ℤ-graded vector bundle over S^{n} × B_{+} representing p^{∗}x̃.

We form two geometric families E^{±} with underlying bundle S^{n} × B_{+} → B_{+} with its standard fibrewise orientation and spin structure, and with the real twisting bundle W_{ℝ}^{±}. Then we have (by Bott periodicity or the index theorem) x = index_{ℝ}(E^{+} ∪_{B_{+}} E^{−})_{|_{B}}. □

3.2 Some properties of d_{B,k−1}^{n}

3.2.1 We approach the study of the natural transformation d_{B,k−1}^{n} from two sides. First, in view of its definition through the analysis of families of Dirac operators we use mainly analytical arguments in order to show some simple properties of this transformation. This is the subject to the present section.

A finer study in Section 4 leading to a complete understanding of the transformation uses methods from topology and the observation, that a natural transformation comes from an universal one between suitable classifying spaces. It should be noted that most results of this section, the important exception being Corollary 3.7, will also follow from the topological description, and will not be needed to derive this description.

3.2.2 There is a natural transformation

r_{B} : K^{∗}(B) → KO^{∗}(B) .

It is determined by the special case r_{B} : K^{0}(B) → KO^{0}(B) which associates to a class represented by a complex vector bundle the class represented by the underlying real vector bundle. It is easy to see that

r_{B} ∘ c_{B} = 2

(multiplication by 2).

3.2.3

Proposition 3.5 We have 2d_{B,k−1}^{n} = 0.

Proof. Fix x ∈ U_{k}^{n}(B). The homology class d_{B,k−1}^{n}(x) ∈ H^{k−1}(B, ℝ/ℤ) is determined by its values on all smooth cycles Z ∈ Z^{k−1} on B.

Given a (k − 1)-cycle Z there exists a manifold X which is homotopy equivalent to a (k − 1)-dimensional CW-complex, a smooth map f : X → B, and a (k − 1)-cycle Z′ in X such that f_{∗}Z′ = Z (compare 2.1.15). By the naturality of d_{·,k−1}^{n} we have

2d_{B,k−1}^{n}(x)(Z) = 2f^{∗}d_{B,k−1}^{n}(x)(Z′) = d_{X,k−1}^{n}(2f^{∗}x)(Z′) .

It thus suffices to show that 2f^{∗}x = 0.

Since f^{∗}x ∈ U_{k}^{n}(X) and X is (up to homotopy equivalence) (k − 1)-dimensional we have c_{X}(f^{∗}x) = 0. This implies 0 = r_{X} ∘ c_{X}(f^{∗}x) = 2f^{∗}x. □

3.2.4 We have defined d_{B,k−1}^{n} in terms of eta forms of families of Dirac operators. In the following we show that it suffices to understand eta forms for zero-dimensional families. Note that the latter are essentially objects of linear algebra.

3.2.5 If x ∈ U_{∞}^{−n}(B), then there exist a real ℤ/2ℤ-graded vector bundle over Σ^{n}B_{+} which represents the class x̃ ∈ K̃O^{0}(Σ^{n}B_{+}) corresponding to x under the identification KO^{−n}(B) ≅ K̃O^{0}(Σ^{n}B_{+}). Let W_{ℝ} = W_{ℝ}^{+}⊕W_{ℝ}^{−} be the pull-back of this bundle to S^{n} ×B under the natural map S^{n} × B → Σ^{n}B_{+}.

The bundle π: S^{n} × B → B has a natural fibrewise orientation and spin structure. The round metric of S^{n} induces a fibrewise metric g^{T^{v}π}. The canonical decomposition T (S^{n} ×B) = pr_{S^{n}}^{∗}T S^{n} ⊕pr_{B}^{∗}T B yields the horizontal distribution pr_{B}^{∗}T B. After choosing geometric bundles W_{ℝ}^{±} = (W_{ℝ}^{±}, ∇^{W_{ℝ}^{±}}, h^{W_{ℝ}^{±}}) we obtain the geometric family E^{±} and E := E^{+} ∪_{B} (E^{−})^{op} over B, with underlying bundle S^{n} × B, such that index_{ℝ}(E) = x.

3.2.6 Since c_{B}(x) = 0 by assumption we know that the complexification W of W_{ℝ} represents the trivial element 0 = [W] ∈ K^{0}(S^{n} × B). Thus (possibly after adding a trivial bundle of formal dimension zero) the bundle W admits an odd unitary selfadjoint (not necessary parallel) automorphism U.

3.2.7 The bundles W_{ℝ}^{±} give rise to geometric families F^{±} over S^{n} × B where the underlying zero-dimensional fiber bundle is id: S^{n} × B → S^{n} × B, and the twisting bundle is W_{ℝ}^{±}. We let F := F^{+} ∪_{S^{n}×B} (F^{−})^{op}. Then we have index_{ℝ}(F) = [W_{ℝ}] ∈ KO^{0}(S^{n} × B) and index(F) = [W] = 0 ∈ K^{0}(S^{n} × B). The automorphism U gives a taming F_{t} of F. In particular, we have a well-defined form η(F_{t}) ∈ A(S^{n} × B) such that dη(F_{t}) = Ω(F) = ch(∇^{W}).

3.2.8 For r >> 0 the operator rU can be considered as a sort of taming of the family E. It is not a taming in the strong sense since U is not smoothing along the fibers of π. Rather it is a local taming in the sense of [7]. Local index theory works for local tamings as well. We let E_{t}(r) be the geometric family E tamed with rU and let η(E_{t}(r)) be the associated η-form.

3.2.9

Proposition 3.6 We have

lim_{r→∞} η(E_{t}(r)) = ∫_{S^{n}×B/B} η(F)

Proof. This assertion is proved using the adiabatic limit techniques developed e.g. in [8]. The general method gives

lim_{r→∞} η(E_{t}(r)) = ∫_{S^{n}×B/B} Â(∇^{T^{v}π})η(F_{t}) .

The result now follows from Â(∇^{T^{v}π}) = pr_{S^{n}}^{∗}Â(∇^{T S^{n}}) = 1, since Â(∇^{T S^{n}}) = 1 for round metric (which is also locally conformally flat). □

3.2.10 A closed form ω ∈ A^{k−1}(B) represents a class [ω] ∈ H^{k−1}(B, ℝ). By [ω]_{ℝ/ℤ} ∈ H^{k−1}(B, ℝ/ℤ) we denote its natural image. Let m be determined by 2m = k or 2m = k+1. Then definition (2.4) together with Proposition 3.6 implies the following corollary.

Corollary 3.7 With the notation above

d_{B,k−1}^{−n}(x) = [(−1)^{m−1}(m − 1)! ∫_{S^{n}×B/B} η^{k+n−1}(F)]_{ℝ/ℤ} .

In fact, in view of Proposition 3.5 we could also omit the sign (−1)^{m−1}.

3.2.11 We consider the sequence

0 → ℤ → ℝ → ℝ/ℤ → 0

and let β′′: H^{k−1}(B, ℝ/ℤ) → H^{k}(B, ℤ) be the associated Bockstein operator. Recall from 2.1.20, that β′′ coincides with the composition

H^{k−1}(B, ℝ/ℤ) → H_{Del}^{k}(B) →^{v} H^{k}(B, ℤ) .

Proposition 3.8 For x ∈ U_{4k+2}^{0}(B) we have

(β′′ ∘ d_{4k+1}^{0})(x) = (c_{4k+2} ∘ c_{B})(x) .

Proof. We write x = index_{ℝ}(E) for a suitable geometric family E over B. Then we have the chain of equalities

(β′′ ∘ d_{4k+1}^{0})(x) = (v ∘ ĉ_{4k+2})(E) = c_{4k+2}(index(E)) = (c_{4k+2} ∘ c_{B})(x) .
□

3.3 Nontrivial examples

In Section 4.4 we will give a complete description of the universal classes d_{·,k}^{n}, which also decides when exactly these classes can be non-trivial. In this section, we want to construct explicit and easy non-trivial examples over low dimensional spheres as base manifolds.

3.3.1 Let MSpin^{∗} be the spin bordism cohomology theory and α : MSpin^{∗} → KO^{∗} be the α-genus introduced by Hitchin [12]. Note that α : MSpin^{∗}(∗) → KO^{∗}(∗) is surjective. If E is a closed spin manifold, then we write α(E) for the result of α applied to the class [E] ∈ MSpin^{∗}(∗) represented by E

The most important common feature of the following examples is that they all come with a trivial twisting bundle. In other words, the respective geometric family E represents an element [E, π] ∈ MSpin^{−n}(S^{i}) (with n ∈ ℕ_{0} and i ∈ {0, 1, 2} depending on the case), such that index_{ℝ}(E) = α([E, π]).

3.3.2 [ n ≡ 1(8), k = 1]

Let E be a closed spin manifold of dimension n ≡ 1(8) with α(E) = 1 ∈ KO^{−1}(∗) ≅ ℤ/2ℤ. Such manifolds exist by the Remark 3.3.1.

We choose a Riemannian metric g^{T E}. Then we consider E as a geometric family E over the point ∗ with the trivial twisting bundle W_{ℝ} = E × ℝ. We claim that ĉ_{1}(E) ≠ 0.

Note that H_{Del}^{1}(∗) = ℝ/ℤ and ĉ_{1}(E)([∗]) = [η^{0}(E_{t})]_{ℝ/ℤ}, where E_{t} is any taming. The degree 0 part η^{0}(E_{t}) is defined even for a pre-tamed manifold in the sense of [7], and if the pre-taming is trivial, then it is one half of the usual Atiyah-Patodi-Singer eta invariant [3]. Now we have η^{0}(E) = 0 since D(E) anticommutes with the real structure and thus has symmetric spectrum. Since by spectral flow (taming essentially means that the underlying operators are invertible)

[η^{0}(E_{t}) − η^{0}(E)]_{ℝ/ℤ} = [\frac{1}{2} dim ker D(E)]_{ℝ/ℤ}

we see that

[η^{0}(E_{t})]_{ℝ/ℤ} = [\frac{1}{2} dim ker D(E)]_{ℝ/ℤ} .

The condition α(E) ≠ 0 says that dim ker D(E) ≡ 1(2). Therefore, d_{∗,0}^{1}(index_{ℝ}(E)) = [η^{0}(E_{t})]_{ℝ/ℤ} = [\frac{1}{2}]_{ℝ/ℤ} ≠ 0 ∈ H^{0}(∗, ℝ/ℤ).

3.3.3 [n ≡ 0(8), k = 2]

We consider a family E → S^{1} of closed spin manifolds with fiber dimension n ≡ 0(8) and α(E) ≠ 0. Indeed for any given spin manifold M of dimension n ≡ 0(8) such a bundle with fiber M exists by [12]. We will in addition assume that α(M) =^{Def} Â(M) = 0.

We equip E with geometric structures and consider the trivial twisting bundle E × W_{ℝ}. Let E denote the corresponding geometric family over S^{1}. Since α(M) = 0 we have index(E) ∈ K_{2}^{0}(S^{1}) = 0 and thus index_{ℝ}(E) ∈ U_{2}^{0}(S^{1}).

We claim that ĉ_{2}(E) ≠ 0. We consider a taming E_{t}. For a > 0 the standard metric g^{T S^{1}} of S^{1} and the horizontal distribution of E induces a Riemannian metric g_{a}^{T E} = g^{T^{v}π}⊕aπ^{∗}g^{T S^{1}} on the total space E. Since E is spin we can consider the total Dirac operator D(a) on E and its perturbation D_{t}(a) which is induced by the taming. In the adiabatic limit a → 0 the operator D_{t}(a) becomes invertible. In other words, for small a the perturbation D_{t}(a) is induced by a local taming. As in [5], we have

[lim_{a→0} η^{0}(D_{t}(a))]_{ℝ/ℤ} = [∫_{S^{1}} η^{1}(E_{t})]_{ℝ/ℤ} = d_{S^{1},1}^{n}(index_{ℝ}(E))([S^{1}]) ∈ ℝ/ℤ .

For sufficiently small a the class [η^{0}(D_{t})(a)]_{ℝ/ℤ} is independent of the adiabatic parameter. As in 3.3.2, since α(E) ≠ 0 and η^{0}(D(a)) = 0 we have [η^{0}(D_{t}(a))]_{ℝ/ℤ} = [\frac{1}{2}]_{ℝ/ℤ}. Thus d_{S^{1},1}^{n} ≠ 0 ∈ H^{1}(S^{1}, ℝ/ℤ).

This result can be interpreted as follows : The holonomy of the determinant line bundle of E is −1 ∈ U(1).

3.3.4 [n = 7(8), k = 3]

We consider a family E → S^{2} of closed spin manifolds with fiber dimension n ≡ 7(8) and α(E) ≠ 0. Such a family exists [12] for any given closed spin manifold M of dimension n ≡ 7(8). We choose geometric structures and consider the trivial twisting bundle E × ℝ. In this way we obtain a geometric family E. Since K^{1}(S^{2}) = 0 we have index(E) = 0 and therefore index_{ℝ}(E) ∈ U_{3}^{7}(S^{2}). We claim, that d_{S^{2},2}^{7}(index_{ℝ}(E)) ≠ 0 ∈ H^{2}(S^{2}, ℝ/ℤ).

We proceed as in 3.3.3. We consider a taming E_{t}. It induces a perturbation D_{t}(a) of the total operator D(a) on E. We have again

[lim_{a→0} η^{0}(D_{t}(a))]_{ℝ/ℤ} = [∫_{S^{2}} η^{2}(E_{t})]_{ℝ/ℤ} = d_{S^{2},2}^{7}(index_{ℝ}(E))([S^{2}]) .

Again, for sufficiently small a the class [η^{0}(D_{t}(a))]_{ℝ/ℤ} is independent of the adiabatic parameter. Since α(E) ≠ 0 and η^{0}(D(a)) = 0 we have [η^{0}(D_{t}(a))]_{ℝ/ℤ} = [\frac{1}{2}]_{ℝ/ℤ}. This implies the claim.

4 Topological universal classes

4.1 Transgression of the Chern classes

4.1.1 As proposed in 3.2.1 we shall understand d_{B,4k−1}^{0} through its universal example. In the present section we start with the definition of this universal class. We will obtain an expression of this class in terms of familiar characteristic classes of real vector bundles. In Theorem 4.10 we show that the transformation d_{B,4k−1}^{0} is indeed induced by the corresponding universal example. In Theorem 4.11 we will then identify d_{B,4k−n}^{−n} in topological terms for all n. Then we will establish some vanishing results and more details about the topological side.

4.1.2 In the present section all spaces have distinguished base points and all maps are base point preserving. Let O and U be the direct limits of O(n) and U(n) induced by the embedding into the left upper corner. The embeddings ℝ^{n} ↪ ℂ^{n}, n ∈ ℕ, induce embeddings O(n) ↪ U(n) and c: O ↪ U. The map c induces the complexification transformation c_{B}. (see 2.2.1).

4.1.3 Let EU → BU be a universal bundle for U. We can consider BO := EU/O and obtain a bundle

U/O ⟶^{i} BO ⟶^{p} BU  (4.1)

with fiber U/O over the base point of BU, compare 12 and 11 on page 51.

In the following, we use the transgression homomorphism for cohomology associated to this fibration (4.1). For the convenience of the reader, we have collected the main definitions and properties of transgression in general (with proofs) in Appendix A, and of transgression and cohomology of the spaces in the fibration U/O → BO → BU in Lemma B.6 in Appendix B.

4.1.4

Definition 4.2 We define the universal transgressed Chern classes

d_{4k+1} := T (c_{4k+2}^{ℚ}) ∈ H^{4k+1}(U/O, ℚ) ,

where c_{4k+2}^{ℚ} is the image of the universal Chern class c_{4k+2} under the natural map

H^{4k+2}(BU, ℤ) → H^{4k+2}(BU, ℚ).

4.1.5 We now consider the following commutative diagram:

\begin{aligned}0 & ⟶ & ℤ & ⟶^{2×} & ℤ & ⟶^{q} & ℤ/2ℤ & ⟶ & 0 \\ & & ↓^{=} & ↓^{\frac{1}{2} ×} & ↓^{l} \\ 0 & ⟶ & ℤ & ⟶ & ℚ & ⟶ & ℚ/ℤ & ⟶ & 0 \end{aligned} .  (4.3)

Definition 4.4 We define d̄_{4k+1} ∈ H^{4k+1}(U/O, ℚ/ℤ) as the image of d_{4k+1} under the natural map H^{4k+1}(U/O, ℚ) → H^{4k+1}(U/O, ℚ/ℤ).

4.1.6 The cohomology ring of BO with coefficients in ℤ/2ℤ is a polynomial ring

ℤ/2ℤ[w_{1}, w_{2}, …] ,

where w_{i} ∈ H^{i}(BO, ℤ/2ℤ) are the universal Stiefel-Whitney classes. It is well known (see [14]) that

β(w_{2k} ∪ w_{2k+1}) = p^{∗}c_{4k+2},  (4.5)

where β is the cohomological Bockstein operator associated to the exact sequence of coefficients in the first row of (4.3). In particular, 2p^{∗}c_{4k+2} = 0 ∈ H^{4k+2}(BO; ℤ), as also stated in Appendix B.

4.1.7 Let l_{∗} : H^{4k+1}(BO, ℤ/2ℤ) → H^{4k+1}(BO, ℚ/ℤ) be induced by l of (4.3). Note that by Lemma B.5 the map l_{∗} factors over the image of β.

Definition 4.6 We define

d̃_{4k+1} := l_{∗}u ∈ H^{4k+1}(BO, ℚ/ℤ),

where u ∈ H^{4k+1}(BO, ℤ/2ℤ) is such that β(u) = p^{∗}c_{4k+2}.

Such an u exists by (4.5) and l_{∗}u is independent of the choice since we have fixed β(u).

4.1.8 Let i: U/O ↪ BO be the inclusion.

Lemma 4.7 We have i^{∗}d̃_{4k+1} = d̄_{4k+1}. Moreover, 2d̃_{4k+1} = 0 and 2d̄_{4k+1} = 0.

Proof. The first assertion is a special case of Proposition A.15, where we use 2p^{∗}(c_{4k+2}) = 0. Note that the homomorphism l is given by division by 2.

The second assertion follows from the fact that 2d̃_{4k+1} = 2l_{∗}u = l_{∗}(2u) = 0 for u of Definition 4.6, and 2\overline{d}_{4k+1} = i^{∗}(2d̃_{4k+1}) = 0. □

4.1.9 By (4.5) we have the following corollary.

Corollary 4.8 We have d̃_{4k+1} = l_{∗}(w_{2k} ∪ w_{2k+1}).

4.2 A topological description of d_{B,4k+1}^{0}

4.2.1 In this subsection we pretend that p: BO → BU is a smooth fiber bundle. To be precise, we should replace this bundle by a N-equivalent finite-dimensional smooth bundle for N sufficiently large.

Let W_{ℝ}^{+} → BO be the universal bundle. Then p: BO → BU classifies its complexification, i.e. if W^{+} → BU is the universal bundle over BU, then we have an isomorphism W_{ℝ}^{+} ⊗_{ℝ} ℂ ≅ p^{∗}W^{+} which induces a real structure Q (complex conjugation) on p^{∗}W^{+}. We can assume that W_{ℝ}^{+} comes with a metric h^{W_{ℝ}^{+}}, and we choose a connection ∇^{W_{ℝ}^{+}}. We set W_{ℝ}^{+} := (W_{ℝ}^{+}, ∇^{W_{ℝ}^{+}}, h^{W_{ℝ}^{+}}) and let W^{+} be its complexification.

4.2.2 We now consider the ℤ/2ℤ-graded bundle W := W^{+} ⊕ W^{−} with W^{−} := \overline{W}^{+}. It admits an odd unitary selfadjoint (not necessary parallel) automorphism

R := ( \begin{matrix}0 & Q \\ Q & 0 \end{matrix} ) .

We form the geometric families G^{±} on BO with underlying fiber bundle id: B → B and twisting bundles p^{∗}W^{±}. Then the family G = G^{+} ∪_{B} G^{−} admits a taming G_{t} induced by R. The associated η-form satisfies dη(G_{t}) = Ω(G) = ch(∇^{p^{∗}W}). By construction we have ch_{4k+2}(∇^{p^{∗}W}) = 2p^{∗}ch_{4k+2}(∇^{W^{+}}) and ch_{4k}(∇^{p^{∗}W}) = 0 (compare 2.1.18).

4.2.3 Let i: U/O → BO be the inclusion of the fiber. Then

di^{∗}η^{4k+1}(G_{t}) = 2(i^{∗} ∘ p^{∗})ch_{4k+2}(∇^{W^{+}}) = 0

since p ∘ i is a constant map. Thus we can consider the class

[i^{∗}η^{4k+1}(G_{t})] ∈ H^{4k+1}(U/O, ℝ).

4.2.4 Let d_{4k+1}^{ℝ} ∈ H^{4k+1}(U/O, ℝ) be the image of d_{4k+1} ∈ H^{4k+1}(U/O, ℚ) under the natural map induced by the map of coefficients r: ℚ ↪ ℝ.

Lemma 4.9 We have

[i^{∗}η^{4k+1}(G_{t})] = \frac{2d_{4k+1}^{ℝ}}{(2k)!} .

Proof. The proof follows from the fact that in the smooth situation there is the alternative description of the transgression T^{ℝ}: H^{4k+2}(BU, ℝ) → H^{4k+1}(U/O, ℝ) given in Proposition A.14. Let x ∈ H^{4k+2}(BU, ℝ) be represented by a closed form X ∈ A^{4k+2}(BU). Then there is some form Y ∈ A^{4k+1}(BO) such that dY = p^{∗}X. The class T^{ℝ}(x) ∈ H_{dR}^{4k+1}(U/O, ℝ) is then represented by i^{∗}Y ∈ A^{4k+1}(U/O).

In our case

x = 2ch_{4k+2}^{ℝ}([W^{+}]), X = 2ch_{4k+2}(∇^{W^{+}}), Y = η^{4k+1}(G_{t})

so that

[i^{∗}η^{4k+1}(G_{t})] = T^{ℝ}(2ch_{4k+2}^{ℝ}([W^{+}])).

Note that

ch_{4k+2}([W^{+}]) = \frac{1}{(2k)!}c_{4k+2}^{ℚ} + decomposable classes,

where for reasons of degree each decomposable summand contains at least one factor c_{4l+2} which is transgressive by the table on page B.2.5. Hence by Proposition A.13

T (2ch_{4k+2}([W^{+}])) = \frac{2}{(2k)!}T (c_{4k+2}) = \frac{2d_{4k+1}}{(2k)!} .

This implies the assertion since T^{ℝ} ∘ r_{∗} = r_{∗} ∘ T by Lemma A.9. □

4.2.5 We now consider a manifold B and x ∈ K̃O^{0}(B). Let X: B → BO be a classifying map for x. We assume that x ∈ U_{∞}^{0}(B). Then we can assume that X factors through the inclusion i: U/O → BO, i.e. without loss of generality we can assume that X: B → U/O.

We define d̄_{4k+1}^{ℝ} ∈ H^{4k+1}(B, ℝ/ℤ) as the image of d_{4k+1}^{ℝ} under the map of coefficients ℝ → ℝ/ℤ, or equivalently, as the image of d̄_{4k+1} under the map of coefficients ℚ/ℤ → ℝ/ℤ. We now come to the main result of this subsection.

Theorem 4.10 d_{B,4k+1}^{0}(x) = X^{∗}d̄_{4k+1}^{ℝ}.

Proof. Let x be of the form [V_{ℝ}^{+}] − [V_{ℝ}^{−}], where V_{ℝ}^{−} := ℝ^{N} × B is trivial. Then we have an isomorphism V_{ℝ}^{+} ≅ X^{∗}W_{ℝ}^{+}. The metric h^{W_{ℝ}^{+}} and the connection ∇^{W_{ℝ}^{+}} induce a metric h^{V_{ℝ}^{+}} and a connection ∇^{V_{ℝ}^{+}} on V_{ℝ}^{+}. In this way we obtain a geometric bundle V_{ℝ}^{+} = (V_{ℝ}^{+}, ∇^{V_{ℝ}^{+}}, h^{V_{ℝ}^{+}}). Furthermore, we equip V_{ℝ}^{−} with the canonical geometry and get V_{ℝ}^{−}.

Set V^{±} := V_{ℝ}^{±} ⊗ ℂ and consider the ℤ/2ℤ-graded bundle V := V^{+} ⊕ V^{−}. Since [V] = 0 in K^{0}(B) we can choose a unitary odd selfadjoint (not necessary parallel) automorphism U of V.

We form the geometric families H^{±} over B with underlying bundle id: B → B and twisting bundle V^{±}. Furthermore we define H := H^{+}∪_{B}(H^{−})^{op}. Then we have index_{ℝ}(H) = x and d_{4k+1,B}^{0}(x) = ĉ_{4k+2}(H). The isomorphism U induces a taming H_{t}. By Corollary 3.7 we thus have

ĉ_{4k+2}(H) = [(2k)!η^{4k+1}(H_{t})]_{ℝ/ℤ} ∈ H^{4k+1}(B, ℝ/ℤ) .

We now consider the bundle Ṽ := V^{+}⊕V̄^{−}⊕V̄^{+}⊕V^{−} with the ℤ/2ℤ-grading diag(1, 1, −1, −1) and the two odd automorphisms

Ũ := ( \begin{matrix}0 & 0 & 0 & U^{−} \\ 0 & 0 & Ū^{+} & 0 \\ 0 & Ū^{−} & 0 & 0 \\ U^{+} & 0 & 0 & 0 \end{matrix} ) , R̃ := ( \begin{matrix}0 & 0 & R^{+} & 0 \\ 0 & 0 & 0 & R^{−} \\ R^{+} & 0 & 0 & 0 \\ 0 & R^{−} & 0 & 0 \end{matrix} ) ,

where R^{±} is the ℂ-linear isomorphism between V_{ℝ}^{±} ⊗ ℂ and its complex conjugate given by complex conjugation.

Note that [R̃, Ũ] = 0. The bundle Ṽ gives rise to a geometric family H̃ = H̃^{+} ∪_{B} (H̃^{−})^{op}, where the underlying fiber bundle of H̃^{±} is again id: B → B, and the twisting bundles are Ṽ^{±}. For each α ∈ [0, π/2] the operator

cos(α)Ũ + sin(α)R̃

defines a taming H̃_{t_{α}}. The family (H̃_{t_{α}})_{α} defines a taming Ĥ_{t} of Ĥ := pr_{B}^{∗}H̃ over [0, π/2]× B. A computation shows that dη(Ĥ_{t}) = pr_{B}^{∗}ch(∇Ṽ) = 0. We conclude the following equality of de Rham cohomology classes

[η^{4k+1}(H̃_{t_{0}})] = [η^{4k+1}(H̃_{t_{π/2}})] .

An inspection of the definitions shows that

η^{4k+1}(H̃_{t_{0}}) = 2η^{4k+1}(H_{t})
η^{4k+1}(H̃_{t_{π/2}}) = (X^{∗} ∘ i^{∗})η^{4k+1}(G_{t}).

We conclude with Lemma 4.9 that

ĉ_{4k+2}(H) = X^{∗}[i^{∗} \frac{(2k)!}{2}η^{4k+1}(G_{t})]_{ℝ/ℤ} = X^{∗}d̄_{4k+1}^{ℝ} .

□

4.3 The topological interpretation of d_{B,4k+1−n}^{−n}

4.3.1 Recall that the classifying space of KO^{−n} is Ω^{n}BO. In view of the fibration

Ω^{n}U/O → Ω^{n}BO → Ω^{n}BU

we see that the classifying map X: B → Ω^{n}BO of an element x ∈ U_{∞}^{−n}(B) factors (up to homotopy) over Ω^{n}U/O, since then the composition B ⟶^{X} Ω^{n}BO → Ω^{n}BU is null homotopic.

Let Ω^{n} : H^{4k+1}(U/O, ℝ) → H^{4k+1−n}(Ω^{n}U/O, ℝ) be the n-fold iteration of the loop map introduced in Definition A.1.

Theorem 4.11 We have

d_{B,4k+1−n}^{−n}(x) = X^{∗}[\frac{(m − 1)!}{(2k)!}Ω^{n}d_{4k+1}^{ℝ}]_{ℝ/ℤ} = X^{∗}[\frac{(m − 1)!}{(2k)!}Ω^{n}T (c_{4k+2}^{ℝ})]_{ℝ/ℤ} ,

where m is determined by 2m = 4k + 3 − n or 2m = 4k + 2 − n.

Proof. We employ Corollary 3.7. Let x̂ ∈ K̃O^{0}(Σ^{n}B) correspond to x ∈ KO^{−n}(B) under the identification KO^{−n}(B) ≅ K̃O^{0}(Σ^{n}B). Let x̃ ∈ K̃O^{0}(S^{n} × B) be the pull-back of x̂ under the natural map S^{n} × B → Σ^{n}B.

Note that the classifying map Y : Σ^{n}B → U/O of x̂ is the adjoint of X : B → Ω^{n}U/O, and that the composition X̃: S^{n} × B → U/O of the projection S^{n} × B → Σ^{n}B and Y is the classifying map of x̃.

Then we have

d_{B,4k+1−n}^{−n}(x) = [(m − 1)! ∫_{S^{n}×B/B} η^{4k+1}(H_{t})]_{ℝ/ℤ} ,

where H_{t} is constructed as in the proof of Theorem 4.10. In that proof we have also shown that

[η^{4k+1}(H_{t})] = [\frac{1}{2}(X̃^{∗} ∘ i^{∗})η^{4k+1}(G_{t})] .

We now apply Lemma 4.9 in order to conclude that

[η^{4k+1}(H_{t})] = \frac{X̃^{∗}d_{4k+1}^{ℝ}}{(2k)!} .

Thus

d_{B,4k+1−n}^{−n}(x) = [\frac{(m − 1)!}{(2k)!} ∫_{S^{n}×B/B} X̃^{∗}d_{4k+1}^{ℝ}]_{ℝ/ℤ} .

The assertion now follows from the general fact that for any z ∈ H^{4k+1}(U/O, ℝ) we have

∫_{S^{n}×B/B} X̃^{∗}z = Σ^{n}Y^{∗}z = X^{∗}Ω^{n}z ,

where Σ is the suspension isomorphism. For the first equality we use that integration over the fiber essentially is the suspension isomorphism in the above construction. The second equality is a special case of the relation between suspension and loop homomorphism proved in Lemma A.6. □

4.3.2 Theorem 4.10 and Theorem 4.11 give a topological description of the value of d_{B,4k+1−n}^{−n}(x) only under the additional assumption that x ∈ U_{∞}^{−n}(B) ⊂ U_{4k+1−n}^{−n}(B). In order to see that this determines d_{B,4k+1−n}^{−n} completely we argue as follows.

Let x ∈ U_{4k+1−n}^{−n}(B). Then the cohomology class d_{B,4k+1−n}^{−n}(x) of degree 4k + 1 − n is determined by its restriction d_{B,4k+1−n}^{−n}(x)_{|B^{4k+1−n}} to the 4k + 1 − n-skeleton B^{4k+1−n} of B. We have x_{|B^{4k+1−n}} ∈ U_{∞}^{−n}(B^{4k+1−n}). Thus we know the topological description of d_{B^{4k+1−n},4k+1−n}^{−n}(x_{|B^{4k+1−n}}), which is equal to d_{B,4k+1−n}^{−n}(x)_{|B^{4k+1−n}} by naturality.

4.4 Explicit calculation of the universal class

Theorem 4.11 does give a topological interpretation of our invariant 3.1. However, we want to be even more precise and explicitly compute the corresponding universal cohomology class

[\frac{(m − 1)!}{(2k)!}Ω^{n}d_{4k+1}]_{ℚ/ℤ} = [\frac{(m − 1)!}{(2k)!}Ω^{n}T (c_{4k+2}^{ℚ})]_{ℚ/ℤ} ∈ H^{4k+1−n}(Ω^{n}U/O, ℚ/ℤ) ,

where m is determined by 2m = 4k + 3 − n or 2m = 4k + 2 − n. In particular, we will show that for half of the parameters n (mod (8)) this class vanishes

We will make use of many of the results about the cohomology of BO, BU and their loop spaces collected in Appendix B.

4.4.1 Consider the map of fibrations

U ⟶ EU ⟶ BU
↓^{p}  ↓  ↓^{id}
U/O ⟶^{i} BO ⟶^{Bc} BU,

where the upper row is the universal principal U-bundle, and the lower row is obtained from the upper by dividing out the subgroup O.

4.4.2 By Lemma B.6, c_{4k+2}^{ℚ} ∈ H^{4k+2}(BU, ℚ) is transgressive in the second (and of course also in the first) fibration. We have to compute the transgression T (c_{4k+2}) ∈ H^{4k+1}(U/O, ℚ). To do this, we observe that the upper fibration is the path space fibration, and therefore by Lemma A.4 the transgression T_{U} of this fibration coincides with the loop homomorphism. By Theorem B.4 we obtain T_{U}(c_{4k+2}) = c_{4k+1} (even in integral cohomology). Moreover, transgression is natural, therefore

p^{∗}(T (c_{4k+2}^{ℚ})) = T_{U}(c_{4k+2}^{ℚ}) = c_{4k+1}^{ℚ} ∈ H^{∗}(U, ℚ).

By Theorem B.2, Theorem B.3 and Table B.2.5, p^{∗}: H^{∗}(U/O, ℚ) → H^{∗}(U, ℚ) is injective and p^{∗}(a_{4k+1}/2) = c_{4k+1}.

Notation 4.12 In order to avoid an inflationary appearance of the exponent ^{ℚ} from now on we will use the same symbol for an integral cohomology class and its image in rational cohomology. It will be clear from the context which meaning the symbol has.

Consequently (with the new convention 4.12 ) we can write

T (c_{4k+2}) = \frac{1}{2}a_{4k+1} ∈ H^{4k+1}(U/O; ℚ).

4.4.3 Our next goal is the calculation of

Ω^{n}T (c_{4k+2}) = Ω^{n}(\frac{1}{2}a_{4k+1}) ∈ H^{4k+1−n}(Ω^{n}U/O, ℚ) .

We consider the fibration

Ω^{n}U/O → Ω^{n}BO → Ω^{n}BU  (4.13)

which is the n-fold loop of the fibration considered above.

4.4.4 In the following, we use the Bott periodicity maps to identify Ω^{n}U/O with the spaces listed in Theorem B.1:

n  0  1  2 3  4  5  6 7 ; Ω^{n}U/O U/O BO × ℤ O O/U U/Sp BSp × ℤ Sp Sp/U

4.4.5 Unfortunately, our knowledge about the map Ω^{n} is not complete enough to calculate Ω^{n}(T c_{4k+2}) directly. We use the following trick:

Using that map α : U/O → U (compare Subsection B.2, Item 15) we have \frac{1}{2}a_{4k+1} = α^{∗} \frac{1}{2}c_{4k+1}. Therefore,

Ω^{n}(\frac{1}{2}a_{4k+1}) = (Ω^{n}α)^{∗}Ω^{n}(\frac{1}{2}c_{4k+1}) .

4.4.6 We shall first compute Ω^{n}(c_{4k+1}). By Theorem B.4

Ω(c_{4k+1}) = (2k)!ch_{4k}.

Note that ΩBU = Ω(BU × ℤ) so that we can iterate the argument.

Next, (2k)!ch_{4k} = 2k · c_{4k} + decomposable. We conclude that

Ω((2k)!ch_{4k}) = 2kΩ(c_{4k}) = 2kc_{4k−1} .

4.4.7 Now, an easy induction allows us to compute Ω^{n}(c_{4k+1}) for each n ∈ ℕ. However, because of the factors appearing in our formulas, we really have to study \frac{(m−1)!}{(2k)!}Ω^{n}(c_{4k+1}), where m = 2k + 1 − \frac{n}{2} if n is even, and m = 2k + 1 − \frac{n−1}{2} if n is odd. In the induction, this factor cancels the factors (like (2k)) which show up in the calculations above, and we get (with m depending on n and k as above)

Ω^{n}(\frac{(m − 1)!}{(2k)!}c_{4k+1}) = \begin{cases}c_{4k+1−n} & n even \\ (2k − \frac{n−1}{2})!ch_{4k+1−n} & n odd . \end{cases}  (4.14)

4.4.8 From this and the calculation of the map in cohomology induced by Ω^{n}α : Ω^{n}U/O → Ω^{n}U, we read off the cohomology classes we are interested in. Since we are really only interested in the image of the class under the map in cohomology induced by the map of coefficients ℚ → ℚ/ℤ, we obtain the following list:

Theorem 4.15

[\frac{(m − 1)!}{(2k)!}T (c_{4k+2})]_{ℚ/ℤ} = [\frac{1}{2}a_{4k+1}]_{ℚ/ℤ} ∈ H^{4k+1}(U/O; ℚ/ℤ)
[Ω(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = \begin{cases}[\frac{1}{2}ch_{0}]_{ℚ/ℤ} ∈ H^{0}(BO × ℤ, ℚ/ℤ) & 4k + 1 = n \\ 0 ∈ H^{4k}(BO × ℤ; ℚ/ℤ) & 4k + 1 > n \end{cases}
[Ω^{2}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = 0 ∈ H^{4k−1}(O, ℚ/ℤ)
[Ω^{3}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = 0 ∈ H^{4k−2}(O/U, ℚ/ℤ)
[Ω^{4}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = 0 ∈ H^{4k−3}(U/Sp, ℚ/ℤ)
[Ω^{5}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = 0 ∈ H^{4k−4}(BSp × ℤ, ℚ/ℤ)
[Ω^{6}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = [\frac{1}{2}y_{4k−5}]_{ℚ/ℤ} ∈ H^{4k−5}(Sp, ℚ/ℤ)
[Ω^{7}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = [\frac{1}{2}(c_{4k−6} + c_{2}c_{4k−8} + ⋯ + c_{2k−4}c_{2k−2})]_{ℚ/ℤ} ∈ H^{4k−6}(Sp/U, ℚ/ℤ)

(4.16)

For n ≥ 8, the answer can be read off from the list by reduction mod 8 by Bott periodicity.

In particular, the natural transformation d_{B,4k+1−n}^{−n} vanishes for n congruent to 2, 3, 4, 5 mod 8, and, if k > 0, also for n ≡ 1 (mod 8). In the other cases, since the universal classes are non-trivial, there are non-trivial examples.

4.4.9 Proof. As observed above, we simply have to take the cohomology classes on the right-hand side of equation (4.14), divide them by 2, and then apply the map from rational cohomology to cohomology with coefficients in ℚ/ℤ. Finally through Ω^{n}α we pull back the result to Ω^{n}U/O. In this step we use the results of Subsection B.2.

4.4.10 Note first that x_{n} := c_{4k+1−n} (n even), and respectively, x_{n} := (2k− \frac{n−1}{2})!ch_{4k+1−n} (n odd) belong to the integral lattice in rational cohomology. Therefore [\frac{1}{2}(Ω^{n}α)^{∗}Ω^{n}x_{n}]_{ℝ/ℤ} = 0 if \frac{1}{2}(Ω^{n}α)^{∗} preserves the integral lattices. This is the case whenever Ω^{n}α maps the Chern classes to twice a generator of the integral cohomology, i.e. if Ω^{n}U/O equals O, O/U, or U/Sp by the table B.2.5. This observation accounts for the zeros for n = 2, 3, 4 in the theorem.

4.4.11 Because of Proposition A.10

[T (Ω^{5}(\frac{(m − 1)!}{(2k)!}c_{4k+2}))]_{ℚ/ℤ} = [ΩT (Ω^{4}(\frac{(m − 1)!}{(2k)!}c_{4k+2}))]_{ℚ/ℤ} = 0 ,

we obtain the zero for n = 5.

4.4.12 We now discuss the case n = 1. We have [T (c_{4k+2})]_{ℚ/ℤ} = l_{∗}(w_{2k} ∪ w_{2k+1}), where l_{∗} is induced by the map of coefficients ℤ/2ℤ → ℚ/2ℤ ⟶^{·1/2} ℚ/ℤ (compare 4.1.5 and Corollary 4.8). If k > 0, then Ω(w_{2k} ∪ w_{2k+1}) = 0 since the loop map is applied to a decomposable class. Note that \frac{(m−1)!}{(2k)!} = 1 in this case. Thus

[Ω(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = l_{∗}Ω(w_{2k} ∪ w_{2k+1}) = 0

for k > 0.

4.4.13 For the calculation of [Ω^{7}(\frac{(m−1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} we proceed as follows. The class (2k − 3)!ch_{4k−6} ∈ H^{4k−6}(Sp/U, ℚ) belongs to the integral lattice. In fact, if we write (2k − 3)!ch_{4k−6} = σ_{4k−6}(c_{2}, c_{4}, …, c_{4k−6}) with the Newton polynomial σ_{4k−6}, then the right-hand side can be interpreted as an integral cohomology class in H^{4k−6}(Sp/U, ℤ). We now have

l_{∗}[σ_{4k−6}(c_{2}, c_{4}, …, c_{4k−6})]_{ℤ/2ℤ} = [Ω^{7}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} .

The cohomology H^{4k−6}(Sp/U, ℤ/2ℤ) is an exterior algebra generated by [c_{2}]_{ℤ/2ℤ}, [c_{4}]_{Z/2ℤ}, . . .. Considered in this algebra, the Newton polynomial

σ_{4k−6}(c_{2}, c_{4}, …, c_{4k−6})

= −(2k − 3) ∑_{i_{1}+2i_{2}+...(2k−3)i_{2k−3}=2k−3}(−1)^{i_{1}+···+i_{2k−3}} \frac{(i_{1} + ⋯ + i_{2k−3} − 1)!}{i_{1}! … i_{2k−3}!}c_{2}^{i_{1}} … c_{4k−6}^{i_{2k−3}}

simplifies considerably and gives exactly the expression asserted, if we use that

l_{∗}[c_{2l}]_{ℤ/2ℤ} = [\frac{1}{2}c_{2l}]_{ℚ/ℤ} .

□

4.5 The relation with ordinary characteristic classes

4.5.1 Let us consider the fibration

Ω^{n}U/O →^{Ω^{n}i} Ω^{n}BO → Ω^{n}BU .

We have constructed and calculated the universal cohomology class

[Ω^{n}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} ∈ H^{4k+1−n}(Ω^{n}U/O, ℚ/ℤ) .

If this class would be of the form (Ω^{n}i)^{∗}u for some u ∈ H^{4k+1−n}(Ω^{n}BO, ℚ/ℤ), then the invariant d_{B,4k+1−n}^{−n}(x), x ∈ U_{4k+1−n}^{−n}(B), could be expressed in terms of familiar characteristic classes of the element x ∈ KO^{−n}(B).

4.5.2 In the case n = 0 we indeed have

[Ω^{n}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = [T (c_{4k+2})]_{ℚ/ℤ} = i^{∗}l_{∗}(w_{2k} ∪ w_{2k+1})

(above we have written w_{l} for i^{∗}w_{l} in order to save notation, but in the current discussion it makes sense not to omit i^{∗}). In particular we can extend d_{B,4k+1}^{0} to all of KO^{0}(B) by setting d_{B,4k+1}^{0}(x) := l_{∗}(w_{2k} ∪ w_{2k+1}).

4.5.3 In the case n = 1 and k = 0 it is obvious that the class comes from Ω^{1}BO. As we have seen in example 3.3.2, if x ∈ KO^{−1}(X) is represented by a family of anti-selfadjoint real Fredholm operators (F_{b})_{b∈B}, then d_{B,0}^{0}(x) is represented by the locally constant ℝ/ℤvalued function b ↦ [\frac{1}{2} dim ker F_{b}]_{ℝ/ℤ}

4.5.4 In the case n = 6 the class

[Ω^{6}(\frac{(m − 1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} = [Ω^{6}(\frac{1}{2k(2k − 1)(2k − 2)}T (c_{4k+2}))]_{ℚ/ℤ}

definitely is not a pull-back from Ω^{6}BO = Sp/U. In fact H^{∗}(Sp/U, ℚ/ℤ) is concentrated in even degrees, while our class is of odd degree. We see that in this case our invariant d_{B,4k−5}^{6} is more exotic and therefore more interesting. Unfortunately, we haven’t been able to produce simple examples of for this invariant in Section 3.3.

4.5.5 The situation is unclear at the moment for n = 7. However, we suspect that the map Sp/U → U/O induces a surjection in cohomology with ℤ/2ℤ-coefficients, which would imply that all our classes pull back from U/O.

4.6 Extendibility

4.6.1 Given x ∈ KO^{−n}(B), in order to define d_{4k+1−n}^{−n}(x) using topology we had to assume that x ∈ U_{∞}^{−n}(B). Our analytic definition however works under the weaker condition that x ∈ U_{4k+1−n}^{−n}(B). Of course, if B^{4k+1−n} ⊂ B denotes a 4k + 1 − n-skeleton, we have x_{|B^{4k+1−n}} ∈ U_{∞}^{−n}(B^{4k+1−n}). We also have seen that d_{4k+1−n}^{−n}(x_{|B^{4k+1−n}}) = d_{4k+1−n}^{−n}(x)_{|B^{4k+1−n}} determines d_{4k+1−n}^{−n}(x) uniquely. The interesting feature of the analytic definition is that it shows that d_{4k+1−n}^{−n}(x_{|B^{4k+1−n}}) admits an extension from B^{4k+1−n} to B.

In the following Lemmas we give an alternative proof of this property.

4.6.2 Let f: B → Ω^{n}BO be a map. Assume that the restriction f^{r} := f_{|B^{r}}: B^{r} → Ω^{n}BO of f to a r-skeleton B^{r} of B factors over a map g^{r}: B^{r} → Ω^{n}U/O (i.e. f^{r} = Ω^{n}i ∘ g^{r}, where i: U/O → BO is as above). Assume further that r + n ≡ 1 (mod 8). Let R be some abelian group.

Lemma 4.17 If x ∈ H^{k}(Ω^{n}U/O, R) satisfies 2x = 0, then the class (g^{r})^{∗}(x) extends from B^{r} to B.

Since the map H^{r}(B; R) → H^{r}(B^{r}; R) is injective, this extension is unique. Note that the Lemma in particular applies to the cohomology classes listed in Theorem 4.15.

4.6.3 Proof. Let K(R; r) denote the Eilenberg-Mac Lane space which represents the functor H^{r}(…, R). We represent the cohomology class x by a map x: Ω^{n}U/O → K(R; r). It suffices to show that (g^{r})^{∗}(x) extends to a r + 1-skeleton B^{r+1} of B (such that B^{r} ⊂ B^{r+1}). In fact, it then further extends to B since the inclusion B^{r+1} → B is a r + 1equivalence.

4.6.4 The universal example is given by the space B = K^{r+1} which is obtained from Ω^{n}U/O by attaching r+1-cells in such a way as to kill the kernel of (Ω^{n}i)_{r} : π_{k}Ω^{n}(U/O) → π_{r}(Ω^{n}BO). Here f is obtained from Ω^{n}i: Ω^{n}(U/O) → Ω^{n}BO, which extends to some map f: K^{r+1} → Ω^{n}BO by the construction of K^{r+1} (and elementary obstruction theory), and g^{r} is the inclusion of the r-skeleton of Ω^{n}(U/O) (and therefore of K^{r+1}) into Ω^{n}(U/O).

In our case we have π_{r}(Ω^{n}U/O) ≅ ℤ and ker(Ω^{n}i)_{r} ≅ 2ℤ (as follows from Bott periodicity and the long exact homotopy sequence (in low degrees) of U/O → BO → BU). If ϕ: S^{r} → Ω^{n}(U/O) represents a generator of π_{r}(Ω^{n}U/O), and if h: S^{r} → Ω^{n}U/O represents twice this generator, i.e. a generator of ker(Ω^{n}i)_{r}, then h^{∗}(g^{r})^{∗}x = 2ϕ^{∗}(g^{r})^{∗}x = 0. Thus the map x ∘ h is null homotopic, and therefore (g^{r})^{∗}x extends to K^{r+1}. □

4.6.5 Let still f: B → Ω^{n}BO be a map and assume that the restriction f^{4k+1−n} := f_{|B^{4k+1−n}}: B^{4k+1−n} → Ω^{n}BO of f to a 4k + 1 − n-skeleton B^{4k+1−n} of B factors over a map g^{4k+1−n}: B^{4k+1−n} → Ω^{n}U/O (i.e. f^{4k+1−n} = Ω^{n}i ∘ g^{4k+1−n}, where i: U/O → BO is as above).

Lemma 4.18 If x = [Ω^{n}(\frac{(m−1)!}{(2k)!}T (c_{4k+2}))]_{ℚ/ℤ} ∈ H^{4k−n+1}(Ω^{n}U/O) is one of the classes of Theorem 4.15, then (f^{4k+1−n})^{∗}x extends from B^{4k+1−n} to B.

4.6.6 Note that half of the cases are already covered by Lemma 4.17, namely whenever the dimension condition is satisfied, i.e. when (4k + 1 − n) + n ≡ 1 (mod 8), in other words, if k is even.

4.6.7 Moreover, the cases n ≡ 0 (mod 8) as well as n ≡ 1 (mod 8) (and 4k + 1 = n) are trivial, because in these cases we have seen that the characteristic class x already pulls back from Ω^{n}BO to Ω^{n}U/O: it is expressed in terms of Stiefel-Whitney classes in the first case, and in terms of the dimension of the bundle in the second case.

4.6.8 We use the proof Lemma 4.17 to deal with the remaining cases. This proof shows that it suffices to treat the case B^{4k+1−n} = S^{4k+1−n} and to show that the pullback class (f^{4k+1−n})^{∗}x vanishes for arbitrary f: S^{4k−n+1} → Ω^{n}U/O (and therefore extends over the disc D^{4k+2−n}).

4.6.9 Observe that, by Equation (4.14), the cohomology class x is obtained as pull back of \frac{1}{2}c_{4k+1−n} or \frac{1}{2}(2k −(n−1)/2)!ch_{[4k−(n−1)]} from U or BU, respectively (depending on the parity of n). However, on all spheres the Chern character is integral, i.e. for an arbitrary map f: S^{k} → BU, f^{∗}ch ∈ H^{∗}(S^{k}; ℤ).

4.6.10 If n ≡ 7 (mod 8) and k odd (and 4k > n) then 4k+1−n ≥ 4. This implies \frac{1}{2}(2k− (n − 1)/2)! ∈ ℤ. Therefore by 4.6.9 the cohomology class \frac{1}{2}(2k − (n − 1)/2)!ch_{[4k−(n−1)]} pulls back to 0 in H^{∗}(S^{4k−(n−1)}; ℚ/ℤ) for an arbitrary map S^{4k−(n−1)} → BU. As observed in 4.6.6, n ≡ 7 (mod 8) and k even is covered by Lemma 4.17.

4.6.11 For n ≡ 6 (mod 8) and an arbitrary map f: S^{4k−n+1} → U, by Lemma A.6 and Theorem B.4

f^{∗}(\frac{1}{2}c_{4k+1−n}) = Σ^{−1}F^{∗}(\frac{1}{2}Ωc_{4k+1−n}) = Σ^{−1}(F^{∗} \frac{1}{2}(2k − \frac{n}{2})!ch_{[4k−n]}).

Here, Σ: H^{4k−n}(S^{4k−n}) → H^{4k−n+1}(S^{4k−n+1}) is the suspension isomorphism and

F: S^{4k−n} → ΩU = BU × ℤ

is the adjoint of f: ΣS^{4k−n} = S^{4k−n+1} → U. Again, if 4k − n ≥ 4 then (2k − n/2)! ∈ 2ℤ and therefore by 4.6.9 F^{∗}(\frac{1}{2}(2k − n/2)!ch_{[4k−(n−1)]} = 0. However, if 4k − n = 2 then, since n ≡ 6 (mod 8) k is even and therefore (4k + 1 − n) + n ≡ 1 (mod 8), such that this case is covered by Lemma 4.17.

4.6.12 If n ≡ 2, 3, 4, 5 (mod 8), then x = 0, which trivially extends. This concludes the proof of Lemma 4.18. □

A Transgression

A.1 Transgression in cohomology

A.1.1 In this section, we want to recall the general definition of transgression and its basic properties. Special cases are “suspension” or “looping”. All of this is well known, and included here for the convenience of the reader.

A.1.2 The situation is the following: let f: E → B be a map, and b ∈ B a point. Write F_{b} := f^{−1}(B). Let i: F_{b} → E be the inclusion. Let H^{∗} be any (generalized) cohomology theory. In the following, the loop spaces ΩB are defined with respect to the basepoint b. Suspensions are reduced suspensions.

A.1.3 The adjoint of the identity map ΩB → ΩB gives a canonical map ΣΩB → B. This induces H^{∗}(B) → H^{∗}(ΣΩB).

Definition A.1 We define the loop map

Ω: H^{∗}(B) → H^{∗−1}(ΩB).

as the composition of H^{∗}(B) → H^{∗}(ΣΩB) with the suspension isomorphism H^{∗}(ΣΩB) → H^{∗−1}(ΩB)

By construction and functoriality of the suspension isomorphism, the loop map is functorial, too

A.1.4 Given the map f: E → B, consider the cofibration sequence E → Zf → Cf, where Zf is the mapping cylinder and Cf the mapping cone. The inclusion B → Zf is a homotopy equivalence. The long exact sequence in cohomology of this cofibration gives

H^{k}(B)
∥
⋯ → H^{k}(Cf) → H^{k}(Zf) ⟶^{f^{∗}} H^{k}(E) → ⋯

In particular, H^{k}(Cf) maps surjectively onto ker(f^{∗}) ⊂ H^{k}(B).

A.1.5 Consider now ΩCf. Since the composition F ⟶^{i} E ⟶^{f} B is the constant map to b, we can define a canonical map

l : F → ΩCf , x ↦ c_{x},

where c_{x} is the loop in Cf with c_{x}(0) = b ∈ B, c_{x}(t) = (x, t) ∈ E × (0, 1) ⊂ Cf, and c_{x}(1) is the cone point in Cf.

Mapping B to the second cone point gives the second map j in the cofibration sequence B → Cf →^{j} ΣE. From this we conclude that the kernel of H^{k}(Cf) → H^{k}(B) equals im(j^{∗}). The composition F → ΩCf → ΩΣE can be factored as F ⟶^{i} E → ΩΣE, where the second map is the adjoint of the identity map ΣE → ΣE.

A.1.6

Definition A.2 We define the transgression T: H^{k}(B) ⊃ ker(f^{k}) → H^{k−1}(F)/ im(i^{∗}) as the composition

ker(f^{∗}) ≅ H^{k}(Cf)/ im(j^{∗}) ⟶^{Ω} H^{k−1}(ΩCf)/ im((Ωj)^{∗}) → H^{k−1}(F)/ im(i^{∗}) .

Note that for the second map we used the factorization which shows that im((Ωj)^{∗}) goes into im(i^{∗}).

It is clear from the construction that transgression is natural with respect to the map f : E → B, i.e. given a diagram

\begin{matrix}E′ & →^{H} & E \\ f′ ↓ & & f ↓ \\ B′ & →^{h} & B \end{matrix}

we have an equality of the form T′ ∘ h^{∗} = H^{∗} ∘ T.

A.1.7

Definition A.3 The elements of ker(f^{∗}) ⊂ H^{∗}(B) are called transgressive. These are the classes whose transgression is defined.

A.1.8

Lemma A.4 The transgression in the fibration ΩB → P B → B, where P B is the (contractible) space of paths ending at b coincides with the loop map.

Proof. Carry out the construction. If f: P B → B is the start point projection, use the homotopy equivalence Cf → B which maps (p, s) ∈ P B × (0, 1) ⊂ Cf to p(s) (recall that s = 1 corresponds to the cone point). □

A.1.9

Lemma A.5 The transgression of B ↪ CB → ΣB is the suspension isomorphism Σ: H^{k+1}(ΣB) → H^{k}(B).

Proof. Carry out the construction. Use the “folding” homotopy equivalence Cf → ΣB (where f: CB → ΣB is the projection). The composition of ΣB → Cf with this homotopy equivalence is the identity map. Starting with H^{∗}(ΣB), we have to pull back with this map and then use the suspension isomorphism (by naturality of the latter). □

A.1.10 Let f: ΣX → Y be a map with adjoint F: X → ΩY. Then we have a commutative diagram of fibrations

\begin{matrix}X & ⟶ & CX & ⟶ & ΣX \\ ↓^{F} & & ↓ & & ↓^{f} \\ ΩY & ⟶ & P Y & ⟶ & Y \end{matrix} .

Lemma A.6 Then for each a ∈ H^{∗}(Y),

F^{∗}(Ωa) = Σf^{∗}a .  (A.7)

Proof. This follows from naturality of transgression and the fact that both Ω and Σ are transgression homomorphisms by Lemma A.4 and Lemma A.5. □

A.1.11 In the construction of the transgression, we consider in particular the following commutative diagram of maps:

\begin{matrix}F & ⟶^{i} & E & ⟶^{f} & B \\ ↓^{id} & ↓ & & ↓ \\ F & ⟶ & CE & ⟶ & Cf \\ ↑^{id} & ↑ & & ↑ \\ F & ⟶ & CF & ⟶ & ΣF \end{matrix}  (A.8)

By naturality, the transgression homomorphism in E ⟶^{f} B is determined by the transgression in CF → ΣF (this is of course, what we used in the construction), since H^{∗}(Cf) → H^{∗}(B) surjects onto the transgressive classes.

A.1.12

Lemma A.9 Let Φ: H^{∗} → h^{∗} be a natural transformation between generalized cohomology theories. Transgression commutes with this natural transformation.

Proof. Let f: E → B be a continuous map. First observe that by naturality Φ maps ker(H^{∗}(f)) to ker(h^{∗}(f)) and im(H^{∗}(i)) to im(h^{∗}(i)), so that the assertion makes sense.

The construction of the transgression homomorphism only uses maps induced from continuous maps between topological spaces (and their inverses) and the suspension isomorphism. By definition, a natural transformation between cohomology theories is compatible with such homomorphisms, and therefore also with transgression. □

A.1.13 Let F ⟶^{i} E ⟶^{f} B be a sequence of maps as above. This gives rise to the transgression homomorphism T: H^{∗}(B) ⊃ ker(f^{∗}) → H^{∗−1}(F)/ im(i^{∗}).

Applying the loop space functor we also get the sequence of maps

ΩF ⟶^{Ωi} ΩE ⟶^{Ωf} ΩB,

with associated transgression homomorphism

T_{Ω} : H^{∗}(ΩB) ⊃ ker(Ωf^{∗}) → H^{∗−1}(ΩF)/ im((Ωi)^{∗}) .

Proposition A.10 If x ∈ H^{n}(B) is transgressive, then

ΩT (x) = T_{Ω}(Ωx) ∈ H^{n−2}(ΩF)/ im((Ωi)^{∗}).

Proof. We obtain the following commutative diagram

ΣΩF ⟶^{ΣΩi} ΣΩE ⟶^{ΣΩf} ΣΩB
↓^{p_{F}}  ↓  ↓^{p_{B}}
F ⟶^{i} E ⟶^{f} B

(A.11)

where the vertical maps are adjoints of the identity maps Ω· → Ω·. Since we work with the reduced suspension, the inclusion ΣΩF ↪ ΣΩE is the fiber of ΣΩf. By naturality of the transgression, p_{F}^{∗} (T x) = T_{ΣΩ}(p_{B}^{∗}x) for each transgressive class x ∈ H^{∗}(B). The suspension isomorphism maps by definition p_{F}^{∗} (T x) to Ω(T x) and p_{B}^{∗}x to Ωx. By Lemma A.9 transgression commutes with the suspension isomorphism (indeed the suspension isomorphism can be interpreted as a natural transformation between cohomology theories). Therefore we have Ω(T x) = T_{Ω}(Ωx). □

A.2 Transgression and products

A.2.1 Let Δ: B → B × B be the diagonal map. We still consider the map f : E → B.

Definition A.12 1. A class x ∈ H^{∗}(B) is called a non-trivial product, if x = Δ^{∗}y for some y ∈ H^{∗}(B × B) such that (id_{B} × {b})^{∗}y = 0 = ({b} × id_{B})^{∗}y.

2. We say that the first factor of a non-trivial product x is transgressive, if x = Δ^{∗}y for an y such that (f ×id_{B})^{∗}y = 0 ∈ H^{∗}(E ×B), similarly we define that the second factor is transgressive.

A.2.2 Note that if one of the factors of a product is transgressive, then so is the product.

Proposition A.13 The transgression of a non-trivial product with at least one transgressive factor is zero.

Proof. This follows from naturality of the transgression. Consider the diagram

F ⟶ E ⟶ B
↓^{id} ↓^{(id_{E}×f)∘Δ} ↓^{Δ}
F ⟶ E × B ⟶ B × B
↑^{id} ↑^{id_{B}×{b}} ↑^{id_{B}×{b}}
F ⟶ E ⟶ B

Let us assume that x is transgressive in the first factor. By T_{i} we denote the transgressions associated to the corresponding rows. Then we have T_{1}(x) = T_{2}(y) = T_{3}((id_{B} ×{b})^{∗}y) = 0, since (id_{B} × {b})^{∗}y = 0. □

A.2.3 Let us consider the following example. Define N := T^{2} \ (D^{2})^{∘}, i.e. N is the two torus with an open disc removed. Let f: N → T^{2} be the map which collapses the boundary of N to one point. On the one hand, the fundamental class [T^{2}] ∈ H^{2}(T^{2}, ℤ) is transgressive. On the other hand, [T^{2}] is a non-trivial product of 1-dimensional cohomology classes, and none of the factors is transgressive.

Collapsing the complement of an open disc in T^{2} to a point gives a degree 1 map g: T^{2} → S^{2}. If we write S^{2} = ΣS^{1}, we then get a diagram

S^{1} ⟶ CS^{1} = D^{2} ⟶ ΣS^{1} = S^{2}
↑^{id}_{S^{1}}  ↑  ↑^{g}
S^{1} ⟶  N  ⟶^{f}  T^{2}

Here, g^{∗}[S^{2}] = [T^{2}], where [S^{2}] ∈ H^{2}(S^{2}, ℤ) is the fundamental class. By naturality, T ([T^{2}]) = T ([S^{2}]) = [S^{1}] ∈ H^{1}(S^{1}, ℤ) is the fundamental class of S^{1}, in particular nonzero.

This shows that at least one of the factors in Proposition A.13 has to be transgressive for the assertion to hold.

A.3 Transgression in ordinary cohomology and the relation with the Bockstein

A.3.1 We now want to describe how one can construct the transgression in ordinary singular cohomology with coefficients on the level of chains. Let f: E → B be a map with fiber i : F ↪ E over b ∈ B. Let R be an abelian group. Assume that x ∈ H^{k}(B, R) is transgressive, i.e. we have f^{∗}x = 0. We choose a cocycle c ∈ C^{k}(B, R) representing x. Then the cocycle f^{∗}c is a boundary, i.e. there exists a chain c_{0} ∈ C^{k−1}(E, R) with dc_{0} = f^{∗}c. The restriction of c_{0} to F is closed, since di^{∗}c_{0} = i^{∗}dc_{0} = i^{∗}p^{∗}c. It follows that i^{∗}c_{0} represents a cohomology class [i^{∗}c_{0}] ∈ H^{k−1}(F, R). The cocycle c_{0} is well defined only up to closed cocycles in E. It follows that the class [c_{0}] is well defined only up to the image of i^{∗}. Hence we get a well-defined class T̃(x) ∈ H^{k−1}(F)/ im(i^{∗}).

A.3.2

Proposition A.14 We have T̃(x) = T (x).

Proof. The recipe described in the proposition defines a transformation T̃ which is again natural with respect to the map f : E → B. As explained in the Remark A.1.11 it must coincide with the transgression T if it does so in the special case of the cofibration B → CB → ΣB. But in this case the above description produces exactly the suspension isomorphism which is by definition the transgression map T. □

A.3.3 If f : E → B is a map of smooth manifolds and R = ℝ, then we could replace the singular cochains by differential forms and construct T̃ on the level of forms. Again we get T̃ = T.

A.3.4 For a cohomology class x ∈ H^{∗}(X, ℤ) let x^{ℚ} ∈ H^{∗}(X, ℚ) denote the image of x under the canonical coefficient homomorphism ℤ → ℚ. Let n ∈ ℕ and β be the Bockstein transformation associated to the sequence

0 → ℤ ⟶^{·n} ℤ → ℤ/nℤ → 0 .

A.3.5 Let x ∈ H^{k}(X, ℤ) be such that nx is transgressive. Note that then q^{ℚ} is transgressive, too. Since nf^{∗}x = 0 by the Bockstein exact sequence there exist u ∈ H^{k−1}(E, ℤ/nℤ) with β(u) = f^{∗}x. Recall that T (nx) is an equivalence class of cohomology classes.

Proposition A.15 1. We have

T (nx) ∋ i^{∗}u,  (A.16)

2.

nT (x^{ℚ}) ∋ i^{∗}u^{ℚ}.  (A.17)

Proof. We use the description of the transgression on the singular cochain level given in Proposition A.14. Let c be an integral cocycle representing x. Let c_{0} be an integral cochain of E with dc_{0} = nf^{∗}c. Then i^{∗}c_{0} represents T (nx).

The reduction of c_{0} modulo nℤ becomes closed and therefore represents a cohomology class u ∈ H^{k−1}(E, ℤ/nℤ). By the explicit construction of the Bockstein homomorphism, β(u) = f^{∗}x. Equation (A.16) follows.

Since transgression commutes with the passage to rational coefficients by Lemma A.9, Equation (A.17) follows from Equation (A.16). □

B COHOMOLOGY OF BO, BU AND THEIR LOOP SPACES 47

B Cohomology of BO, BU and their loop spaces

B.1 The cohomology

B.1.1 In this appendix, we summarize the main results about the cohomology of BO, BU and their loop spaces, and the relations between them, including the determination of the transgression homomorphisms. These results are all classical, and almost all of them can be found in Cartan’s [9], where these calculations were essential in his cohomological proof of Bott periodicity. Since they are scattered over these papers, we collect them here in more convenient form. All results without a proof or a different reference can be found in [9].

B.1.2 Bott periodicity gives canonical (up to homotopy) homotopy equivalences between Ω^{n}BO and other classical spaces summarized in the following list.

Theorem B.1

n − 1  −1  0  1  2  3  4  5  6  7
Ω^{n}U/O U/O BO × ℤ  O  O/U  U/Sp BSp × ℤ Sp Sp/U U/O
or  SO × ℤ/2 SO/U × ℤ/2

This extends 8-periodically.

In the following, we will frequently identify the (cohomology of) different loop spaces of spaces in this table using the corresponding homotopy equivalence without further mentioning it. Note that we have done so already throughout the body of the paper.

B.1.3 In the following, L(x_{i_{1}}, x_{i_{2}}, …) denotes a polynomial algebra in the generators x_{i}, where by convention x_{i} has (cohomological) degree i, and E(y_{i_{1}}, y_{i_{2}}, …) denotes an exterior algebra, with similar degree conventions for the generators.

B.1.4 In the following list, we describe the cohomology of the connected component of the base point in Ω^{k}BO. Note that we “rename” some of the usual characteristic classes like the Pontryagin classes: p_{4} is a cohomology class in H^{4} etc.

Theorem B.2

k  H^{∗}(Ω^{k}BO_{0}, ℤ)  H^{∗}(Ω^{k}BO_{0}, ℤ/2ℤ) H^{∗}(Ω^{k}BO_{0}, ℤ[\frac{1}{2}])
0 L(p_{4}, p_{8}, …) ⊕ 2-Tors L(w_{1}, w_{2}, …)  L(p_{4}, p_{8}, …)
1 ∗ L(d_{1}, d_{3}, …) E(v_{3}, v_{7}, …)
2  L(u_{2}, u_{6}, …)
3  E(a_{1}, a_{5}, …)
4  L(y_{4}, y_{8}, …)
5  E(y_{3}, y_{7}, …)
6 L(u_{2}, u_{6}, …)^{∗} E(c_{2}, c_{4}, …) L(c_{2}, c_{6}, …)
6 \frac{L[c_{2}, c_{4}, …]}{∑_{i+j=2k}(−1)^{i}c_{2i}c_{2j}}
7 E(a_{4k+1}) ⊕ 2-Tors  E(w_{1}, w_{2}, …)  E(a_{1}, a_{5}, …)

We add the following detailed explanations, using the description of the loop spaces as in Theorem B.1.

1. (a) H^{∗}(BO; ℤ) contains a subalgebra isomorphic to the quotient by its torsion. (b) This is a polynomial algebra L(p_{4}, p_{8}, …).

(c) The torsion is annihilated by 2, it is the image of Bockstein.

(d) Reduction mod 2 maps p_{4k} to (w_{2k})^{2}.

(e) The classes w_{2k+1} ∈ H^{2k+1}(BO, ℤ/2ℤ) have unique lifts to H^{2k+1}(BO, ℤ) which we also denote by w_{2k+1}.

(f) The same is true for every class in degree k for k not divisible by 4, since in these degrees H^{k}(BO, ℤ) is annihilated by 2.

2. Most complicated is the cohomology of SO with ℤ-coefficients, for reasons of space simply denoted ∗ in the list (case k = 1). We can say the following about it.

(a) The torsion in H^{∗}(SO, ℤ) is annihilated by 2, it is the image of Bockstein.

(b) The quotient of H^{∗}(SO, ℤ) by its torsion is an exterior algebra E(\overline{v_{3}}, \overline{v_{7}}, …). It does not split back to H^{∗}(SO, ℤ) because of the product structure (compare with H^{∗}(SO, ℤ/2ℤ)).

(c) But of course, each monomial \overline{v_{i_{1}}} … \overline{v_{i_{s}}} has an inverse image v_{i_{1}} … v_{i_{s}} ∈ H^{∗}(SO, ℤ) (only additive! no multiplicative structure) which is well defined up to torsion, and the products are correct up to torsion.

3. The integral cohomology of Sp/U (case k = 6) is the dual of L(u_{2}, u_{6}, …). This shows in particular, that it is torsion-free. As a ring, it is the quotient of L(c_{2}, c_{4}, …) by the ideal generated by the elements ∑_{i+j=2k}(−1)^{i}c_{2i}c_{2j}.

4. (a) H^{∗}(U/O, ℤ) contains a subalgebra isomorphic to the quotient by its torsion,

this is an exterior algebra E(a_{1}, a_{5}, …).

(b) The torsion is annihilated by 2, it is the image of Bockstein.

(c) Reduction mod 2 maps a_{4k+1} to w_{2k}w_{2k+1}+β(w_{4k}+w_{2}w_{4k−2}+⋯+w_{2k−2}w_{2k+2}).

B.1.5 We also need the complex case, i.e. BU (and will later relate BO to BU). The case of BU is of course much easier because of 2-periodicity, and since the cohomology does not contain torsion.

Theorem B.3

k natural homotopy equivalence of Ω^{k}BU to H^{∗}(Ω^{k}BU_{0}, ℤ),
0  BU  L(c_{2}, c_{4}, …)
1  U  E(c_{1}, c_{3}, …)
2  BU × ℤ.

B.1.6 We now describe the effect of the loop map Ω: H^{∗}(X) → H^{∗−1}(ΩX) for integral cohomology and some of the spaces in B.1.

Theorem B.4

Space X ΩX x ∈ H^{∗}(X, ℤ) Ω(x) ∈ H^{∗−1}(ΩX, ℤ)
BU  U  c_{2k}  c_{2k−1}
U  BU × ℤ  c_{2k−1}  (k − 1)!ch_{2k−2}
BSp  Sp  y_{4k}  y_{4k−1}
BO  O  p_{4k}  2v_{4k−1} + Tors

Proof. We only have to prove that Ω(c_{2k−1}) = (k − 1)!ch_{k−2}. For this, observe that

c_{2k−1} = Ω(c_{2k}) = Ω((k − 1)!ch_{2k}) = (k − 1)!Ω(ch_{2k}),

since the other summands in ch_{2k} are decomposable (and the loop map applied to a decomposable class is zero by Proposition A.13. Now the Chern character is compatible with Bott periodicity, and therefore Ω^{2}(ch_{2k}) = ch2k − 2. Consequently

Ωc_{2k−1} = (k − 1)!Ω^{2}(ch_{2k}) = (k − 1)!ch_{2k−2}.

B.1.7

Lemma B.5 The natural map

l_{∗} : H^{4k+1}(BO, ℤ/2ℤ) → H^{4k+1}(BO, ℚ/ℤ)

of Equation (4.3) factors through the image of the Bockstein homomorphism

β : H^{4k+1}(BO, ℤ/2ℤ) → H^{4k+2}(BO, ℤ).

Proof. We have the following map of long exact sequences

\begin{aligned}→^{2×_{∗}} & H^{4k+1}(BO, ℤ) & →^{q_{∗}} & H^{4k+1}(BO, ℤ/2ℤ) & →^{β} & H^{4k+2}(BO, ℤ) & → \\ & \frac{1}{2}×_{∗} ↓ & & l_{∗} ↓ & & ∥ \\ → & H^{4k+1}(BO, ℚ) & → & H^{4k+1}(BO, ℚ/ℤ) & →^{β′} & H^{4k+2}(BO, ℤ) & → \end{aligned} .

The assertion now follows from the fact that H^{4k+1}(BO, ℚ) = 0. □

B.2 Maps between loop spaces of BO

B.2.1 There is a large number of canonical maps between the different spaces in Theorem B.1 and in Theorem B.3 which are important for us and which are described in the following list.

B COHOMOLOGY OF BO, BU AND THEIR LOOP SPACES 51 1. The inclusion c: O → U (given by complexification).

2. The induced map Bc: BO → BU, which gives rise to Bc × id_{ℤ}: BO × ℤ → BU × ℤ 3. The inclusion q: U → Sp (given by tensoring with the quaternions)

4. the induced map Bq: BU → BSp

5. the inclusion f: U → O (given by forgetting the complex structure)

6. the induced map Bf: BU → BO

7. the inclusion j: Sp → U (given by forgetting the quaternionic structure)

8. the induced map Bj: BSp → BU, which gives rise to Bj×id_{ℤ}: BSp×ℤ → BU ×ℤ.

9. the projection p: U → U/O

10. the projection P: U → U/Sp

11. the inclusion of the fiber i: U/O → BO obtained by dividing the total space of the universal principal U-fibration U → EU → BU by O (here we use the fact that EU/O is a model for BO)

12. The fibration BO → BU constructed in 11

13. the (similar) inclusion of the fiber I: Sp/U → BU.

14. the (similar) inclusion of the fiber ϕ: O/U → BU.

15. a map α: U/O → U given as composition

U/O ⟶^{∼} ΩSp/U ⟶^{ΩI} ΩBU ⟶^{∼} U

where the first map is the Bott periodicity homotopy equivalence, and the third is the usual homotopy equivalence (which is also part of (complex) Bott periodicity).

16. a similar map β: U/Sp → U, given as composition

U/Sp ⟶^{∼} ΩO/U ⟶^{Ωϕ} ΩBU ⟶^{∼} U

B.2.2 The following relations hold between these maps. As usual, we will freely use the Bott periodicity homotopy equivalences of Theorem B.1 and Theorem B.3 to identify certain loop spaces with other spaces (therefore, strictly speaking, the following assertions are true up to homotopy).

1. It is a general fact in the theory of classifying spaces that one way to construct Bc in 2 is the fibration map of 12, which therefore can be identified with Bc. Reason: the identity map EU → EU, where the domain is considered as contractible Oprinciple bundle and the target as contractible U-principle bundle intertwines, using the inclusion c: O → U the structures as principle bundles. Therefore the induced map on the quotients is the map Bc.

2. The map 1 is obtained from 2 by applying the loop space transformation (and using the Bott periodicity identifications of ΩBO ⟶^{∼} O and ΩBU ⟶^{∼} U).

3. Similarly, 7 is obtained by applying the loop space functor to 8.

4. By construction, looping 13 gives 15.

5. By construction, looping 14 gives 16.

6. Cartan [9] proves that O/U → BU, i.e. 14 is obtained by applying the loop space functor to the inclusion 1, O → U.

7. Cartan [9] proves that looping 7 gives 13. This requires to check that his explicitly given maps Sp/U → BU and O/U → BU are the fiber inclusions we claim they are.

8. Cartan [9] also checks that looping 16 gives 8.

9. Cartan [9] proves that looping 15 gives 2. Strictly speaking, in this and the previous case he considers the corresponding maps of universal coverings SU/Sp → SU which loops to BSp → BU, and SO/SU → SO which loops to BO → BU. Since we know that U/Sp → U and O/U → O induce isomorphisms on π_{1}-level (all isomorphic to ℤ), the claim follows.

B.2.3 To conclude, we have shown that in the sequence

Bc × id: BO × ℤ → BU × ℤ
c: O → U
ϕ: O/U → BU
β: U/Sp → U
Bj × id: BSp × ℤ → BU × ℤ
j: Sp → U
I: Sp/U → BU
α: U/O → U
Bc × id: BO × ℤ → BU × ℤ

each map is obtained by looping the previous one (and applying Bott periodicity to identify the loop spaces with the next spaces in the list).

B.2.4 In the following table, we list the effect of the maps in cohomology. Again, this is due to Cartan [9], with a few exceptions easily obtained from his work. In these cases, the reason is indicated in the last column of the following table. Recall that we always only consider the cohomology of the connected component of the base point. “By looping” means that we know that certain maps are obtained from each other by applying the loop space functor (and some canonical homotopy equivalences), and that we know the effect of the natural loop map functor Ω: H^{∗}(X) → H^{∗−1}(ΩX) by Theorem B.4.

B COHOMOLOGY OF BO, BU AND THEIR LOOP SPACES 54 B.2.5

f: X → Y  x ∈ H^{∗}(Y, ℤ) f^{∗}(x_{0}) ∈ H^{∗}(X_{0}, ℤ) reason
BSp → BU c_{4k}  y_{4k}
c_{4k+2}  0
O/U → BU c_{4k+2}  2u_{4k+2}
c_{4k}  −2 · ∑_{0<i<2k}(−1)^{i}f^{∗}(c_{2i})/2 · f^{∗}(c_{4k−2i})/2
Sp/U → BU c_{2k}  \overline{c_{2k}}
c_{2k} mod 2 c_{2k}
Sp → U  c_{4k+1}  0  by looping
c_{4k+3}  y_{4k+3}  by looping
U/Sp → U c_{4k+1}  2a_{4k+1}  (since true dually in homology)
c_{4k+3}  0  (since true dually in homology)
U → Sp  y_{4k−1}  2c_{4k−1}  by looping (since products suspend to

zero)

BU → BSp y_{4k}  ∑_{i+j=2k}(−1)^{i}c_{2i}c_{2j}
SO → U c_{4k+3} 2v_{4k+3} + 2-Tor
c_{4k+1}  2-Tor
U → U/O  a_{4k+1}  2c_{4k+1}
U/O → U  c_{4k+1}  a_{4k+1} + Tors
c_{4k+3}  Tors
U → O v_{4k+3} c_{4k+3}
BO → BU c_{4k}  p_{4k}
c_{4k+2}  w_{2k+1}^{2}
c_{4k} mod 2  w_{2k}^{2}
BU → BO p_{4k}  ∑_{i+j=k}(−1)^{i}c_{2i}c_{2j}
w_{2k+1}  0
w_{2k}  c_{2k} mod 2
U/O → BO w_{k} mod 2  w_{k} mod 2

p_{4k} 0 mod 2 it maps to 0, and pulled back further to U it is also 0, i.e. no 2-torsion and no free part

B.2.6 In two cases, we have to take the different components into account:

f: X → Y  x ∈ H^{0}(Y, ℚ) f^{∗}(x) ∈ H^{0}(X, ℚ)
BSp × ℤ → BU × ℤ ch_{0}  2ch_{[0]}^{ℍ}
BO × ℤ → BU × ℤ ch_{0}  ch_{[0]}^{ℝ}

B.2.7

Lemma B.6 In the fibration U/O ⟶^{i} BO ⟶^{Bc} BU, the classes c_{4k+2}^{ℚ} ∈ H^{4k+2}(BU; ℚ) are transgressive.

Proof. The pull back of c_{4k+2} to BO is 2-torsion in integral cohomology, therefore vanishes in rational cohomology. □

References

[1] M. F. Atiyah and F. Hirzebruch. Vector bundles and homogeneous spaces. Proceedings of Symp. in pure Math, Vol.3, 1961. 2.1.6

[2] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Index theory for skew-adjoint Fredholm operators. Publ. I.H.E.S, Vol. 37 (1969), 5–26. 2.2.3

[3] M. F. Atiyah, V. K. Patodi, and I. M. Singer. Spectral asymmetry and Riemannian geometry I. Math.Proc.Camb.Phil.Soc., 77(1975), 43–69, 1975. 3.3.2

[4] N. Berline, E. Getzler, and M. Vergne. Heat Kernels and Dirac Operators. Springer-Verlag Berlin Heidelberg New York, 1992. 1.0.1, 2.1.11, 2.1.16

[5] J. M. Bismut and J. Cheeger. η-invariants and their adiabatic limits. J. AMS, 2(1989), 33–70. 3.3.3

[6] J. L. Brylinski. Loop spaces, characteristic classes, and geometric quantization. Birkhäuser, Progress in Math. 107, 1993.

[7] U. Bunke. η-form, index theory and Deligne cohomology. Preprint 2002. arXiv:math.DG/0201112 2.1.14, 2.1.15, 2.1.16, 3.2.8, 3.3.2

[8] U. Bunke and X. Ma. Index and secondary index theory for flat bundles with duality. Preprint 2002 : http://www.uni-math.gwdg.de/bunke/sample3.pdf 3.2.9

[9] H. Cartan. Démonstration homologique des théorèmes de périodicité de Bott. I, II, III Périodicité des Groupes d’Homotopie stables des Groupes classiques, d’apres Bott, Sém. H. Cartan 12 (1959/60), No.16, 16 p., No.17, 32 p., No.18, 9 p. (1961) 1.0.9, B.1.1, 6, 7, 8, 9, B.2.4

[10] J. Cheeger and J. Simons. Differential characters and geometric invariants. In LNM1167, pages 50–80. Springer Verlag, 1985. 2.1.8, 2.1.9, 2.1.20

[11] D. Freed. Dirac charge quantization and generalized differential cohomology. In: Surveys in differential geometry, International Press, Somerville, 2000, 129-194. arXiv:hep-th/0011220

[12] N. Hitchin. Harmonic spinors. Adv. in Math. 14 (1974), 1-55. 3.3.1, 3.3.3, 3.3.4

[13] M. Karoubi. K-theory. Springer-Verlag, 1978. 2.2.3

[14] J.W. Milnor and J.D. Stasheff. Characteristic classes. Annals of Mathematics Studies, Vol 76, Princeton University Press, Princeton, 1974. 4.1.6