The e-Invariant and Transfer Map Dissertation zur Erlangung des Doktorgrades vorgelegt von Yi-Sheng Wang betreut von Prof. Dr. Sebastian Goette an der Fakult¨atf¨urMathematik und Physik der Albert-Ludwigs-Universit¨at Freiburg Dezember 2016 Dekan: Prof. Dr. Gregor Herten I Gutachter: Prof. Dr. Sebastian Goette II Gutachter: Prof. Dr. Wolfgang Steimle Datum der Promotion: 22.3.2017 1 Contents 1 Introduction 3 Acknowledegment 10 2 The e-invariant 11 2.1 Topological K-theory with coefficients . 11 2.1.1 Homology sphere cobordisms . 11 2.1.2 Topological K-theory with Z=m-coefficients . 17 2.1.3 Topological K-theory with Q=Z-coefficients . 22 2.1.4 Topological K-theory with C=Z-coefficients . 24 2.1.5 Relative K-theory . 34 2.2 The e-invariant . 39 0 n 2.2.1 The e-invariant and Tor(KgaC (S )) . 40 2.2.2 A C=Z-valued invariant and the ξ~-invariant . 48 2.2.3 The ξ~-invariant of Seifert homology spheres . 50 3 Homotopy liftings and e∗ 63 3.1 Moore prespectra and rationalization . 64 3.1.1 Moore spaces and prespectra . 64 3.1.2 Rationalization of spaces and CW -spectra . 68 3.2 Homotopy liftings . 71 \ 3.3 Homotopy liftings, e∗, t∗, and eh .................. 78 3.4 The Adams e-invariant . 85 4 Summary and future works 87 4.1 An index theorem . 87 4.2 Delooping and algebraic K-theory machines . 88 4.3 Realizing the Becker-Gottlieb transfer . 89 Appendix A The stable homotopy category 91 A.1 The category of prespectra . 91 A.1.1 The model structure . 91 A.1.2 Homotopy fiber and cofiber . 93 A.1.3 Stability of P ......................... 97 A.2 Relation with the Adams category . 99 Bibliography 103 2 Chapter 1 Introduction The early use of topological K-theory with F=Z-coefficients (F = Q; R or C) can be traced back to the study of the index theorem for flat vector bundles in Atiyah, Patodi and Singer's seminal papers [APS75] and [APS76] where a ge- ometric approach to topological K-theory with F=Z-coefficients was presented. At a similar time, in a letter to Milnor [Qui76], Quillen gave a homotopy- theoretic description of this kind of K-theory and utilized it to obtain an equiv- alent definition of the Adams e-invariant. Later, a more algebraic point of view of F=Z-type topological K-theory was considered in Karoubi's book [Kar87]. Over the past forty years, topological K-theory with F=Z-coefficients has been used more extensively in many research areas of Mathematics. Its trace now can be found not only in index theory but also in algebraic K-theory, differential K-theory and KK-theory. For example, it is recognized as the secondary data after the Atyiah-Singer index theorem (see [Lot00, p.2]), and its corresponding R=Z index theorem for compact smooth bundles with vertical tangent bundle equipped with a spinc structure is also proved in [Lot94]. Parallel to this, Lott, in [Lot00], constructs another group K¯R(−), which is considered as the secondary data after the index theorem for flat vector bundles (with respect to the Borel classes), and he conjectures (see [Lot00, p.23-4]) that there exists an analogous index theorem for K¯R(−). Some recent development of Lott's conjecture can be found in the article by Bunke and Tamme [BT15, p.5]. In addition, its connection with differential K-theory is also known for years. As a matter of fact, topological K-theory with F=Z-coefficients is part of the differential cohomology diagram (see [BT15, p.3-5]). Recently, the idea of using von Neumann algebras to construct topological K-theory with coefficients is carried out in [Bas05] and [AAS14]. Especially, a purely K-theoretic description of the APS index theorem for flat vector bundles is discovered in [AAS14]. There are basically two different ways to construct this kind of K-theory. Those concerned more about geometry usually define it in terms of vector bun- dles with a connection or trivialization|called geometric models. The advan- tage of this approach is that it makes topological K-theory with F=Z-coefficients more concrete and sometimes more accessible, yet the drawback is that it usu- ally works only for finite CW -complexes. Besides, there are currently quite a number of different geometric models being used in the literature, but whether or not they are equivalent is not always clear. The constructions along this line can be found in [APS75],[APS76],[Lot94] and [JW95]. As opposed to it, those 3 more topology-minded tend to define it as the smash product of topological K-spectrum and the Moore spectrum MF=Z; or as the suspension of the homo- topy fiber of the Chern character. One can find this point of view in [Qui76] or [Wei84]. As a result, there are several different constructions in the literature, and hence it is important to have a comparison theorem for them. One of the main purposes of the second chapter is to identify some different constructions of topological K-theory with F=Z-coefficients in the literature. Especially, we obtain the following theorem (Theorem 2.1.22): Theorem 1.0.1. Let GAP S(M) be the geometric model constructed by Atiyah, Patodi and Singer in [APS76, p.88] and [APS75, p.428-9], and GJW (M) the geometric model used in [JW95], respectively, then there are isomorphisms ~ −1 ∼ ∼ [M; Ft;C=Z] =: K (M)C=Z = GAP S(M) = GJW (M); where M is a (pointed) compact smooth manifold, Ft;C=Z the homotopy fiber of the Chern character, and [M; Ft;C=Z] the homotopy classes of pointed maps from M to Ft;C=Z. Many studies of topological K-theory with F=Z-coefficients have been con- nected, in one way or another, to its interplay with flat vector bundles. In fact, in many cases, one wants to construct a homomorphism δ ~ −1 [X; BGL(C )] ! K (X)F=Z (1.1) or a homomorphism + δ ~ −1 [X; BGL (C )] ! K (X)F=Z; (1.2) where BGL(Cδ) is the colimit of the classifying spaces of the n-dimensional δ general linear groups of complex numbers with discrete topology fBGLn(C )g, X is a pointed (finite) CW -complex and (−)+ is the plus construction. Ho- momorphism (1.2) is usually a direct outcome of homomorphism (1.1) via the δ universal property of the plus construction. Replacing GLn(C ) with some other subgroups such as Σn, the symmetric groups, or GLn(Z), the general lin- ear groups over the integers Z, we could obtain some interesting invariants. For instance, the Adams e-invariant can also be defined this way (see [Qui76]). The significance of homomorphisms (1.1) and (1.2) can be viewed from two different angles: From the geometric point of view, it is related to flat vector bundles, which the Chern character is not able to distinguish. On the other hand, from the topology perspective, it is connected to algebraic K-theory, which is not easy to calculate in general, yet it contains some very interesting geometric topology information. In the literature, there are several different ways to define homomorphisms (1.1) and (1.2), for example, Bunke's 1-categorical method ([BNV16]); An- tonini, Azzali and Skandlis' K-theoretic approach ([AAS14]); and the geometric constructions given via Atiyah, Patodi and Singer's geometric model and Jones and Westbury's model, respectively. Some of them are identical, yet the relation between the others is not clear. In this thesis, we are mainly concerned with the construction given by Atyiah, Patodi and Singer in [APS76], and the one by Jones and Westbury in [JW95]. Although they use different geometric models, the ideas are similar: One first fixes a geometric model of topological K-theory 4 with F=Z-coefficients of a compact manifold M and then assigns a geometric cocycle in this geometric model to each flat vector bundle over M. Once the functoriality of this assignment is verified, the representability theorem (Adams' variant) gives a map (unique up to weak homotopy) δ e¯ : BGL(C ) ! Ft;C=Z: In view of the universal property of the plus construction, one can further obtain + δ e : BGL (C ) ! Ft;C=Z: As a corollary of the comparison theorem (see 1.0.1), these two constructions ofe ¯ in [APS76] and [JW95], respectively, are identified. Namely, they yield the same homomorphism (1.1) (see Remark 2.2.2). As shown in [APS76],e ¯∗ is closely related to the (AP S) ξ~-invariant. More precisely, we have the following commutative diagram ([APS76, p.87]): ξ~(0; M; A) e¯ ind δ ∗ A [M; BGL(C )] [M; Ft;C=Z] C=Z where indA is the index map associated to a self-adjoint elliptic operator A, and M is an odd-dimensional smooth closed manifold (see [APS76, p.87]). In fact, the main motivation behind the thesis is to establish the commutative diagram below for every compact smooth fiber bundle E ! B with E, B odd- dimensional and closed (If necessary, we assume E and B are spin, and A and A0 are the associated Dirac operators): ξ~(−; E; A) e¯ ind δ ∗ A [E; BGL(C )] [E; Ft;C=Z] C=Z π! ? ? e¯ ind 0 δ ∗ A [B; BGL(C )] [B; Ft;C=Z] C=Z ξ~(−; E; A0) where π! is induced by taking fiber-wise homology groups. A potential candidate for the left question mark is the Becker-Gottlieb transfer trBG, whereas the right question mark might be related to the adiabatic limits.