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 {BGLn(C )}, 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 ∞-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. In this thesis, we mainly focus on the left square. In fact, this square fits into the following prism-like diagram:

5 e¯ δ ∗ [E, BGL(C )] [E,Ft,C/Z] e∗

[E, BGL+(C)] ∗ π! trBG

∗ trBG e¯ δ ∗ [B, BGL(C )] [B,Ft,C/Z] e∗

[B, BGL+(C)]

Now as the commutativity of the left parallelogram has been proved in the DWW index theorem ([DWW03]), if we can show the right parallelogram also commutes, then the commutativity of the square behind follows. This is partic- ularly so when there exists an infinite loop map weakly homotopic to e. However, it is not easy to check if e is weakly homotopic to an infinite loop space map directly from the geometric definition given in [JW95] or [APS76]. In fact, from these geometric constructions, we only know the map e is a H-map. Besides, the map e is defined only up to weak homotopy—the universal property of the plus construction preserves weakly homotopic maps 1. These difficulty and observation lead us to the following questions: Can we find a homotopy- theoretic description of the map e so that one can easily see whether or not the map e is an infinite loop space map, or can we use homotopy-theoretic method to construct some infinite loop space maps from algebraic K-theory of complex numbers to topological K-theory with F/Z-coefficients that resemble the map e—meaning, sharing some properties of the map e? The strategy is first to extract as many homotopy properties of the map e as possible from its geometric definition and then try to approximate it with some homotopy-theoretically defined maps. Some homotopy properties of the map e have been studied in [JW95]. In particular, the following commutative diagram is shown in [JW95, Theorem 3.1]:

Bo

e ∗ ch ◦J odd [M,KaC] [M,Ft,C/Z] H (M, R) where J :[M,Ft,C/Z] → [M, ΩKt,R] is induced by taking the imaginary part, namely

C/Z → R [a] + ib 7→ b, ch is the odd Chern character, and Bo is the Borel regulator, a linear combina- tion of Borel classes as described in [JW95]. This gives us information about the free elements. As for the torsion part, we use a generalized Hausmann-Vogel’s

1A similar proof used in [CCMT84, Lemma 1.1] can be applied to this case

6 geometric model for K-groups with coefficients to show the following theorem2 (see Theorem 2.2.9): Theorem 1.0.2.

∼ e∗|Tor : Tor(π∗(KaC)) −→ Tor(π∗(Ft,C/Z)) ' Q/Z. Now the homotopy-theoretically defined maps used to approximate the map e are homotopy liftings (dashed arrow) in the diagram below—note that e can be thought of as a weak homotopy lifting of the following diagram:

Ft,F/Z

KaC Kt

Kt,F where the spaces are the infinite loop spaces of the 0-connective covers of the al- gebraic K-prespectrum of complex numbers (KaC), topological K-prespectrum with F/Z-coefficients (Ft,F/Z), topological K-prespectrum (Kt) and topologi- cal K-prespectrum with F-coefficients (Kt,F), respectively. In particular, the sequence on the right-hand side

Ft,F/Z → Kt → Kt,F is a homotopy fiber sequence. Note the map from Kt to Kt,F can be identified with the Chern character. The results of our investigation on these “e-like maps” are summarized in the following theorem: Theorem 1.0.3. There exist infinite many different homotopy liftings (as infi- nite loop maps) and all of them restrict to the same isomorphism on the torsion subgroups, namely ∼ Tor[X,KaC] −→ Tor[X,Ft,C/Z], where X is a pointed topological space. Furthermore, in the category of infinite loop spaces, there exists a homotopy lifting unique up to phantom maps:

\ eh : KaC → Ft,C/Z such that \ e∗ = e∗ : π∗(KaC) → π∗(Ft,C/Z), for ∗ ∈ N. The implication is that, if e¯ is a homotopy lifting—not just weak \ homotopy lifting—and induces an infinite loop map e, then e and eh differ only by a phantom map. As an immediate consequence of this theorem, we obtain a refined BL index theorem: 2This result has been observed in [JW95, Corollary 2.4]. The key ingredient in their proof −1 is the claim: “Rep(π (M), GL( )) generates K] (M) in exactly the same way that 1 C Z/m aC Z/m 0 Rep(π1(M), GL(C)) generates K]aC (M)” ([JW95, Proof of Theorem 2.3]). It is however not clear to us if this works. In fact, Rep(π (M), GL( )) seems to be zero in all cases. 1 C Z/m

7 e¯\ ˜ ∗ [E,F ] K(E, C) t,C/Z

∗ π! trBG

\ e¯∗ K˜ (B, ) [B,Ft, / ] C C Z (1.3)

\ wheree ¯∗ is the composition: ˜ e∗ K(−, C) → [−,KaC] −→ [−,Ft,C/Z]. It refines the BL index theorem in the sense that, after composing with ch ◦J, we recover the BL index theorem with respect to the regulator Bo:

Bo odd K˜ (E, C) H (E, R)

∗ π! trBG

Bo K˜ (B, ) Hodd(B, ) C R (1.4) The thesis is organized in the following way: In Chapter 2, the main focus is put on topological K-theory with coefficients and the e-invariant. In the first section, we unify various constructions of topological K-theory with coefficients and present a generalized Hausmann and Vogel’s homology sphere model for algebraic K-groups with coefficients. Based on this model, we obtain a geometric proof of Theorem 2.3 in [JW95] in the second section (see Theorem 2.2.9). In the same section, the relation between the ξ˜-invariant and the e-invariant and how this relation can be applied to the computation of the e-invariant of algebraic K-groups are also explained. In the end of this section, in order to do some concrete computation for the e-invariant, we study the 4-dimensional cobordism W represented by the following relative Kirby diagram:

Figure 1.1:

where (a1, ..., an) are pairwise coprime integers, bi is chosen in such a way that it is relatively prime to ai for every i, and they together satisfy n X bi a ...a (−b + ) = 1 1 n a i=1 i

8 with b ∈ Z. We examine the spinnability of this cobordism and obtain the following proposition, which is of interest in its own right:

Proposition 1.0.4 (Proposition 2.2.21). W is spin when the product a1...an is c even, whereas when the product a1...an is odd, W admits a spin structure that restricts to the canonical3 spinc structure on the boundary ∂W . With this cobordism W and the invariance of the ξ˜-invariant under spinc (spin) cobordism, we reobtain the formula for the real part of the e-invariant of 4 Σ(a1, ..., an) given in [JW95, Theorem C] except for a sign change. The investigation into the homotopy properties of other e-like maps is carried out in Chapter 3. We collect the required preliminary knowledge of this chap- ter, including the rationalization of CW -spectra (spaces) and Suslin’s theorem [Sus84, Corollary 4.8] in the first section. The proof of the first part of Theo- rem 1.0.3 occupies the second section and, in the next section, some structure rel theorems of KaC and K , the infinite loop space of relative K-theory (see Corollary 3.3.1 and Corollary 3.3.3) are presented. Particularly, using these structure theorems and the homotopy fiber sequence of relative K-theory, we \ identify the special infinite loop space map eh. This chapter is closed with a short discussion about how the e-invariant generalizes the Adams e-invariant. Conclusions and some possible future projects are discussed in Chapter 4.

Notations/Conventions:

1. By Z/m, Q, R and C, we understand the cyclic group of order m, the fields of rational numbers, real numbers and complex numbers, respectively.

2. We are mainly working in Top(∗), the category of (pointed) k-spaces (or Kelly spaces, see [Lew82]), and it is endowed with the Quillen model structure. Given two (pointed) topological spaces X,Y ,[X,Y ] always

means the homotopy classes of maps from X to Y in Top(∗). 3. In this thesis, we use both category of CW -spectra A, and category of prespectra P. The former is handy when dealing with the smash product, S-duality and rationalization, whereas the latter is more convenient when we need the homotopy fiber construction and infinite loop functor. The relevant notions and the comparison between A and P can be found in Appendix A.

4. By the ξ˜-invariant of a flat vector bundle over a spin manifold (α, M), we always understand the ξ˜-invariant with respect to the Dirac operator associated to the spin structure. Therefore, we omit the third entry and denote the ξ˜-invariant of (α, M) simply by ξ˜(α, M).

3The spinc structure induced by the trivial complex line bundle. 4In [JW95], a 4-dimensional cobordism is also constructed for the computation of ξ˜- invariants, yet it is not clear to us if the cobordism defined there is spin (see Remark 2.2.22 for more details).

9 Acknowledgement

I would like to thank the people who contributed in some way to this thesis work. My special thanks go to my academic advisor Prof. Sebastian Goette, who introduced me to the world of index theory. I thank him for his patience, support and encouragement throughout my PhD studies. His guidance and expert advice were so crucial to the completion of this thesis. I would also like to express my sincere gratitude to Prof. Wolfgang Steimle for agreeing to act as the second referee for my thesis and providing his insightful comments and useful feedback. I thank Matthias Wendt for his interest in my project and helpful suggestions. I also owe a debt of gratitude to Prof. Ulrich Bunke, Georg Tamme and Georgios Raptis for taking the time to understand my work and giving me valuable comments and advice during my visit to Regensburg. I would like to express my heartfelt thanks to other colleagues at the insti- tute for their support, discussions, and feedback over these years. Especially, I want to mention Prof. Victor Bangert, Oliver Br¨aunling, Anda Degeratu, Felix Grimm, JProf. Nadine Große, Prof. Stefan Kebekus, Prof. Annette Huber- Klawitter, BlaˇzMramor, Jørgen Olsen Lye, Michael Rottmaier, Maximilian Schmidtke, Prof. Katrin Wendland, Anja Wittmann, Elmiro Vetere and the late Alex Koenen. I thank Konrad V¨olkel for proofreading a part of the thesis and Florian Beck for all his help in the preparation of this dissertation and valuable comments on my thesis. I am also indebted to Daniela Behringer, the former administrator of the Graduiertenkolleg, for the kind assistance and help she provided me especially when I just came to Freiburg. Thanks are also due to Prof. Yng-Ing Lee, Prof. Ai-Nung Wang and Prof. Su-Win Yang for teach- ing and guiding me during my Master’s studies at NTU. I thank them for their patience and encouragement. The research of this thesis was carried out within the framework of the GK 1821 “Cohomological Methods in Geometry” and was founded by Deutsche Forschungsgemeinschaft. I also gratefully acknowledge the financial aid from Universit¨atFreiburg in the last year of my PhD. The project was made possible through their generous support. My sincere thanks also go to my other friends. In particular, I would like to thank Zbyszek and Witek as well as their welcoming families for the hospitality, kindness and selfless love they have shown me. I thank my loving family, my Mom, Dad and sister, who have been my great spiritual support throughout these years abroad. I cannot thank them enough for their prayer and love, which are so important to me. Finally, I want to give my deepest thanks to Jesus, who has been my ever-present help. He gave me hope and strength when I faced difficulties or felt frustrated. Without Him, this thesis would not have been completed or written.

10 Chapter 2

The e-invariant

2.1 Topological K-theory with coefficients

Various constructions of topological K-theory with coefficients are examined in this subsection. In particular, we prove the equivalence between the geometric model defined in [JW95, p.938-9] and the one constructed in [APS76, p.88]. We also extend the Hausmann and Vogel’s homology sphere model of algebraic K- groups ([HV78, Section 4]) to the topological, algebraic and relative K-groups with coefficients. The main purpose of this generalized homology sphere models is to provide a unified framework so that one can compare these K-groups and realize the homomorphisms between them concretely.

2.1.1 Homology sphere cobordisms

Here we recall Hausmann and Vogel’s theory on H∗-cobordisms of homology spheres and the plus construction (see [HV78, Sec.4] for more details). Some ap- plications and an extension of their theory are presented in this section. Though most theorems and definitions presented here also apply to PL, the category of PL manifolds, we are primarily working in Top, the category of topological man- ifolds. In addition, all manifolds are assumed to be connected and compact.

Definition 2.1.1 ([HV78, p.72-73]). Let M#(b)N be the oriented (boundary) connected sum of two oriented n-dimensional manifolds M and N. −M means the opposite orientation on M.

• By a homology n-sphere, we understand a pointed oriented n-manifold which has the same integral homology groups as a n-sphere. The notations 0 Σ, Σ , Σ0, Σˆ and the other decorated Σ’s are reserved for homology spheres. • By an acyclic n-manifold, we understand a pointed oriented n-dimensional manifold which has trivial reduced integral homology groups. We also ask 0 the base point is on the boundary. The notations W , W , W0, Wˆ and the other decorated W ’s are reserved for acyclic manifolds.

0 • A H∗-cobordism between two homology n-spheres Σ and Σ is an oriented manifold H with ∂H = Σ t −Σ0

11 such that the inclusions Σ ,→ H and Σ0 ,→ H inducing isomorphisms of integral homology groups. We specify a neat embedding I,→ H, called base arc, that connects the bases points of Σ and Σ0, where I = [0, 1], the interval. • Given a pointed topological space X, the homology sphere bordism set HS Ωn (X) is given by the equivalence classes of the pairs (Σ, f), where Σ is a ho- mology n-sphere and f a pointed continuous map. Two pairs (Σ, f) and (Σ0, f 0) are equivalent if and only if there is another pair (W, F ), where W is an acyclic n-manifold and F a continuous map W → X such that ∂W = Σ# − Σ0 and F restricts to f#f 0 on ∂W , or equivalently, there 0 exists a pair (H,F ) such that H is a H∗-cobordism between Σ and Σ , and F restricts to f t f 0 on ∂H. • Given a pair of pointed topological spaces (X,A), meaning ∗ ∈ A ⊂ X, the relative homology sphere bordism set HS Ωn (X,A) is given by the equivalence classes of the pairs (B, f) with B an acyclic n-manifold and f a pointed continuous map from (B, ∂B) to (X,A). Two pairs (B, f) and (B0, f 0) are equivalent if and only if there exists a triple 0 (W, T, F ), where T is a H∗-cobordism between ∂B and ∂B , W an acyclic 0 (n+1)-manifold with ∂W = B ∪∂B −T ∪∂B0 −B , and F a pointed contin- 0 uous map from (W, T ) to (X,A) with F |BtB0 = f tf . In addition, we ask that F sends the base arc (in T ) to the base point in X, or alternatively, there exists a pair (W, T, F ), where W is an acyclic (n + 1)-manifold, T 0 an acyclic n-manifold with ∂T = ∂(B#b − B ), and F :(W, T ) → (X,A) 0 a pointed continuous map such that ∂W = −T ∪∂T (B#b − B ) and 0 0 F |B#b−B = f#bf . 0 Remark 2.1.2. Taking connected sum f#(b)f needs some care. one can as- sume that some small disk neighborhoods of the base points of Σ and Σ0 are sent to the base point of X and then using these neighborhoods to perform connected sum. HS Lemma 2.1.3 ([HV78, p.72-73]). 1. When n ≥ 2, Ωn (X) can be made into an abelian group with the additive operation given by the oriented connected sum of homology n-spheres. The zero element is (Sn, c), where c is the constant map, and (−Σ, f) represents the inverse element of (Σ, f).

HS 2. When n ≥ 2, Ωn (X,A) admits a group structure induced by the ori- ented boundary connected sum of acyclic n-manifolds. The zero element is (Dn, c), where c is the constant map, and the inverse element of (B, f) is (−B, f). If n > 2, the operation is commutative and hence it is an abelian group. 3. Given a plus map X → X+ with respect to the largest locally perfect subgroup—the union of all finitely generated perfect subgroup of π1(X), then there is a natural map HS + α :Ωn (X) → πn(X )

12 defined by the universal property of the plus construction. Let A be the homotopy fiber of X → X+, then there is a natural map

HS + Ωn (X,A) → πn(X ). The construction of these homomorphisms are illustrated as follows:

Σn X (Bn, ∂Bn) (X,A)

Sn X+ Sn X+

We collect some facts about topological homology n-spheres and PL homol- ogy n-spheres.

Proposition 2.1.4. 1. For n > 4, every PL homology n-sphere bounds a contractible PL manifold. There are homology 3-spheres which bound no acyclic or contractible PL manifold. 2. For n 6= 4, every homology n-sphere has a smooth structure. 3. Every topological homology n-sphere bounds a topological contractible man- ifold, for every n ∈ N. Proof. The first statement can be deduced from [Ker69, p.71, Corollary] where the statement is for combinatorial homology n-spheres, yet it is known that every PL manifold is PL-homeomorphic to a combinatorial manifold (see [Hus94, Chp.3]). For the second assertion, we note, by smoothing theory [Mil11, Theorem 2], every contractible PL manifold has a smooth structure, and, by the Kirby- Siebenmann obstruction, every topological homology n-sphere has a unique PL structure when n ≥ 5. Hence, when n ≥ 5, the second assertion follows from the first as every homology n-sphere is the boundary of a smooth (contractible) manifolds. For the last claim, when n ≥ 5, it is the consequence of the first and the second statements, and when n = 3 and 4, we refer to [FQ90, 11.1A and 9.3 C]. Corollary 2.1.5.

HS Ωn (∗) = 0 n ≥ 4, in Top and PL HS Ωn (∗) = 0 n = 3, in Top

HS Remark 2.1.6. The H∗-cobordism group Ω3 (∗) in PL appears to be quite large and very complicated. To the author’s knowledge, it is not yet fully understood HS (see [Sav02, Chap.7]). Notice also, Ω3 (∗) in PL is equivalent to the smooth H∗-cobordism group as when dimension lower than 7, there is no distinction between PL and smooth category (see [Mil67, Theorem 2] for more detail). As for non-smooth homology 4-spheres, the author does not know whether or not they exist.

13 Before restating Lemma 4.3 and Proposition 4.4 in [HV78], we first fix some notations and conventions. By a plus map X → X+, we understand it is the plus map with respect to the perfect radical (maximal perfect subgroup) of π1(X). In order to make use of the theory in [HV78], we assume the perfect radical of π1(X) is always locally perfect. In addition, to avoid cohomology groups with coefficients in non-abelian groups, we also ask the perfect radical is the commutator of π1(X). Note all examples in this thesis have this property.

Proposition 2.1.7. Given (X,A) a pair of pointed topological spaces, then 1. The inclusion A,→ X and (X, ∗) ,→ (X,A) induces a long exact sequence:

HS HS HS HS HS → Ωn (A) → Ωn (X) → Ωn (X,A) → Ωn−1(A) → ... → Ω2 (X,A)

2. Let ι : X → X+, be the plus map and A its homotopy fiber, then there is a natural homo- morphism HS + µ :Ωn (X,A) → πn(X ) making the following diagram commute

HS α + Ωn (X) πn(X )

µ

HS Ωn (X,A).

Furthermore, when n ≥ 2, µ is an isomorphism, where α is the homomor- phism defined in 2.1.3.

HS 3. Suppose A is acyclic with π1(A) locally perfect, then Ωn (A) = 0, for n 6= 3. Proof. (1) is proved directly from the definition. The commutativity of the diagram in (2) is also clear. To see µ is an iso- morphism, for n ≥ 2, it is sufficient to show λ in the following diagram is an isomorphism. Now its injectivity follows from the commutative diagram:

HS Ωn (X,A)

λ µ

∼ + πn(X,A) πn(X ).

14 and the surjectivity can be proved by obstruction theory. More precisely, let HS n (B, f) be any element in Ωn (X,A) and D ,→ intB an embedding disk in ∗ n the interior of B, then because H (B \ intD , ∂B, π∗(X,A)) = 0, for all ∗, via n obstruction theory, we can deform f|B\intDn : B \ intD → X into A. By letting f 0 be the union of the restriction of the homotopy of the deformation n−1 n n 0 on S × I and f on D , we obtain an element (D , f ) ∈ πn(X,A) which is HS equivalent to (B, f) in Ωn (X,A) as the homotopy of the deformation gives us the cobordisms between (Dn, f 0) and (B, f). As for the third statement, we have, for n ≥ 4, every homology sphere Σn bounds a contractible manifold Cn+1, and thus, by the existence of the semi- s-cobordism for (A, A+), we obtain the following commutative diagram (see [HV78, p.74,p.68-69])

lifting n+2 ∂1W A

inclu.

W n+2 lifting inclu.

n n+1 n+2 + Σ ,→ C = ∂0W A = pt ext. of ι ◦ f where W n+2 is the semi-s-cobordism of Cn+1. Therefore, given any element n ∗ n+2 (Σ , f), the H -cobordism ∂1W shows it is equivalent to zero. For the case of n = 2, we refer to [HV78, p.74]. Remark 2.1.8. In [HV78, p.72-74], the proof is done in the category PL, but the same method works for Top as well. The following theorem ([HV78, Theorem 4.1]) is a corollary of Proposition 2.1.7. It plays a fundamental role in constructing the geometric models later. Theorem 2.1.9. Assume that X → X+ is the plus map as in the previous theorem and A the homotopy fiber of X → X+, then we have • For n ≥ 5, the homomorphism

HS + α :Ωn (X) → πn(X ) is an isomorphism. • When n = 3, we have the following exact sequence:

HS α + δ HS θ HS α + 0 → Ω4 (X) −→ π4(X ) −→ Ω3 (A) −→ Ω3 (X) −→ π3(X ) → 0, where δ is the composition

+ HS π4(X ) → π3(A) → Ω3 (A), or alternatively

+ ∼ HS HS π4(X ) −→ Ω4 (X,A) → Ω3 (A).

15 Proof. These two assertions follow from the previous proposition. We present some examples and the applications in the following. Some of them are used later. 0 n n + δ Example 2.1.10. 1. Let K]aC (S ) = [S , BGL (C )] be the n-th alge- braic K-group of complex numbers, where BGL(Cδ) is the classifying space of flat vector bundles over C, and BGL+(Cδ) its plus construc- tion with respect to the perfect radical. Now because the perfect radical of δ π1(BGL(C )) is locally perfect (see [HV78, Corollary 4.2]) and equal to δ the commutator of π1(BGL(C )), we can apply Theorem 2.1.9 to the plus map δ + δ BGL(C ) → BGL (C ), and obtain an isomorphism

0 HS δ ∼ n α :Ωn (BGL(C )) = K]aC (S ), n ≥ 5. This implies that one can use complex flat vector bundles over homology spheres to represent the elements in the algebraic K-groups of complex numbers, when n ≥ 5. 2. For n = 3 and given a plus map X → X+ with A its homotopy fiber, we HS have the quotient Ω3 (X)/im(θ) is equivalent to the abelian group

HS A Ω3 (X) := {(Σ, f) | f :Σ → X}/ ∼,

where two elements are equivalent

(Σ, f) ∼ (Σ0, f 0)

if and only if there exists a 4-tuple (W, F ;Σ0, f0) such that

0 ∂W = Σ#Σ0# − Σ

f0 :Σ0 → A 0 F |∂W = f#f0#f .

The connected sum of homology spheres is the additive operation. In par- ticular, when X = BGL(Cδ), α induces an isomorphism

0 HS δ A ∼ 3 Ω3 (BGL(C )) = K]aC (S ).

+ HS ∼ 3. In Top, when X = X , we have α induces an isomorphism Ωn (X) = πn(X), for every n ∈ N. In the category PL, however, due to the presence HS of the non-trivial group Ω3 (∗), α might not yield an isomorphism, when HS ∼ n = 3. Nevertheless, it is still true that, for n ≥ 5, Ωn (X) = πn(X). The examples we have in mind are X = BU, the classifying space of complex Q Q vector bundles; and X = K(C, 2i) or K(C, 2i − 1), the product or i i Eilenberg-Maclane spaces.

16 2.1.2 Topological K-theory with Z/m-coefficients In this subsection, we present some constructions of topological K-theory with ˜ Z/m-coefficients, denoted by K(−)Z/m. Especially, the geometric models for the topological K-groups and algebraic K-groups of complex numbers with Z/m- coefficients, which extend Hausmann and Vogel’s theory (Theorem 2.1.9), are given here. We let Kt be the Ω-CW -spectrum that represents complex topo- logical K-theory.

Lemma 2.1.11. 1. Let Fm be the homotopy fiber of the map of prespectra m Kt ∧na MZ −→ Kt ∧na MZ induced by the degree m map:

φ :z7→zm S1 −−−−−−→m S1,

then there is a natural π∗-isomorphism in P, the category of prespectra:

Fm → Ω(Kt ∧na MZ/m), where MZ/m is the Moore CW -spectrum of Z/m (see 3.1.1 for more about CW -Moore spectra), and ∧na is the naive smash product given by

Kt ∧ M(Z/m, k)[−k], where M(Z, k) is the Moore space of the abelian group Z/m of degree k. 2. Assume we are given the S-duality

∗ MZ/m ∧ M Z/m → S, where S is the sphere spectrum. Then there is an 1-1 correspondence

∞ ∗ 1−1 ∞ [Σ X ∧ M Z/m, Kt]Ho(A) ←−→ [Σ X, Kt ∧ MZ/m]Ho(A), for any pointed topological space X, where ∧ stands for the smash product of CW -spectra (see [Swi02, p.255-p.258], and A is the Adams category of CW -spectra. Proof. For the first assertion, we refer to Lemma A.1.4 (see [MMSS01, Sec.7] or [LMS86, p.128] also). The second assertion is the definition of the S-duality (see, for example, [Swi02, Definition 14.20]). Remark 2.1.12. The comparison between the Adams category of CW -spectra A (used in [Swi02]) and the category of prespectra P (used in [LMS86]) has be carried out in Section A.2. We need the former for various constructions such as S-duality, whereas the latter is large enough to have homotopy fiber construction.

Definition 2.1.13. Topological K-theory with coefficients in Z/m is defined by the prespectra Kt ∧ MZ/m. Namely, for any pointed topological space X, ˜ ∗ ∞ ∗ K (X)Z/m := [Σ X, Σ Kt ∧ MZ/m]Ho(A).

17 Remark 2.1.14. Following from the constructions in [Swi02, 14.30-33], we −1 ∞ −2 ∞ have the suspension CW -spectra Σ Σ C(φm) and Σ Σ C(φm) are models for MZ/m and M∗Z/m, respectively. Now when X is a pointed finite CW -complex, a smooth manifold or a n- sphere, one can utilize vector bundles over X to construct the geometric models ˜ −1 for K (X)Z/m. Theorem 2.1.15 (Geometric Models). Let X be a pointed finite CW complex, ˜ −1 then we have the following geometric models for K (X)Z/m: 1.

˜ −1 K (X)Zm = {(V, φ) | V a vector bundle over X and φ : mV ⊕ ? −→∼ ? an isomorphism}/ ∼, where (V, φ) ∼ (V 0, φ0) if and only if they are stably isomorphic via ψ : V ⊕ ? −→∼ V 0 ⊕ ?, such that the following diagram commutes up to homotopy (of isomor- phisms of vector bundles)

mψ ⊕ id mV ⊕ ? mV 0 ⊕ ?

φ ⊕ id φ0 ⊕ id

?

where, by ?, we understand a trivial vector bundle of suitable dimension. The additive operation is direct sum of vector bundles, the zero element is the trivial bundle with identity, and the inverse element of (V, φ) is (V ⊥, φ⊥), where V ⊥ is the complement of V , namely, V ⊕ V ⊥ = ?, with dimV ⊥ > dimB. φ⊥ is constructed as follows: Without loss of generality, we can assume φ : V −→∼ ?

= ⊥ m(V ⊕ V ⊥) mdimV +dimV

φ−1

dimV

Because the dimension of V ⊥ is large enough, there is always an isomor- phism φ⊥ : mV ⊥ → ? such that φ⊥ ⊕ φ homotopic to id (see [Kos07, IX.1.1]).

18 2. When X = Sn, one can further define the following geometric model via homology spheres:

HS Ωn+1(BU)Z/m := {[(Σ, f), (W, F )] | f :Σ → BU; F : W → BU such that ∂W = −#Σ; F |∂W = #f}/ ∼, m m

where (Σ, f; W, F )) ∼ (Σ0, f 0; W 0,F 0) if and only if there exists (W,Gˆ ) such that

G : Wˆ → BU   ˆ W is an acyclic (n + 1)-manifold  ∂Wˆ = Σ# − Σ0 , 0 G| ˆ = f#f  ∂W  0 ˆ HS (W #b − W ) ∪#(Σ#−Σ0) #bW = 0 ∈ Ωn+1(BU). m m

Later we often use ∂(W,Gˆ ) = (Σ# − Σ0, f#f 0) to denote the third and forth conditions above. The abelian group structure is induced by the (oriented) connected sum of homology spheres and (oriented) boundary connected sum of acyclic manifolds. The zero element is represented by (Sn, c; Dn+1, c), where c is the constant map. The inverse element of (Σ, f) is given by changing the orientation of Σ.

˜ −1 −1 ˜ −1 0 Proof. Firstly, note K (X)Z/m = K (X)Z/m as K (S ) = 0, and secondly, observe that the second statement in Lemma 2.1.11 and the Bott periodicity imply the following isomorphisms

∞ 3 ∗ 1−1 ∞ ∗ −1 [Σ X+ ∧ Σ M Z/m, Kt] ←−→ [Σ X+ ∧ M Z/m, Σ Kt] 1−1 ∞ −1 −1 ←−→ [Σ X+, Σ Kt ∧ MZ/m] =: K (X)Z/m. The rest then follows from the construction performed in [APS75, p.428-429]. Note, in [APS75], they consider the pair of vector bundles (E,F ) over X with an isomorphism φ : mE → mF , whereas, in our model, F is always trivial, a kind of normalization. Now, in the case of Sn, we have, by Definition 2.1.13, the following identifi- cations ˜ −1 n n−1 K (S )Z/m = [S ∧ C(φm),Kt] = πn(BU)Z/m (compare [Wei13, Section 4.2]). With this, one can construct a well-defined map

HS n−1 p :Ωn+1(BU)Z/m → πn+1(BU)Z/m = [S ∧ C(φm),Kt], which is given by the plus maps from homology spheres to Sn and acyclic manifolds to Dn+1. Then we observe the following commutative diagram of abelian groups

19 # # HS m HS ∂ HS π HS m HS Ωn+1(BU) Ωn+1(BU) Ωn+1(BU)Z/m Ωn (BU) Ωn (BU)

o o p o o

# # m ∂ π m πn+1(BU) πn+1(BU) πn+1(BU)Z/m πn(BU) πn(BU)

ˆ ˆ HS ˆ n+1 n Given an element (Σ, F ) ∈ Ωn+1(BU), we let W := Σ \ D , where D is a ˆ ˆ small disk neighborhood of the base point in Σ with F |Dn+1 = c, the constant map, then we define ∂(Σˆ, Fˆ) := (Sn, c; W, F ), (2.1) where F := Fˆ|W . The homomorphisms p and π are given, respectively, by ( p(Σ, f; W, F ) := (Σ+, f +; W +,F +), (2.2) π(Σ, f; W, F ) := (Σ, f).

˜ ∗ n Hence, by the five lemma, we have shown it models K (S )Z/m, for n ∈ N. Notice the H-structure on BU and the co-H-structure on Sn induce the same additive operation on π∗(BU). In the geometric models, the former can be realized by connected sum of vector bundles, whereas the latter is useful when one considers the homology sphere construction (see [Swi02, Prop.2.25]) for the equivalence of these two induced operations). In the next theorem, we present the geometric model for algebraic K-groups of complex numbers with Z/m-coefficients. The idea is basically the same, yet, HS because the abelian group Ω3 (A) might not be trivial, we have to consider the short exact sequence:

HS α + δ HS θ HS α + 0 → Ω4 (X) −→ π4(X ) −→ Ω3 (A) −→ Ω3 (X) −→ π3(X ) → 0.

Definition 2.1.16. 1. For n > 5,

HS δ δ δ Ωn+1(BGL(C ))Z/m := {(Σ, f; W, F ) | f :Σ → BGL(C ),F : W → BGL(C ) such that W is acyclic , ∂W = −#Σ; F = #f}/ ∼, m m where two 4-tuples are equivalent

(Σ, f; W, F ) ∼ (Σ0, f 0; W 0,F 0)

if and only if there exists (W,Gˆ ) such that

 ˆ δ G : W → BGL(C )  Wˆ is acyclic . ∂(W,Gˆ ) = (Σ# − Σ0, f#f 0)   0 ˆ HS W #b − W ∪#(Σ#−Σ0) #bW ∼ 0 ∈ Ωn+1(BU). m m

20 HS δ 2. For n = 3, Ω3+1(BGL(C ))Z/m is defined by (Σ , f :Σ → A),  0 0 0  δ δ ∂W = −#Σ#Σ0, {[(Σ, f), (W, F )] | f :Σ → BGL(C ),F : W → BGL(C ) s.t. there exists m }/ ∼,  F |∂W = #f#f0 m where two elements are equivalent ((Σ, f), (W, F )) ∼ ((Σ0, f 0), (W 0,F 0)) if and only if there exists (W,Gˆ ) and (Σ1, f1) such that

G : Wˆ → BGL( δ),  C  f1 :Σ1 → A,  Wˆ is acyclic , . ˆ 0 0 ∂(W,G) = (Σ# − Σ #Σ1, f#f #f1)   0 ˆ 0 HS (W #b − W ∪#(Σ#−Σ0) #bW, −Σ0#Σ0#Σ1) = 0 ∈ Ωn+1(X,A) m m m

+ Note A is the homotopy fiber of BGL(Cδ) → BGL (Cδ) In both cases, the oriented (boundary) connected sum of homology spheres induces the additive operation, the n-sphere and (n + 1)-disc together with the constant maps give us the zero element, and inverse elements are given by chang- ing orientations. Theorem 2.1.17.

HS δ + δ Ωn+1(BGL(C ))Z/m = πn+1(BGL (C ))Z/m, + δ for n ≥ 3 and n 6= 4. Recall πn+1(BGL (C ))Z/m is given by ∞ n+1 ∗ n−1 + δ [Σ S ∧ M Z/m, KaC] = [S ∧ C(φm), BGL (C )]

(see Lemma 2.1.11), where KaC is the Ω-CW -spectrum that represents algebraic K-theory of complex numbers. Proof. For the case n ≥ 5, it is proved in the same way as in the case of BU. Namely, observe the following commutative diagram of abelian groups:

# # HS δ m HS δ ∂ HS δ π HS δ m HS δ Ωn+1(BGL(C )) Ωn+1(BGL(C )) Ωn+1(BGL(C ))Z/m Ωn (BGL(C )) Ωn (BGL(C ))

o o p o o

# # + δ m + δ ∂ + δ π + δ m + δ πn+1(BGL (C )) πn+1(BGL (C )) πn+1(BGL (C ))Z/m πn(BGL (C )) πn(BGL (C )) where ∂, π and p are defined as in (2.1) and (2.2)—One only needs to replace BU by BGL(Cδ). The commutativity of the diagram is clear, and hence, by the five lemma, the claim for n ≥ 5 follows. For the case n = 3, we recall in Example 2.1.10 and Proposition 2.1.7, there are the isomoprhisms

HS A + Ω3 (X) = π3(X ) HS + Ω4 (X,A) = π4(X ).

21 These isomorphisms, combing with the following commutative diagram, im- ply the claim

# # HS δ m HS δ ∂ HS δ π HS δ A m HS δ A Ω4 (BGL(C ),A) Ω4 (BGL(C ),A) Ω3+1(BGL(C ))Z/m Ω3 (BGL(C )) Ω3 (BGL(C ))

o o p o o

# # + δ m + δ ∂ + δ π + δ m + δ π4(BGL (C )) π4(BGL (C )) π3+1(BGL (C ))Z/m π3(BGL (C )) π3(BGL (C )) where p and π are given as above except ∂(W, Σ; F, f) := (Sn, c; W, F ).

2.1.3 Topological K-theory with Q/Z-coefficients To construct topological K-theory with Q/Z-coefficients, we first recall the cat- egory of prespectra is endowed with a model structure (see A.1.1), and then ob- serve that a model for the Moore CW -prespectrum of Q/Z can be constructed via the following homotopy colimit

hocolim M /m = M / , m Z Q Z where the direct system is given by

n MZ/m −→ MZ/nm

1 1 and n is the map induced by the degree n map φn : S → S . Lemma 2.1.18.

Kt ∧ M / = hocolim Kt ∧ M /m Q Z m Z

Proof. By cofinality (see [Dug, Theorem 6.7]), we can assume the index set of the homotopy colimit is isomorphic to N, for example, considering k 7→ C(φk!). 0 N Now let {C (φm)} be a cofibrant replacement in Top∗ , then the lemma follows quickly from the equivalence below:

−1 0 −1 0 Σ Kt ∧ colim C (φm) = colim Σ Kt ∧ C (φm),

0 as smash product is a left adjoint functor, and {Kt ∧ C (φm)} is a cofibrant replacement of {Kt ∧ C(φm)} in PN (see Theorem A.1.2). Now, by the S-duality, there is a morphism

n∗ MZ/nm −→ MZ/m, which can be thought of as induced by the branch cover of C(φm) with order n. Thus, combining with the models in Theorem 2.1.15, we obtain the following −1 n geometric description for K (X)Q/Z, where X is finite CW -complex or S .

22 ˜ −1 Corollary 2.1.19. 1. Let X be a pointed finite CW -complex, K (X)Q/Z is isomorphic to the following abelian group.

∗ ∼ ∗ {(V, φ) | There exists m ∈ N s.t. φ : mV ⊕  −→  }/ ∼, where (V, φ) ∼ (V 0, φ0) if and only if there exist n and n0 such that nm = 0 0 0 0 −1 n m and (V, nφ) ; (V , n0 φ ) represent the same element in K (X)Z/nm, 0 where (V, nφ) (resp. (V, nφ)) is the image of (V, φ) (resp. (V, φ )) under the homomorphism ˜ −1 ˜ −1 K (X)Z/m → K (X)Z/nm induced by n∗ MZ/nm −→ MZ/m. (2.3)

n −1 n 2. If X = S , K (S )Q/Z can be identified with ∂W = −#Σ HS  m Ωn+1(BU)Q/Z := {[(Σ, f; W, F )] | There exists m s.t. }/ ∼, F |∂W = #f m where (Σ, f; W, F ) ∼ (Σ0, f 0; W 0,F 0) if and only if there exist n and n0 such that nm = n0m0 and

0 0 0 0 HS (Σ, f;#W, #F ) ∼ (Σ , f ;#W , #F ) ∈ Ω (BU)Z/nm. n n n0 n0

Proof. The isomorphism nφ is constructed as follows: Firstly, observe φ induces an isomorphism

1 ∗ 1 ∼ 1 ∗ ? 1 (1X × S φn) (mV ⊗ (H ⊕ (n − 1) )) −→ (1X × S φn) ( ⊗ (H ⊕ (n − 1) )),

2 1 where H is the Hopf bundle over S , and φn the degree n map from S to itself. So mV ⊗ (H ⊕ (n − 1)1) is a vector bundle over X × S2. Secondly, recall that there is a unique isomorphisms (up to homotopy)

h : Hn ⊕ (n − 1)1 ' nH.

? Then nφ is the isomorphism nmV →  such that the following commutes

(1 × S1φ )∗(φ ⊗ id) 1 ∗ 1 X n 1 ∗ ? 1 (1X × S φn) (mV ⊗ (H ⊕ (n − 1) )) (1X × S φn) ( ⊗ (H ⊕ (n − 1) ))

id ⊗ h id ⊗ h

nφ ⊗ id V ⊗ nmH nmV ⊗ H ? ⊗ H ? ⊗ nmH

Comparing with the construction in [APS75, p.428-9], we see this assignment

(V, φ) 7→ (V, nφ) realizes the homomorphism ˜ −1 ˜ −1 K (X)Z/m → K (X)Z/nm

23 induced by (2.3). The first statement is then proved. For the second statement, the geometric picture is clearer as the homomor- phism induced by (2.3),

HS HS Ωn+1(BU)Z/m → Ωn+1(BU)Z/nm, can be realized as follows:

(Σ, f; W, F ) → (Σ, f;#W, #F ). n n

Remark 2.1.20. 1. One might hope nφ is homotopic (through isomorphisms) to nφ, yet the author could not show this as nφ might not make the diagram in the proof commute. Nevertheless, nφ and nφ are not far away. For instance, if we pick up a connection ∇v of V , then we have 1 ∗ ∗ n Tch(nm∇v, nφ d) = Tch(m∇v, φ d)—This can be seen from the com- mutative diagram in the proof.

˜ −1 2. We later denote the element in K (X)Q/Z by [(V, φ)m] (resp. [(Σ, f; W, F )m]) ˜ −1 to indicate the element (V, φ) (resp. (Σ, f; W, F )) is living in K (X)Z/m.

2.1.4 Topological K-theory with C/Z-coefficients In this subsection, we deal with topological K-theory with C/Z-coefficients, ˜ ∗ denoted by K (−)C/Z. We prove the equivalence of several constructions in literature. Particularly, the geometric models constructed by Atyiah, Patodi and Singer in [APS76, p.88]1 and the model used by Jones and Westbury in [JW95, Sec.2], which they credit to Karoubi [Kar87], are identified here. To fit into our framework, another geometric model via homology spheres is also discussed.

Theorem 2.1.21. 1. There is a natural π∗-isomorphism in P, the category of prespectra: 1 Kt ∧na MC/Z ' Σ Fib(ch), where Fib(ch) is the homotopy fiber of the Chern character Y ch : Kt → HC[2i], i

and HA[j] means the Eilenberg-Maclane spectrum for the abelian group A with degree shift j. We define

˜ ∗ ∞ ∗ ∞ ∗+1 K (X)C/Z := [Σ X, Σ Kt ∧ MC/Z] = [Σ X, Σ Fib(ch)],

for X ∈ Top∗ 1The model presented in [APS76, p.88] is attributed to Segal. Yet it is the geometric model induced by this model and the Atyiah, Patodi and Signer’s model for topological K-theory with Z/m we want to compare with Jone and Westbury’s construction.

24 2. There is an isomorphism

˜ ∗ (−j∗,p∗) ˜ ∗ ˜ ∗ ˜ ∗ Coker[K (X)Q −−−−−→ K (X)C ⊕ K (X)Q/Z] = K (X)C/Z,

for any pointed topological space X, where j∗ is the map induced by MQ → MC, and p∗ the natural map from MQ to the cofiber of MZ → MQ (see 3.1.1 for more about Moore CW -spectra and spaces). Proof. Observe the natural inclusions

Z ,→ Q ,→ C give us the following maps of Moore CW -prespectra

MZ → MQ → MC.

These maps are uniquely determined because MQ =∼ HQ and MC =∼ HC, and hence they are rational (see 3.1.12). Now observe, given an injective homomorphism A → B, the homotopy cofiber of the induced map of Moore CW -prespectra for the abelian groups A and B MA → MB is a model for the Moore spectrum of MB/im(A). This can be shown by the exact sequence of the homotopy cofiber sequence of prespectra and the fact that the homotopy cofiber can be written as a suspension prespectrum (see 3.1.1, the discussion after Corollary 3.1.4). Combining with these observations and the following commutative diagram: Q Q HQ[2i] HC[2i] i i

ch ch ch Q C

Kt ∧ MZ Kt ∧ MQ Kt ∧ MC we obtain Y Cofib(ch : Kt → HC[2i]) ' Kt ∧ MC/Z, n where Cofib(ch) is the homotopy cofiber of ch. The first assertion then follows from Lemma A.1.6)(see also [LMS86, p.126]), which says there is a natural π∗-isomorphism from Fib(f) → ΩCofib(f) for any map of prespectra f. 2. Observe there is a natural map of prespectra

i : MQ/Z → MC/Z making the following commutes

k p MZ MQ MQ/Z j i π M / MZ MC C Z (2.4)

25 Then, to see the second claim, it suffices to show the following sequence is exact

˜ ∗ (−j∗,p∗) ˜ ∗ ˜ ∗ π∗+i∗ ˜ ∗ K (X)Q −−−−−→ K (X)C ⊕ K (X)Q/Z −−−−→ K (X)C/Z → 0.

For the surjectivity of π∗ + i∗, we consider the following commutative diagram of exact sequences. We have dropped the topological space X and (−˜ ) from the notations to simplify the presentation. K∗ C π∗

K∗ C/Z i∗

∗ ∗+1 ∗+1 K / K K Q Z Z Q i K∗+1 C

0

Given an element x ∈ K∗ , we have its image in K∗+1 is zero. Since K∗+1 → C/Z C Q K∗+1 is injective, we know the image of x in K∗+1 is also zero. Therefore there C Q is an element y ∈ K∗ such that its image in K∗+1 coincides with the image of Q/Z Z ∗ ∗ x. Now the difference x − i∗(y) in K can be corrected by an element z ∈ K . C/Z C That is π∗(z) + i∗(y) = x, and hence π∗ + j∗ is onto. Next, we note commutative diagram (2.4) implies

im(−j∗, p∗) ⊂ ker(π∗ + i∗), and the other direction follows from the following commutative diagram of exact sequences: K∗ Z

∗ k∗ K C −j∗ π∗

K∗ K∗ Q C/Z

p∗ i ∗ ∗ ∗+1 K / K Q Z Z

More precisely, suppose (z, y) is in the kernel of π∗ +i∗, namely, π∗(z) = −i∗(y). Then by the exactness of

K∗ → K∗ → K∗+1, Q Q/Z Z there is an element u in K∗ which has its image equal to y in K∗ . Therefore Q Q/Z ∗ the element z + j∗(u) has its image in K is zero. Now, by the exactness of C/Z K∗ → K∗ → K∗ , Z C C/Z

26 ∗ ∗ one can find an element v in K such that its image in K equal to z + j∗(u). Z C ∗ ∗ Finally, one can check u − k∗(v) has its image equal to (z, y) in K ⊕ K via C Q/Z (−j∗, p∗), and hence we have

im(−j∗, p∗) ⊃ ker(π∗ + i∗).

˜ −1 The next theorem is about geometric models for K (X)C/Z, where X is a finite CW -complex, a compact smooth manifold or a n-sphere. Theorem 2.1.22. 1. Given X a finite pointed CW -complex, we have

˜ −1 ? ∼ ? K (X)C/Z = GAP S(X) := {(ω, [(V, φ)m]) | φ : mV ⊕  −→  ; ω a closed odd cocycle}/ ∼,

0 0 0 where two elements are equivalent (ω, [(V, φ)m]) ∼ (ω , [(V , φ )m0 ]) if and only if there exists two natural numbers l and l0 and an isomorphism ψ : ? → ? such that  ml = m0l0;  0 0 −1 0 0 ˜ (V, lφ)ml ∼ (V , ψ ◦ l0 φ )m l ∈ K (M)Z/ml;  0 1 ∗ ˜ odd [ω] − [ω ] = −[ ml Tch(ψ d, d)] ∈ H (X, C).

0 0 0 Given two elements (ω, [(V, φ)m]) and (ω , [(V , φ )m0 ])—Without loss of generality, one can assume m = m0, the addition is give by

0 0 0 0 0 0 (ω, [(V, φ)m]) + (ω , [(V , φ )m0 ]) := (ω + ω ,V ⊕ V , φ ⊕ φ )m.

? (0, ( , id)1) represents the zero element, and the inverse element of ([ω], [(V, φ)m]) ⊥ ⊥ is (−ω, [(V , φ )m]). 2. If X = M, a compact smooth manifold, then

˜ −1 K (M)C/Z = GJW (M) := {(V, ∇, ω) | ∇ is a connection on V and ω an odd differntial form such that dω = ch(∇)}/ ∼,

0 0 0 where (V, ∇, ω) ∼ (V , ∇ , ω ) if and only if there exists (U, ∇U ), a vector bundle with connection over X × I, such that (U, ∇U ) restricts to (V, ∇) and (V 0, ∇0) stably on X × {1} and X × {0} and Z 0 ch(∇U )dt = ω − ω + dη, t where η is an even differential form. Given two elements (V, ∇, ω), (V 0, ∇0, ω0), the addition is given by (V ⊕ V 0, ∇ ⊕ ∇0, ω + ω0). The zero element is the trivial bundle with trivial connection and zero form, (∗, d, 0), and the inverse element of (V, ∇, ω) is (V ⊥, ∇⊥, ω⊥), ⊥ ⊥ ⊥ R where ∇ is any connection on V , and ω := −ω + ch(∇U ) with U a vector bundle over X × I that restricts to V ⊕ V ⊥ and ? on X × 1 ⊥ and X × 0, and ∇U a connection on U that restricts to ∇ ⊕ ∇ and d on X × 1 and X × 0.

27 3. If X = Sn, there is a homology sphere model:

˜ −1 n HS odd K (S )C/Z = Ωn+1(BU)C/Z := {([ω], [(Σ, f; W, F )m]) | [ω] ∈ H (Σ, C); − #(Σ, f) = ∂(W, F )}/ ∼, m where ([ω], (Σ, f; W, F )) ∼ ([ω0], (Σ0, f 0; W 0,F 0)) if and only if there exist 0 ˆ HS two natural numbers l and l and (Σ,G) ∈ Ωn+1(BU) such that  ml = m0l0;   0 1 R ˆ HS Q [ω] ∼ [ω ] − ml [ch(Σ,G)] ∈ Ωn ( K(C, 2i − 1)); i ,  0 0 0 00 0 00 HS (Σ, f;#W, #F )) ∼ (Σ , f ;#W #bW , #F #bG ) ∈ Ωn+1(BU)Z/ml.  l l l0 l0 (2.5) where W 00 := Σˆ \ Dn+1, with Dn+1 a disk neighborhood of the base point ˆ 00 in Σ that has G|Dn+1 = c, the constant map, and G := G|W 00 . The additive structure is induced by the (oriented) connected sum. The n-sphere and (n + 1)-disc along with the constant maps constitute the zero element, and inverse elements is given by changing orientations. Before starting the proof, we explain some notations used in the statement.

Remark 2.1.23. • By H˜ odd(−, C), we mean the (reduced) cohomology the- ory of graded Eilenberg-Maclane spectrum Y HC[2i − 1]. i

• Applying Hausmann and Vogel’s theorem (Theorem 2.1.9) to the product of Eilenberg-Maclane spaces, we have the following geometric model: ( ˜ odd n HS Y Σ is a homology sphere H (S , C) = Ωn ( K(C, 2i−1)) := {[ω]Σ | }/ ∼, [ω] ∈ Hodd(Σ, ) i Σ C 0 where [ω]Σ ∼ [ω ]Σ0 if and only if ∗ 0∗ 0 odd 0 p [ω]Σ − p [ω ]Σ0 = 0 ∈ H (Σ# − Σ , C), (2.6) with p, p0 the natural collapsing maps p : Σ# − Σ0 → Σ; p0 : Σ# − Σ0 → −Σ.

HS Q Namely, Ωn ( K(C, 2i − 1)) is the abelian group generated by [ω]Σ and i subject to the relation defined in (2.6). • In the statement of the theorem, we have used the notations ch(∇) and ch(Σˆ,G). The former is the differential form constructed via Chern-Weil theory, whereas the latter is the class determined by the composition Y ch ◦ G : Wˆ → BU → K(C, 2i), i where K(C, 2i) is the Eilenberg-Maclane space of degree 2i.

28 R even 2n odd 2n−1 • The isomorphism : H (Σˆ , C) → H (S , C) can be viewed as the composition

Z −1 even 2n ∼ even 2n σ odd 2n−1 : H˜ (Σˆ , C) −→ H˜ (S , C) −−→ H˜ (S , C),

where σ is the suspension. This explain the second equivalence in (2.5). After the introduction of rela- tive K-theory in 2.1.5, a better and neater description of R ◦ch will be given. Proof. Step 1.: The first geometric model can be deduced from the second statement in Theorem 2.1.21 and the geometric model described in Corollary 2.1.19. Step 2.: Recall that given two connections ∇ and ∇0, there is a Chern-Simon class, denoted by Tch(∇, ∇0) well-defined in Ωodd(M)/d(Ωeven(M)) such that ch(∇)−ch(∇0) = d Tch(∇, ∇0), where Ωeven(odd)(M) is the graded group of even (odd) differential forms of M and d is the differential. The explicit construction for Tch(∇, ∇0) used here is given by the following integral Z ch(∇t)dt, t

0 where ∇t = t∇ + (1 − t)∇ . These Chern-Simon classes satisfy the following equality in Ωodd(M)/d(Ωeven(M)):

Tch(∇, ∇0) + Tch(∇0, ∇00) = Tch(∇, ∇00). (2.7)

Step 3.: When X = M a compact smooth manifold, we construct a homo- morphism from the first geometric model to the second:

Φ: GAP S(M) → GJW (M) 1 (ω, [(V, φ) ]) 7→ [(V, ∇ , Tch(m , φ∗d) − ω)] m v m Ov where a connection ∇v on V is chosen. To see it is well-defined, one observes that choosing different connections on V does not change the class. That is because we can define

0 (U := V × I, ∇u := t∇v + (1 − t)∇v)

0 which restricts to (V, ∇v) and (V, ∇v) on M ×{1} and M ×{0}. Then, by (2.7), we have the identity in Ωodd(M)/d(Ωeven(M)): 1 1 Tch(∇ ) := Tch(∇ , ∇0 ) = Tch(m∇ , φ∗d) − Tch(m∇0 , φ∗d). u v v m v m v Hence using different connections on V yields the same equivalence class in GJW (M). 0 0 0 Now let (ω , [(V , φ )m0 ]) be another representative—Without loss of gener- ality (see 2.1.20), one can assume m0 = m, then, by definition, they differ by an ? ˜ −1 element ( , ψ) in K (M)Q. That is there is an equivalence: 0 0 ˜ −1 (V, φ) ∼ (V , ψ ◦ φ ) ∈ K (M)Z/m, (2.8)

29 for some m, and 1 1 Tch(ψ∗d, d) = ch(ψ) + ω = ω0 in Ωodd(M)/d(Ωeven(M)). m m Now this means there is an isomorphism of vector bundles ρ : V 0 → V such that mρ is homotopic (through isomorphisms of vector bundles) to

φ−1 ◦ ψ ◦ φ0 : V 0 → V.

Now ρ induces a vector bundle with connection (U, ∇U ) which restricts to 0 (V, ∇v) and (V , ∇V 0 ) on M ×{1} and M ×{0}, respectively, and since mρ is ho- motopic to φ−1 ◦ψ◦φ0, we have the following equation in Ωodd(M)/d(Ωeven(M)):

Z ch(∇u)dt t 1 1 ∗ 1 0,∗ ∗ ∗,−1 = ch(mρ) = Tch((mρ) m∇ , m∇ 0 ) = (Tch(φ ◦ψ ◦φ m∇ , m∇ 0 ) m m v v m v v 1 0,∗ ∗ ∗,−1 0,∗ ∗ 0,∗ ∗ 0,∗ 0,∗ = (Tch(φ ◦ψ ◦φ m∇ , φ ◦ψ d)+Tch(φ ◦ψ d, φ d)+Tch(φ d, m∇ 0 )) m v v 1 ∗ 0,∗ 0 = (Tch(m∇ , φ d) − Tch(m∇ 0 , φ d)) + ω − ω + dη. m v v Hence these two elements 1 (V, ∇ , Tch(m∇ , φ∗d) − ω) v m v and 0 1 0,∗ 0 (V , ∇ 0 , Tch(m∇ 0 , φ d) − ω ) v m v are equivalent in GJW (M). Step 4. Now we construct the inverse map of Φ:

Ψ: GJW (M) → GAP S(M) 1 [(V, ∇ , ω)] 7→ ( Tch(m , φ∗d) − ω, [(V, φ) ]) v m Ov m where a (stable) trivialization φ : mV ⊕ ? −→∼ ? is chosen. Suppose there is another trivialization φ0 : m0V ⊕ ? −→∼ ?—Notice that, without loss of generality, we can assume m = m0. Then, via the following identities in Ωodd(M)/Ωeven(M):

1 ∗ 1 0∗ 1 0∗ ∗ Tch(m∇ , φ d) − ω − ( Tch(m∇ 0 , φ d) − ω) = Tch(φ d, φ d) m v m v m 1 1 = Tch(d, (φ ◦ φ0,−1)∗d) = − Tch(φ ◦ φ0,−1). m m and (V, φ0) + (?, φ ◦ φ0−1) = (V, φ).

Therefore they induce the equivalent elements in GAP S(M).

30 Suppose we are given another representative (V 0, ∇0, ω0), then, by definition, 0 there exists (U, ∇u) over M × I such that it restricts to (V, ∇v) and (V , ∇v0 ) on M × {1} and M × {0}, respectively, with Z 0 d ch(∇u)dt = ω − ω + dη, t where η is an even differential form. Now let ψ : V 0 → V be the isomorphism induced from the homotopy, then we have the following identities:

(?, φ ◦ mψ ◦ φ0,−1) + (V 0, φ0) ' (V 0, φ ◦ mψ) ' (V, φ) and

1 1 Tch(φ ◦ mψ ◦ φ−1) = (Tch(φ) − Tch(φ0)) + Tch(ψ) m m 1 0,∗ 1 ∗ 0 = Tch(m∇ 0 , φ d) − Tch(m∇ , φ d) + ω − ω m v m v in Ωodd(M)/Ωeven(M). Hence we have shown the two elements

∗ (Tch(m∇v, φ d), [(V, φ)m]) and 0,∗ 0 0 (Tch(m∇v0 , φ d), [(V , φ )m]) are equivalent in GAP S(M). Step 5.: For the last geometric model, we observe the following commutative diagram. Notice that the cokernel of the lower map is equivalent to the second definition for topological K-theory with C/Z-coefficients in Theorem 2.1.21, where

n−1 n+1 ∞ 0 πn+1(BU) / = invlim[S ∧ C(φm),BU] = [Σ Σ S , Kt ∧ M / ], Q Z n Q Z and the cokernel of the upper map gives us the third geometric model in this theorem.

(−¯j , p¯ ) HS ∗ ∗ HS Q HS Ωn+1(BU) ⊗ Q Ωn ( i K(C, 2i − 1)) ⊕ Ωn+1(BU)Q/Z

(−j∗, p∗) π (BU) ⊗ π (Q K( , 2i − 1)) ⊕ π (BU) n+1 Q n i C n+1 Q/Z (2.9)

The vertical homomorphisms are induced by the plus construction and are all isomorphisms (see Example 2.1.10). The homomorphism

¯ HS HS Y −j∗ :Ωn+1(BU) ⊗ Q → Ωn ( K(C, 2i − 1)) i

31 is defined by the assignment p p Z (Σˆ,F ) ⊗ 7→ ch(Σˆ,F ) q q whereas the homomorphism

HS HS p¯∗ :Ωn+1(BU) ⊗ Q → Ωn+1(BU)Q/Z is given by p n (Σˆ,F ) ⊗ 7→ (S , c;#W, F |)q, q p where W := Σˆ \Dn+1 such that Dn+1 is a disc neighborhood of the base point in ˆ n n+1 Σ with F |Dn+1 = c, the constant map, F | := F |W , and S = ∂D . Because the homomorphisms induced by the plus construction are compatible with the desuspension, the firs half of the diagram in (2.9)

−¯j HS ∗ HS Q Ωn+1(BU) ⊗ Q Ωn ( i K(C, 2i − 1))

−j∗ Q πn+1(BU) ⊗ Q πn( i K(C, 2i − 1)) commutes, whereas the commutativity of the other half follows from the defini- tion.

Remark 2.1.24. 1. Observe that K˜ 0(Σ) = 0, for any odd dimensional ho- mology sphere, and hence every vector bundle over an odd dimensional homology sphere is stably trivial. This allows us to have the following ˜ −1 n simplified geometric model for K (S )C/Z when n is odd:

ΩHS (BU)0 := {(ω, [(Σ, f; W, F ) ]) | (−Σ, f) = ∂(W, F ) n+1 C/Z 1 HS Y and [ω] ∈ Ω ( K(C, 2i − 1))}/ ∼, i

0 0 0 0 where (ω, [(Σ, f; W, F )1]) ∼ (ω, [(Σ , f ; W ,F )1]) if and only if there ex- ˆ HS ists (Σ,G) ∈ Ωn+1(BU) such that Z 0 ˆ HS Y [ω] = [ω ] − ch(Σ,G) ∈ Ωn+1( K(C, 2i − 1)). i

The abelian group structure on ΩHS (BU)0 is constructed as before, and n+1 C/Z there is a natural isomorphism induced by the inclusion

ΩHS (BU)0 −→∼ ΩHS (BU) . n+1 C/Z n+1 C/Z

32 2. In the same way, when X is a homology sphere, the first geometric models in Theorem 2.1.22 can also be improved. Meaning, we define  V a vector bundle over Σ,  0 ? ∼ ? GAP S(X) := {(ω, [(V, φ)1]) | φ : V ⊕  −→  ; }/ ∼,  [ω] ∈ H˜ odd(Σ, C). 0 0 0 where (ω, [(V, φ)1]) ∼ (ω , [(V , φ )1]) if and only if

0 1 ch odd [ω ] − [ω] ∈ im[K˜ (Σ) −→ H˜ (Σ, C)]. There is also a canonical isomorphism 0 GAP S(X) → GAP S(X). In the following, we recall some basic properties of the topological K-theory ∼ with C/Z-coefficients for later reference. Let Kt and KC/Z = Kt ∧MC/Z be the Ω-CW -spectra that represent topological K-theory and topological K-theory with C/Z-coefficients. Proposition 2.1.25. 1. Topological K-theory with C/Z-coefficients fits into the following long exact sequence: ˜ −1 ˜ odd ˜ −1 ˜ 0 ˜ even ... → K (X) → H (X) → K (X)C/Z → K (X) → H (X) → ..., for any topological space X (in fact, any CW -spectrum).

2. KC/Z is a module CW -spectrum over Kt. Namely, there is a map of CW -spectra:

KC/Z ∧ Kt → KC/Z subject to certain compatibility conditions (see [Rud08, Definition 2.13]).

Proof. The long exact sequence in the first assertion follows from Proposition A.1.8 because we have

Kt → Kt ∧ MC → Kt ∧ MC/Z is isomorphic to a cofiber sequence in P. The second follows from the fact that Kt is a CW -ring spectrum. Corollary 2.1.26. 1. ( = / n = odd ; K˜ −1(Sn) C Z C/Z = 0 n = even .

2. If M is an odd dimensional spin manifold and [D] is the Dirac operator associated with the spin structure, then there is a Thom isomorphism:

˜ −1 ⊗[D] ˜ −1 ν TD : K (M)C/Z −−−→ K (M )C/Z, where M ν is reduced Thom space of the normal bundle of M. Proof. The first assertion follows from the long exact sequence in Proposition 2.1.4, and the second statement follows from the fact that [D] is a Thom class ˜ ν in K(M ) ([LM89, Appendix C]) and KC/Z is a module CW -spectrum over Kt (see [Rud08, Section 5.1] for more on Thom isomorphisms for module CW - spectra).

33 2.1.5 Relative K-theory In this subsection, we consider the homology sphere model for relative K-theory of complex numbers KrelC, which is the homotopy fiber of the natural map

KaC → Kt. Relative K-theory has appeared in Karoubi’s work in [Kar87] and later used in [JW95, Sec.4]. One of its important features is its connection with the trans- gression classes. Our aim here is to provide a geometric model of the relative K-groups of complex numbers and explain how the transgression classes can be realized by this model. To better understand the construction of relative K-theory, we first recall Theorem 1.1(a) in [Ber82]. Let PG be the perfect radical of the group G, the maximal perfect subgroup, then we have

Lemma 2.1.27. Assuming Pπ1B is trivial, then, given a homotopy fiber se- quence F → E → B, the sequence F + → E+ → B is a homotopy fiber sequence. Recall that by the plus map X → X+ we mean the plus map with respect to the perfect radical of π1(X). Proof. See [Ber82, p.150-151]. Berrick considers F 0 the homotopy fiber of E+ → B and then shows the induced map F → F 0 is a plus map. Now let δ ι : BGL(C ) → BU be the canonical (topology-changing) map, and the sequence

δ I∗ δ BGL(C ) ×i BU → BGL(C ) → BU the associated fiber sequence, where BU I∗ is the function space of pointed maps from (I, 0), the interval [0, 1] with base point 0, to (BU, ∗), and

δ I∗ δ I∗ BGL(C ) ×ι BU := {(x, γ(t)) ∈ BGL(C )) × BU with γ(1) = ι(x)}. Then, applying Berrick’s theorem to this case, we obtain the following map of homotopy fiber sequences:

δ I∗ δ BGL(C ) ×ι BU BGL(C ) BU

+ δ I∗ + δ BGL (C ) ×ι BU BGL (C ) BU

34 where all vertical maps are the plus maps. In particular, we see BGL+(C) × BU I∗ is a model for the infinite loop space of relative K-theory. Now assume the plus map

δ + δ BGL(C ) → BGL (C ), is a fibration and A is its homotopy fiber, then we have A × ∗I∗ = A is also a homotopy fiber of the map

δ I∗ + δ I∗ BGL(C ) ×ι BU → BGL (C ) ×ι BU . They together form a commutative diagram whose rows and columns are ho- motopy fiber sequences:

A × ∗I∗ A ∗

δ I∗ δ BGL(C ) ×ι BU BGL(C ) BU

+ δ I + δ BGL (C ) ×ι BU BGL (C ) BU

Definition 2.1.28.

rel δ I∗ BGL (C) := BGL(C ) ×ι BU

rel,+ + δ I∗ BGL (C) := BGL (C ) ×ι BU , where BGLrel(C) → BGLrel,+(C) is the plus map (with respect to the perfect radical).

Applying Theorem 2.1.9 and Example 2.1.10 to this case, one obtains the following: Theorem 2.1.29. The following homomorphisms are isomorphisms

HS rel + rel,+ Ωn (BGL (C)) −→ πn(BGL (C)) n ≥ 5 HS rel A + rel,+ Ω3 (BGL (C)) −→ π3(BGL (C)).

rel δ Proof. The only thing we need to check is the perfect radical Pπ1(BGL (C )) rel δ is locally perfect and equal to the commutator of π1(BGL (C )). This follows from the commutative diagram:

35 rel δ δ 0 Pπ1(BGL (C ) Pπ1(BGL(C )

rel δ 0 π2(BU) π1(BGL (C)) π1(BGL(C )) 0

rel,+ + δ 0 π2(BU) π1(BGL (C)) π1(BGL (C )) 0 because all its columns and the two bottom rows exact, so by the nine (or 3×3) lemma, we have the top row is also exact. Namely,

rel δ δ Pπ1(BGL (C )) = Pπ1(BGL(C ).

δ rel δ Since Pπ1(BGL(C )) is locally perfect (see [HV78, Proof of Cor.4.2])), Pπ1(BGL (C )) rel δ is locally perfect as well. To see π1(BGL (C )) is abelian, we recall it is the homotopy fiber of the infinite loop space map BGL+(Cδ) → BU.

δ I∗ Now observe that the space BGL(C ) ×ι BU classifies flat vector bundles with trivialization, and, given a flat vector bundle with trivialization over a ∼ ? compact smooth manifold M,(V, ∇v, φ : V −→  ) say, one can define the Chern-Simon form by: Z ∗ Tch(∇v, φ d) := ch(∇v,t)dt, t where ∇v,t is the connection on V × I given by

∗ ∇v,t := t∇v + (1 − t)φ d.

It determines a cohomological class as ∇v is flat. This assignment induces a homomorphism of monoids

δ I∗ Y [M, BGL(C ) ×ι BU ] → [M, K(C, 2i − 1)]. i That is it respects the monoidal structures on both sides. Furthermore, it is also functorial with respect to smooth maps. Thus, via the facts that every finite CW -complex is homotopy equivalent to a compact smooth manifold, and every continuous map of smooth manifolds can be approximated by a smooth map— without changing the homotopy class, we have, according to the representability theorem (the Adams’ version, see [Ada71]), that there is a map

rel Y t¯: BGL (C) → K(C, 2i − 1). i

Initially, t¯ is determined only up to weak homotopy. However, since the space Y K(C, 2i − 1) i

36 is rational, there is no phantom map. Therefore, t¯is, in effect, determined up to homotopy. With the universal property of the plus construction, one can factor t¯ through BGLrel,+(C) and obtain the map rel,+ Y t : BGL (C) → K(C, 2i − 1). i In the case of X = Sn , one can realize this homomorphism concretely via the homology sphere models. Proposition 2.1.30. The following diagrams commute

t HS rel ∗ HS Q Ωn (BGL (C)) Ωn ( i K(C, 2i − 1))

t¯ rel,+ ∗ Q πn(BGL (C)) πn( i K(C, 2i − 1)) for n ≤ 5; t¯ HS rel A ∗ HS Q Ω3 (BGL (C)) Ω3 ( i K(C, 2i − 1))

t rel,+ ∗ Q π3(BGL (C)) π3( i K(C, 2i − 1))

Proof. This follows from the universal property of the plus construction. Namely, there is a homotopy commutative diagram:

n rel Q Σ BGL (C) i K(C, 2i − 1)

+ +

Sn BGLrel,+(C)

HS rel Remark 2.1.31. 1. By definition, we know that an element in Ωn (BGL (C)) is represented by a map f δ Σ −→ BGL(C )

ft and a homotopy Σ × I −→ BU such that f1 = ι ◦ f and f0 is the constant map. Since we are working in the category Top, one always can find a contractible manifold C with ∂C = Σ × 0 to close up the bottom of Σ × I,

namely, considering the contractible manifold Σ\× I = C ∪Σ×{0} Σ × I. This leads us to a more flexible geometric model for relative K-groups of complex numbers. For n ≥ 5, we consider

 δ f :Σ → BGL(C ), HS rel δ 0  Ωn (BGL (C )) := {(Σ, f; W, F ) | F : W → BU, }/ ∼, ∂(W, F ) = (Σ, ι ◦ f).

37 where (Σ, f; W, F ) ∼ (Σ0, f 0; W 0,F 0) if and only if there exists (W 00,G; W,ˆ Gˆ) such that  G : W 00 → BGL(Cδ),  ∂(W 00,G) = (Σ# − Σ0, f#f 0), Gˆ : Wˆ → BU,   00 0 0 ∂(W,ˆ Gˆ) = (−W ∪Σ#−Σ0 W #b − W , ι ◦ G ∪Σ#−Σ0 F #bF ).

The abelian group structure is defined as before, and, by obstruction theory, there is an isomorphism

HS rel ∼ HS rel 0 Ωn (BGL (C)) −→ Ωn (BGL (C)) .

For n = 3, we have to take into account the homotopy fiber A × ∗I∗ and define

 δ f :Σ → BGL(C ), HS rel δ A,0  Ω3 (BGL (C )) := {(Σ, f; W, F ) | F : W → BU, }/ ∼, ∂(W, F ) = (Σ, ι ◦ f)

0 0 0 0 00 where (Σ, f; W, F ) ∼ (Σ , f ; W ,F ) if and only if there exists (W , Σ0, G, g; W,ˆ Wˆ 0, G,ˆ Gˆ0) such that

 G : W 00 → BGL( δ),  C  g :Σ0 → A,  00 0 0 ∂(W ,G) = (Σ0#Σ# − Σ , g#f#f ),  Gˆ : Wˆ → BU, ˆ ˆ G0 : W0 → ∗,  ∂(Wˆ , Gˆ ) = (Σ , ι ◦ g),  0 0 0  ˆ ˆ ∂(W, G) =  00 0 0  0 ˆ 0 ˆ (−W ∪Σ0#Σ#−Σ W0#bW #b − W , ι ◦ G ∪Σ0#Σ#−Σ G0#bF #bF ).

Similarly, it admits an abelian group structure via connected sum, and there is an isomorphism

HS rel A ∼ HS rel A,0 Ω3 (BGL (C)) −→ Ω3 (BGL (C)) by obstruction theory.

2. Finally, the map t¯∗ can be extended to these modified geometric models. Meaning, there are homomorphisms:

¯ HS rel 0 HS Y t∗ :Ωn (BGL (C)) → Ωn ( K(C, 2i − 1)), for n ≥ 5 i ¯ HS rel A,0 HS Y t∗ :Ω3 (BGL (C)) → Ω3 ( K(C, 2i − 1)). i

38 3. The relation between Z HS HS Y ch :Ωn+1(BU) → Ωn ( K(C, 2i − 1)) i and ¯ HS rel (A),0 HS Y t∗ :Ωn (BGL (C)) → Ωn ( K(C, 2i − 1)), i can be identified via the homomorphism

HS HS rel (A),0 i :Ωn+1(BU) → Ωn (BGL (C)) ; (Σˆ, Fˆ) 7→ (Sn, c; W, F ),

n+1 n n+1 where W := Σˆ \ intD , F := Fˆ|W , S = ∂W , and D is a disc ˆ neighborhood of the base point with F |Dn+1 the constant map. That is we R n have ◦ch ◦ i(Σˆ, Fˆ) = t¯∗(S , c; W, F )

2.2 The e-invariant

The section is divided into two parts. In the first half, we (re)prove Corollary 2.4 in [JW95]—or rather Theorem 2.3, which says the e-invariant (see Construction 2.2.1 and Lemma 2.2.5) restricts to an isomorphism on the torsion subgroup of the algebraic K-groups of complex numbers

0 n ∼ ˜ −1 n e∗ : Tor(K]aC (S )) −→ Q/Z ,→ K (S )C/Z, (2.10) for every odd number n ≥ 3. Via Suslin’s theorem [Sus84, Corollary 4.6], it suffices to show the commutativity of following diagram:

−1 ι n ˜ −1 n K (S ) /m K]aC (S )Z/m Z

b j∗

0 e∗ n K˜ −1(Sn) K]aC (S ) C/Z where the algebraic K-groups with coefficients is given by

−1 n n−1 + δ K]aC (S )Z/m := [S ∧ C(φm), BGL (C )], ∗ and K]aC (−) is the reduced cohomology of algebraic K-theory of complex num- bers (see 2.2.10 for more detail about the diagram). Now the commutativity of the diagram above is claimed in [JW95, Theorem 2.3]. The proof there is however not complete. Thus we use the homology sphere models developed in the previous section to (re)prove it. Our idea is to use these homology sphere models to concretely realize all these homomorphisms in the diagram above (see Lemma 2.2.5, Lemma 2.2.7, Lemma 2.2.10). In the second part of the section, following Jones and Westbury’s idea, we define a C/Z-valued invariant induced from the e-invariant and describe its

39 relation with the ξ˜-invariant in [APS76]. The main observation is Lemma 2.2.12 which explains why one can use the ξ˜-invariants of homology spheres to compute the e-invariant of algebraic K-groups. In the last subsection, we use relative Kirby diagram to construct a 4-dimensional cobordism between #L(ai, bi) and 1≤i≤n Σ(a1, ...an). Via this cobordism and Lemma 5.4 of [JW95], we obtain a formula for the real part of the e-invariants of Seifert homology spheres Σ(a1, ..., an) when a1...an is even (see 2.2.22 and 2.2.24 for the remarks on the results in [JW95]).

0 n 2.2.1 The e-invariant and Tor(KgaC (S )) In this subsection, we first recall the construction of the e-invariant from [JW95, p.938] and then explain how to realize this homomorphism via the homology sphere models constructed in the previous section. A complete proof of Theorem 2.3 in [JW95] is given in the end of the subsection. Throughout the subsection, all the spaces are assumed to be pointed. Construction 2.2.1. For any compact smooth manifold M, one has the as- signment

δ ˜ −1 [M, BGL(C )] → K (M)C/Z (V, ∇v) 7−→ (V, ∇v, 0), where (V, ∇v) is a flat vector bundle over M and (V, ∇v, 0) is an element in ˜ −1 K (M)C/Z (see Theorem 2.1.22). One can check it is a homomorphism of commutative semigroups for every compact smooth manifold M, and it is also functorial with respect to smooth maps. Again, because every finite CW -complex is homotopy equivalent to a compact smooth manifold, and every continuous map of smooth manifolds is homotopic to a smooth map, one obtains a natural trans- formation of semigroup-valued functors on the category of finite CW -complexes

δ [−, BGL(C )] → [−,Ft,C/Z]. (2.11) The representability theorem in [Ada74, Addendum 1.5] then implies there exists a map unique up to weak homotopy,

δ e¯ : BGL(C ) → Ft,C/Z, (2.12) where Ft,C/Z is the zero component of the Ω-prespectrum Fib(ch) (see Theorem 2.1.21). Applying the universal property of the plus construction, one further obtains a map unique up to weak homotopy:

+ δ e : BGL (C ) → Ft,C/Z. Definition 2.2.2. We call the homomorphism

0 + ˜ −1 e∗ :[X, BGL (C)] = K]aC (X) → K (X)C/Z, induced by the map e constructed above, the e-invariant.

40 Note the algebraic K-theory used here is the 0-connective cover of the usual −1-connective algebraic K-theory of complex numbers—Namely, we disregard the dimension of a vector bundles (see Chapter 3 for more homotopy theoretic discussion and justification of using the 0-connective cover instead of the usual one). Remark 2.2.3. In view of Theorem 2.1.22, one can use another geometric model GAP S to realize the homomorphism e¯∗, namely, considering the following assignment δ e¯∗ :[M, BGL(C )] → [M,Ft,C/Z] 1 (V, ∇) 7→ ([ Tch(m∇, φ∗d)], [(V, φ) ]), m m where φ is a stable trivialization of mV for some m ∈ N. This definition is the one used in [APS76, p.89] Now the difficulty is that, after applying the plus construction, the geometric picture is no longer clear. This obstructs the study of the map e. However, the good news is that, when X = Sn and n 6= 4, one can still use the homology sphere models to realize this map geometrically. In fact, we are going to use homology sphere models to construct a homomorphism, called the e0-invariant, 0 HS δ (A) HS e∗ :Ωn (BGL(C )) → Ωn+1(BU)C/Z such that the following commutes

e0 HS δ (A) ∗ HS Ωn (BGL(C )) Ωn+1(BU)C/Z

o o

0 e∗ n K˜ −1(Sn) K]aC (S ) C/Z

Meaning, the e0-invariant can be identified with the e-invariant. Before con- structing the e0-invariant, we recall some facts from Remark 2.1.24 (see also Corollary 2.1.5). HS Facts 2.2.4. 1. Because Ωn (∗) = 0 in Top, for all n ∈ N, given a homology n-sphere Σ, there always exists an acyclic (n + 1)-manifold C such that ∂C = Σ. One can actually find a contractible one (see 2.1.4). 2. As explained in Remark 2.1.24, there are simplified geometric models ΩHS (BU)0 , n+1 C/Z 0 when n is odd, and GAP S(Σ) , when Σ is an odd-dimensional homology sphere. With these in mind, we now proceed to define the e0-invariant. Lemma 2.2.5. Let n be an odd number, then the assignment

(Σ, f) 7→ (t¯∗(Σ, f; Σ\× I, Φ), [(Σ, f; −Σ\× I, Φ)]) yields the following well-defined homomorphisms

( HS δ HS 0 Ωn (BGL(C )) → Ωn+1(BU) , n ≥ 5, e0 : C/Z ∗ ΩHS(BGL( δ))A → ΩHS (BU)0 , n = 3 3 C 3+1 C/Z

41 Construction 2.2.6. We need to explain the notations used in the assignment

(Σ, f) 7→ (t¯∗(Σ, f; Σ\× I, Φ), [(Σ, f; −Σ\× I, Φ)]).

HS δ Given an element (Σ, f) in Ωn (BGL(C )), we have ι ◦ f :Σ → BU is ho- motopic to the constant map, where ι is the canonical map BGL(Cδ) → BU. Hence we can assume φt :Σ × I → BU is a homotopy such that φ1 = ι ◦ f and φ0 is the constant map. We close the manifold Σ×I from the bottom by a contractible manifold C with ∂C = Σ×{0} to obtain another contractible manifold

Σ\× I := C ∪Σ0 Σ × I and an extension of φt: Φ: Σ\× I → BU with Φ|Σ×I = φt. The pair (Σ\× I, Φ), together with the element (Σ, f), yields a class (see Proposition 2.1.30 and Remark 2.1.31)

odd t¯∗(Σ, f; Σ\× I, Φ) ∈ H (Σ)

Now we can check the well-definedness of the e0-invariant. Proof. To see the e0-invariant is well-defined, we check the following list: 1. It is independent of the choice of the contractible manifold that closes the bottom of the homotopy φt as we can always find a contractible manifold 0 of one dimension higher such that its boundary is equal to C ∪Σ×{0} C , where C0 is another contractible manifold that closes the bottom of Σ × I. 2. It is independent of the choice of the homotopy φt as the difference of HS two homotopies can be corrected by an element in Ωn+1(BU). Hence they yield the equivalent elements in ΩHS (BU)0 . n+1 C/Z 0 HS δ (A) 3. To see the e -invariants of two equivalent elements in Ωn (BGL(C )) are equivalent, we assume (Σ0, f 0) is another representative and (W, F ) a pair 0 0 0 0 that is bounded by (Σ# − Σ , f#f ) when n ≥ 5, or (Σ#Σ0# − Σ , f#f0#f ) when n = 3, where f0 :Σ0 → A. In the case of n = 3, because A → BU is the constant map, there exists a contractible manifold C0 with ∂C0 = −Σ0 and an extension ¯ ¯ F : W ∪Σ0 C0 =: W → BU that extends ι ◦ F . To simplify notations, we also let W¯ = W and F¯ = ι ◦ F when n ≥ 5. Now because W¯ is contractible, there is a homotopy

ψt : W¯ × I → BU with ψ1 = F¯ and ψ0 the constant map. In this case, one can also “close” the bottom via a contractible manifold C¯ and extend ψt to Ψ: ¯ Ψ: W × I ∪W¯ ×{0} C =: W\× I → BU.

42 In more details, we choose first a contractible manifold C(0) such that ∂C(0) = (0) 0 0 −Σ and hence obtain that W × {0} ∪Σ#−Σ0 C#bC is a homology (n + 1)- HS ¯ sphere. Now because Ωn+1(∗) is trivial in Top, we can find C a contractible (n + 2)-manifold such that

0 0 ∂C = W × {0} ∪Σ#−Σ0 C#bC . Then, for n = 3, the 8-tuple

(W, Σ0, F, f0; −W\× I, −C0, Ψ, Ψ|C0 ), or, for n ≥ 5, the 4-tuple (W, F ; −W\× I, Ψ), gives us the equivalence

0 0 0 0 HS rel (Σ, f; Σ\× I, Φ) ∼ (Σ , f ; Σ\× I, Φ ) ∈ Ωn (BGL (C)),

0 where Σ\× I and Σ\× I are the contractible manifolds Σ × I ∪Σ×{0} −C and 0 0 0 Σ ×I∪Σ0×{0}−C , respectively, and Φ and Φ are the restrictions of Ψ. Applying t¯∗, we further obtain they yield the equivalent elements

¯ ¯ 0 0 0 0 HS Y [t∗(Σ, f; Σ\× I, Φ)] ∼ [t∗(Σ , f ; Σ\× I, Φ )] ∈ Ωn ( K(C, 2i − 1)). i Finally, one can easily verify it is a homomorphism from the definition, and thus we have shown the e0-invariant is well-defined.

Lemma 2.2.7. For n ≥ 3, the e0-invariant can be identified with the e-invariant via the following commutative diagram:

e0 HS δ (A) ∗ ΩHS (BU)0 Ωn (BGL(C )) n+1 C/Z

+ †

0 e∗ n K˜ −1(Sn) K]aC (S ) C/Z where + and † are the homomorphisms induced by the (universal property of the) plus construction (see the remark following the proof for more about †). Proof. 1. In view of Hausmann and Vogel’s theory (Theorem 2.1.9), we know 0 n every element in K]aC (S ) can be represented by a flat vector bundle over a homology sphere Σn. 2. Observe that, with the simplified geometric models in 2.1.24 and 2.2.4, one can construct a natural homomorphism l : K˜ −1(Σn) → ΩHS (BU)0 , C/Z n+1 C/Z by the assignment: G0 (Σn) → ΩHS(BU)0 AP S ∗ C/Z

(ω, [(V, φ)1]) 7→ (ω, [(Σ, f; Σ\× I, Φ)1]),

43 where the acyclic manifold Σ\× I is constructed as follows: The trivialization ? ? φ : V ⊕  →  gives us a homotopy φt :Σ × I → BU such that φ1 is the constant map and φ0 = φ. Then we close the homotopy from the top by an acyclic manifold C with ∂C = −Σ and extend φt to Φ, namely, considering the map

Φ: Σ\× I = Σ × I ∪Σ×{1} C → BU. 3. Now observe that the diagram in the statement fits into the following prism-like diagram:

e0 ΩHS(BGL( δ)) ∗ ΩHS (BU)0 n C n+1 C/Z

l

+ † e¯ n δ ∗ ˜ −1 n [Σ , BGL(C )] K (Σ )C/Z + + e ˜ 0 n ∗ ˜ −1 n Ka(S , C) K (S )C/Z

Except for the square behind, all the other squares and triangles are commuta- tive (see the remark following for the homomorphism †). Hence, by the obser- vation in the step 1, we can conclude the square behind also commutes. Remark 2.2.8. Recall that the homomorphism

HS n Ωn+1(BU)C/Z → GAP S(S ) is given by the following composition (see the proof of Theorem 2.1.15 and Lemma 2.1.22):

HS n Ωn+1(BU)C/Z → πn+1(BU)C/Z → GAP S(S ), (2.13) where πn+1(BU)C/Z is the cokernel of the homomorphism

(−j∗,p∗) Y πn+1(BU) ⊗ Q −−−−−→ πn( K(C, 2i − 1)) ⊕ πn+1(BU)Q/Z, (2.14) i which is used in the proof of Theorem 2.1.22. The second homomorphism in (2.13) follows from the construction in [APS75, p.428-9], where they use the Bott periodicity with a fixed isomorphism Hn ⊕ (n − 1)1 ' nH, for each n, to obtain the first geometric model in Theorem 2.1.15 which leads to the geometric model 2 GAP S(−), where H is Hopf bundle over S . However, the use of the isomorphism Hn ⊕ (n − 1)1 ' nH might obscure the topological picture of the connection ∗ between π∗+1(BU)C/Z and GAP S(S ) in 2.14. Nevertheless, when n = 1, one can choose H = H, and the resulting composition (2.13) is just the homomorphism induced by the plus construction. In addition, the homomorphism l is also much easier to define with the simplified geometric model. Nonetheless, for constructing the e0-invariant, there is no need to use the simplified geometric models. In effect, one can consider the following alternative HS δ (A) construction: Given an element in (Σ, f) ∈ Ωn (BGL(C )) , we choose a homotopy φt :#Σ × I → BU such that φ1 = #f and φ0 is the constant map, m m

44 where m ∈ N, then close it up with an acyclic manifold C from the bottom, and define the e0-invariant to be

0 1 ¯ e∗(Σ, f) = ( t∗(Σ, f; #\Σ × I, Φ), (Σ, f; −#\Σ × I, Φ)). m m m It is not difficult to see it is well-defined and identical to the definition given in 2.2.5. Now we are at the position to prove theorem 2.3 in [JW95]. Theorem 2.2.9. The e-invariant restricted to an isomorphism on the torsion 0 n subgroup of K]aC (S ): 0 n ∼ ˜ −1 n e∗|Tor : Tor(K]aC (S )) −→ Q/Z ⊂ C/Z = K (S )C/Z, for every odd natural number n. As mentioned in the beginning of the section, the theorem can be reduced to the following lemma. Lemma 2.2.10. The following diagram is commutative.

−1 ι n ˜ −1 n K (S ) /m K]aC (S )Z/m Z

b j∗

0 e∗ n K˜ −1(Sn) K]aC (S ) C/Z where j∗ is the composition: ˜ −1 n ˜ −1 n ˜ −1 n K (S )Z/m → K (S )Q/Z → K (S )C/Z, induced by the maps of prespectra

j : MZ/m → MQ/Z → MC/Z, ι the map induced by the canonical (topology-changing) map

δ BGL(C ) → BU, and b the Bockstein homomorphism. Proof of the Theorem 2.2.9. For completeness, we recall the proof given in [JW95, Corollary 2.4]: By [Sus84, Corollary 4.6], we know the upper horizontal homo- morphism is an isomorphism, for every m. Now since the torsion subgroup of ˜ −1 n K (S )C/Z is the image of the injection ˜ −1 n ˜ −1 n K (S )Q/Z ,→ K (S )C/Z, which is Q/Z (see the second step in the proof of 3.2.4 for more general cases), we obtain the surjectivity. The injectivity follows from the fact that every torsion 0 −1 n n element in K]aC (S ) is the images of an element in K]aC (S )Z/m via the homomorphism b, for some m.

45 Proof of the Lemma 2.2.10. We first explain how to realize the homomorphisms involved in the diagram via the geometric models developed previously. In particular, we will see the following diagram is commutative.

HS δ ι HS Ωn+1(BGL(C ))Z/m Ωn+1(BU)Zm

−1 ι n ˜ −1 n K (S ) /m K]aC (S )Z/m Z

0 b b j∗ j∗

0 e∗ n K˜ −1(Sn) K]aC (S ) C/Z

e0 HS δ ∗ HS Ωn (BGL(C )) Ωn+1(BU)C/Z

Explanation: Here we define the homomorphisms in the outer square.  (  [Σ, ι ◦ f; W, ι ◦ F ], n ≤ 5 ι([Σ, f; W, F ]) =  [Σ, ι ◦ f; W,c ι[◦ F ], n = 3 b([Σ, f; W, F ]) = [Σ, f]   0 j∗([Σ, f; W, F ]) = [0, [(Σ, f; W, F )m]],

where Wc = C ∪Σ0 W with C a contractible manifold with ∂C = Σ0 and ι[◦ F is the extension of ι ◦ F by the constant map c : C → ∗ ∈ BU—This can be done f0 δ as the composition Σ0 −→ A → BGL(C ) → BU is the constant map, where A is the homotopy fiber of BGL(Cδ) → BGL+(Cδ). With these geometric realizations, we see the four trapezoids around the inner square are commutative and hence the question is reduced to showing the 0 commutativity of the outer square. We write down the compositions j∗ ◦ ι and 0 e∗ ◦ b via the geometric models: ( 0 (0, [(Σ, ι ◦ f; W, ι ◦ F )m]), n ≥ 5 j∗ ◦ ι([Σ, f; W, F ]) = (0, [(Σ, ι ◦ f; W,ˆ ι[◦ F )m]), n = 3 0 1 ¯ e∗ ◦ b([Σ, f; W, F ]) = ( t∗(#Σ, #f; #\Σ × I, Φ), [(Σ, ι ◦ f; −#\Σ × I, Φ)m]) m m m m m where (#\Σ × I, Φ) is constructed as (Σ\× I, Φ) in Lemma 2.2.5. The remaining m task is then to show these two geometric representatives represent the same HS element in Ωn+1(BU)C/Z, and the main tool is obstruction theory. We start with the case n ≥ 5 and consider the following diagram:

W × ∂I ∪ x0 × I ⊂ W × I

ι ◦ F ∪ c x0×{1} Ψ

BU

46 where c is the constant map:

c : {x0} × I ∪x0×{0} W × {0} → ∗ ∈ BU with x0 the base point, and

ι ◦ F : W × {1} → BU.

Because ∗ H (W × I,W × ∂I ∪ x0 × I, π∗−1(BU)) = 0, for all ∗, Ψ can be obtained by obstruction theory. If we define Φ to be the restriction of Ψ on the submanifold

#\Σ × I := #Σ × I ∪ −W × {0} ⊂ W × I, m m

HS rel then we have (−W, F ; W ×I, Ψ) is bound by (#Σ, #f; #\Σ × I, Φ) in Ωn (BGL (C)), m m m and thus

1 ¯ HS Y t∗(#Σ, f; #\Σ × I, Φ) ∼ 0 ∈ Ωn ( (K(C, 2i + 1))). m m m i Furthermore, by the following identification

∂(W × I, Ψ) = (#\Σ × I ∪#Σ×{1} W, Φ ∪#Σ×{1} ι ◦ F ), m m m we obtain

HS [(Σ, ι ◦ f; W, ι ◦ F )m] ∼ [(Σ, ι ◦ f; −#\Σ × I, Φ)m] ∈ Ωn+1(BU)Q/Z, m

0 0 and hence j∗ ◦ ι([Σ, f]) = e∗ ◦ b([Σ, f]). As for the case n = 3, due to the extra homology sphere Σ0 and f0 :Σ0 → A, it is technically more complicated; the idea, however, is the same. We consider the following diagram and use obstruction theory to obtain Ψ:

W × ∂I ∪Σ0×{1}tΣ0×{0} C × I ⊂ (W ∪Σ0 C) × I

ι ◦ F ∪ c Ψ

BU

where C is a contractible manifold with ∂C = Σ0; c : C × I ∪Σ0 W × {0} → ∗ ∈ BU, the constant map; and ι ◦ F : W × {1} → BU. Now the following 8-tuple

(−W, Σ0, F, f0;(W ∪Σ0 C) × I,C, Ψ, c) implies HS rel (#Σ, #f; #\Σ × I, Φ) ∼ 0 ∈ Ω3 (BGL (C)), m m m

47 where Φ the restriction of Ψ on

#\Σ × I := #Σ × I ∪#Σ×{0} −(W ∪Σ0 C) × {0} ⊂ (W ∪Σ0 C) × I. m m m

Hence, on applying t¯∗, we obtain the following:

1 ¯ HS Y t∗(#Σ, #f; #\Σ × I, Φ) ∼ 0 ∈ Ω3+1( K(C, 2i + 1)). m m m m i Furthermore, because of the identification

∂(Wc × I, Ψ) = (Wc × {1} ∪#Σ×{1} #\Σ × I, ι[◦ F ∪#Σ Φ), m m m we see

HS [(Σ, ι ◦ f; W,c ι[◦ F )m] ∼ [(Σ, ι ◦ f; −#\Σ × I, Φ)m] ∈ Ω3+1(BU)Q/Z, m and hence

(0, [(Σ, ι ◦ f; W,c j[◦ F )m]) ∼ 1 ¯ HS ([ t∗(#Σ, #f; #\Σ × I, Φ)], [(Σ, ι ◦ f; −#\Σ × I, Φ)m]) ∈ Ω3+1(BU)C/Z m m m m m

2.2.2 A C/Z-valued invariant and the ξ˜-invariant In this subsection, we construct a C/Z-valued invariant, which is induced from the e-invariant and has a close relationship with the ξ˜-invariant ([APS75] or [APS76]). Note we have the following homomorphism

0 ˜ −1 e∗ : K]aC (X) −→ K (X)C/Z, where X is a pointed finite CW -complex. If X = M, a pointed compact spin manifold, we can further obtain

˜ −1 ⊗[D] ˜ −1 ˜ TD : K (M)C/Z −−−→ K (TνM )C/Z; ∗ ˜ −1 ˜ ˜ −1 N ∼ c : K (TνM )C/Z −→ K (S )C/Z = C/Z, where D is the Dirac operator induced by the spin structure on M, N is an N ˜ odd number large enough so that M can be embedded into S , and TνM is the reduced Thom space of the associated normal bundle νM with respect to the embedding. It is known that the Dirac operator gives us a Thom class in ˜ 0 K (TνM ) (see [LM89, Appendix C]). TD denotes the associated Thom isomor- N ˜ phism and c : S → TνM the collapsing map. Then the composition yields a C/Z-valued invariant

∗ δ c ◦ TD ◦ e¯∗ :[M, BGL(C )] → C/Z, for every pointed compact spin manifold M.

48 Corollary 2.2.11 (Relation with ξ-invariant). Given M a spin manifold and a representation (or flat vector bundle)

ρ : π1(M) → GLN (C), we have ∗ ˜ c ◦ TD ◦ e¯∗(M, ρ) = ξ(ρ, M). Proof. This results from the comparison theorem for the geometric models of topological K-theory with C/Z-coefficients (Theorem 2.1.22), the remarks in [APS76, p.87,p. 89-90]. Now we explain how the ξ˜-invariants of homology spheres can help us to determine the e-invariant. Lemma 2.2.12. Let

ρ : π1(Σ) → GLN (C), 0 0 ρ : π1(Σ ) → GLN (C), be two representations and suppose (Σ, ρ) and (Σ0, ρ0) determine the same ele- 0 n ment x in K]aC (S ), via the homomorphism

0 HS + δ n α :Ωn (BGL (C )) → K]aC (S ), then ξ˜(Σ, ρ) = ξ˜(Σ0, ρ0). Proof. Recall first that the ξ˜-invariant is a spin cobordism invariant and any 0 0 H∗-cobordism is a spin cobordism. For n ≥ 5, we know (Σ, ρ) and (Σ , ρ ) 0 0 bound a H∗-cobordism (H,F ), and hence (Σ, ρ) and (Σ , ρ ) are spin cobordant spin via (H,F ). For n = 3, we first note the third spin cobordism group Ω3 (∗) is trivial, and therefore Σ and Σ0 are spin cobordant. Let W be the spin cobordism, and s : W → TSpin the associated map into the Thom space of BSpin with respect to BSpin → BO, the canonical vector bundle. Then observe there is a plus map π : W → Sn × I that restricts to the plus map Σ → Sn and Σ0 → Sn on the boundary of W . n n + Now suppose x ∈ K]aC(S ) is represented by the map f : S → BGL (C). Then the following composition

k+n id∧π n id∧p n s∧f + δ S ∧I+ → TνW ∧W+ −−−→ TνW ∧(S ×I)+ −−−→ TνW ∧S+ −−→ TSpin∧BGL (C )

0 0 spin + δ implies (Σ, ρ) and (Σ , ρ ) are spin cobordant in Ω3 (BGL (C )) and hence spin δ spin cobordant in Ω3 (BGL(C )), where TνW is the Thom space of the normal bundle of W and p : Sn × I → Sn the obvious projection. We have also used the fact:

∞ δ ∞ δ Σ TSpin ∧ BGL(C ) ' TSpin ∧ Σ BGL(C ) ∼ ∞ + δ ∞ + δ −→ TSpin ∧ Σ BGL (C ) ' Σ TSpin ∧ BGL (C ) in Ho(A).

49 Next, we observe there is a commutative diagram

0 e∗ ∼ n K˜ −1(Sn) / K]aC (S ) C/Z C Z

+ +

e T ∗ 0 ∗ −1 n D −1 c −1 N n K˜ (Σ ) K˜ (T˜ ν n ) K˜ (S ) , K]aC (Σ ) C/Z ∼ Σ C/Z ∼ C/Z where + means the homomorphism induced by the universal property of the n n ∗ plus map Σ → S . In particular, in this case, TD and c are isomorphisms. Therefore, by Lemma 2.2.12 and Corollary 2.2.11, we can deduce the following:

Theorem 2.2.13. Given a homology sphere Σ and a representation ρ of π1(Σ), 0 n n if (Σ , ρ) represents an element x ∈ K]aC (S ), then e∗(x) can be identified with ξ˜(ρ, Σ).

2.2.3 The ξ˜-invariant of Seifert homology spheres In this subsection, we use the relation between the ξ˜-invariant and the e- invariant described earlier to do some calculation. In particular, we examine Theorem C, D and E in [JW95] with the correct ξ˜-invariants of lens spaces and a different construction of the spin cobordism between # L(ai, bi) and 1≤i≤n Σ(a1, ..., an). We first recall the formula for the ξ˜-invariants of lens spaces from [Gil84, Sec.2]. Lemma 2.2.14. Let

s ρs(λ) = λ , 2πi < e p = λ > = Z/p = π1(L(p, q)), where q is chosen so that the greatest common divisor g.c.d{q, 2p} is 1, then we have the following formula:

−ds2 ds ξ˜(ρ ,L(p, q)) ≡ − mod , s 2p 2 Z where d is a natural number that satisfies dq ≡ 1 (mod 2p). Note that when p is even, q specifies the spin structure on L(p, q). More precisely, the two numbers q and q + p indicate the two different spin structures on L(p, q).

Proof. As a special case of the formula proved in [Gil84, Theorem 2.5], we have the following identity: −d p ξ(ρ ,L(p, q)) ≡ Aˆ (s + , (1, q)) mod , (2.15) s p 2 2 Z

50 ˆ p ˆ p where q is chosen so that 1 = g.c.d{q, 2pµ(A2(s+ 2 , (1, q)))}, µ(A2(s+ 2 , (1, q))) ˆ p is the denominator of A2(s + 2 , (1, q)), and d is another integer satisfying 1 = ˆ p g.c.d{dq, 2pµ(A2(s + 2 , (1, q)))}. Recall also ˆ X Ab(x) Aˆ (s; x) := sa , k a! a+2b=k

ˆ where x = (x1, ..., xl), for some l ∈ N, and Ab(x) is the component of degree b in Y Aˆ(xi), 1≤i≤l the product of Aˆ(x) polynomials. To apply this to ξ˜-invariant, we consider the difference ˜ ξ(ρs,L(p, q)) := ξ(ρs,L(p, q)) − ξ(ρ0,L(p, q)). Combing formula (2.15), we obtain the formula in the statement as well as the constraints on q and d. Remark 2.2.15. It is stated in [JW95, Lemma 5.2] that

−ds2 ξ˜(ρ ,L(p, q)) = ; dq ≡ 1 mod p, s 2p yet, with this formula, we see different d0s that satisfy dq ≡ 1 mod p yield dif- ferent values in C/Z, but the formula should not depend on d.

51 Seifert homology 3-spheres: Before starting the computation of ξ˜-invariants of Seifert homology 3-spheres, we first review some basic properties of Seifert homology 3-spheres. A Seifert ho- mology sphere Σ(a1, ..., an) is a homology sphere and at the same time a Seifert manifold with ai the order of the i-th exceptional fiber—a Seifert manifold is a manifold admitting a Seifert fibering (see [Hat, p.13]). It can be constructed topologically as follows (see [Sav02, p.2-4 and Figure 1.1]): Let

2 2 2 F = S \ int(D1 ∪ ... ∪ Dn) be the n-punctured 2-sphere and W → F a S1 fiber bundle with Euler number b. Assume the bundle over ∂F has a fixed trivialization such that the boundary 2 1 of W can be identified with the n-tori ∂Dk ×S , k = 1...n. Now we past another 2 1 1 1 n solid tori Dk ×S , k = 1, ..., n in such a way that ak(S ×{1})+bk({1}×S ) in 2 1 the k-th boundary component of W is null homotopic in Dk × S after pasting, n where {(ak, bk)}k=1 are the n relatively prime pairs. In terms of Dehn surgery, we obtain the following figure:

Figure 2.1: The Linking diagram of Σ(a1, a2, ..., an)

It is easy to see from the figure it has the following representation of the funda- mental group:

−b ai −bi π1(Σ(a1, a2, ..., an)) =< h, x1, ...xn|[xi, h] = 1, x1...xn = h , xi = h , ∀i > . (2.16) Since it is a homology sphere, we further have the equation

bi a1 ··· an(−b + Σ( )) = 1. ai Now by the theorem in [Kir78] (see also [Rol90, p.264,p.278 Remark I.7]), we see that the diffeomorphic type of a Seifert homology sphere is completely de- termined by these n relatively prime numbers a1, ..., an. Irreducible representations: Observe first that all complex representations of π1(Σ(a1, ..., an)) factor through SLN (C) as every homology sphere has its fundamental group is perfect.

52 Secondly, as from now on we will restrict our attention only to the representa- tions ρ : π1(Σ(a1, ..., an)) → SLN (C) that has ρ(h) is a scalar multiple of the identity matrix, for example, irreducible representations, we can assume

ρ(h) = λhI; rh λh = ζN ; 2πi/N ζN := e , and if λ1(j), ..., λN (j) are the eigenvalues of ρ(xj), we further have

λ (j) = ζNsk(j)−bj rh , (2.17) k Naj where sk(j) is an integer such that

0 6 sk(j) < aj for every 1 6 j 6 n and 1 6 k 6 N.

We say ρ is the representation of type (sk(j)).

53 Spin cobordisms: Consider (2.1) as a relative Kirby diagram (see [GS99, Section 5.5]) and denote the corresponding 4-dimensional cobordism by W . That is we attach a 2-handle along the circle with the framing coefficient b to the product of #L(ai, −bi) × I: 1≤i≤n

2 2 W := #L(ai, −bi) × I ∪(∂D2)×D2 D × D . (2.18) 1≤i≤n

It is clear that the boundary of W is homeomorphic to the disjoint union of the connected sum of lens spaces #L(ai, bi) and the Seifert homology 3-sphere 1≤i≤n ˜ Σ(a1, ..., an). In order to use W to compute the ξ-invariants of Σ(a1, ..., an), we yet need to check if the 4-dimensional cobordism W is spinnable. We first recall that every closed oriented 3-manifold can be obtained by performing Dehn (rational) surgery on a link L in S3 (see [Rol90, p.273]). We denote the resulting 3-manifold by ML. It is also proved by Kirby that ML and 0 ML0 are diffeomorphic if and only if L can be obtained from L by performing a sequence of Rolfsen moves or introducing or deleting an unknot with coefficient ∞ (see [Rol90, p.278]). In particular, utilizing Rolfsen moves and deleting an unknot with coefficient ∞, we have the following useful move, called slam-dunk by Cochren:

Figure 2.2: Slam-dunk

Suppose p, q both are larger than zero. There are unique integers n and r such that p = nq − r with 0 ≤ r < q. Hence, we obtain a slam-dunk algorithm that turns a rational surgery on a knot into a unique integral surgery on a link when p, q are larger than zero. By an integral surgery, we understand a Dehn surgery with surgery coefficients integers (or framing coefficients). The following illustrates how via slam-dunk algorithm a rational surgery diagram of the lens space L(5, −2) is turned into an integral surgery diagram:

Figure 2.3:

Note the arrow indicates from where the algorithm starts. We also recall that, given an integral surgery diagram L, one can view it as a Kirby diagram, mean- ing, considering ML as the boundary of the 4-manifold (or 2-handlebody) WL.

54 Now the Wu’s formula says the second Stiefel-Whitney class w2(WL) is uniquely determined by the identity in Z/2:

< w2(WL), x >= x.x for every x ∈ H2(WL; Z/2), where x.x is the intersection number. This, when written in terms of linking numbers of the components in L, gives the following definition, which is the key to visualize the spin structures on the 3-manifold ML and the obstructions to extending them over WL. Definition 2.2.16. Given a framed link (integral surgery diagram) L, a charac- teristic sublink L0 ⊂ L is a sublink with lk(L0,K) ≡ lk(K,K)(mod 2) for every component K in L, where lk(L1,L2) stands for the linking number of two links L1 and L2. Lemma 2.2.17. Given a framed link L, there is a 1−1 correspondence between the set of spin structures on ML and the set of characteristic sublinks of L. Proof. The detailed proof can be found in [GS99, Proposition 5.7.11]. We only recall that the bijection is given by assigning a spin structure s on ML with the 2 relative obstruction in w2(WL, s) ∈ H (WL,ML; Z/2) ' H2(WL; Z/2). Since the classes in H2(WL; Z/2) correspond bijectively to the sublinks of L and the 2 image of w2(WL, s) in H (WL; Z/2) must be the second Stiefel-Whitney class, we see the sublink corresponding to w2(WL, s) is a characteristic sublink, in view of Wu’s formula. The following illustrates the unique spin structure on L(5, 2) as well as L(7, 3) and the two spin structures on L(8, 3)—(red) bold circles denote the character- istic sublinks:

Figure 2.4:

Given a lens space L(p, −q), we call the Dehn surgery on an unknot with p surgery coefficient q the canonical rational surgery diagram of L(p, −q) (the left-hand side of the figure below) and the integral surgery on L p , the framed q link obtained by applying slam-dunk algorithm described above to the canonical surgery diagram for L(p, −q), the canonical integral surgery diagram of L(p, −q) (the right-hand side of the figure below).

55 Figure 2.5:

The following three lemmas describe some general properties of the canonical integral surgery diagram of L(p, −q) when p and q both are larger than 0. Lemma 2.2.18. Given L(p, −q) with p even, then the two characteristic sub- links (spin structures) of its canonical integral surgery diagram can be distin- guished by whether the first component is in the characteristic sublink. Proof. Note that every closed oriented 3-manifold is parallelizable and 1 H (L(p, −q); Z/2) = Z/2, so there are two spin structures on L(p, −q). Suppose we are given a charac- teristic sublink L0 and color it. Then, to obtain another characteristic sublink 00 L , we claim that the color of the first component K1 must be changed. If it is not the case, the color of the second component also cannot be changed because otherwise we have 00 0 lk(L ,K1) ≡ lk(L ,K1) + 1 6≡ lk(K1,K1) (mod 2). By induction, we assume, for every i ≤ k, the color of the i-th component remains unchanged, and we want to show the color of the (k + 1)-th component should stay the same. It is easy to see since if its color changes, the following gives a contradiction 00 0 lk(L ,Kk) ≡ lk(L ,Kk) + 1 6≡ lk(Kk,Kk) (mod 2). Therefore we see the first component must change its color, or otherwise L00 = L0. For convenience, we continue to assume components in a given characteristic sublink are colored. Lemma 2.2.19. Given L(p, −q) with p odd and q even, then the first component of L p is not in the characteristic sublink. q Proof. Suppose p = qn − r with 0 < r < q. Then the first component of the canonical surgery diagram has coefficient n. Since q is even, we know the characteristic sublinks of L q are distinguished by the color of its first compo- r nent. Now if n is even, we choose the characteristic sublink of L q that has r its first component uncolored, and it is clear that the characteristic sublink of L q extends to the characteristic sublink of L p . If n is odd, we pick up the r q characteristic sublink of L q whose first component is colored, and it extends to r the characteristic sublink of L p . In either case, we have the first component of q the canonical integral surgery diagram of L(p, −q) is not colored. Before showing the third lemma, we first recall the effect of handle sliding on characteristic sublinks ([GS99, p.190]): Suppose we slide K over K0, then K0 has its color changed if and only if K is colored.

56 Lemma 2.2.20. Given L(p, −q) with p and q both odd, then the first component of L p is in the characteristic sublink. q Proof. We first claim: Given the canonical integral surgery diagram of L(p, −q) as follows:

Figure 2.6:

we have the expression of the canonical integral surgery diagram of L(p, −q − p) looks like:

Figure 2.7:

This can be seen by the following computation: Suppose p = n1q − r with 0 < r < q. Meaning, after the first step of the algorithm (slam-dunk) for the canonical rational surgery diagram of L(p, −q), the framing coefficient of the q second component is r . Now consider the canonical rational surgery diagram of L(p, −q − p) and note the first slam-dunk gives us an unknot with the framing coefficient 1 because p = (p + q)1 − q and 0 < q < p + q. Applying slam-dunk again, we have qm − s = p + q with 0 < s < q. Since p = n1q − r, we obtain m = n1 + 1 and s = r. That means, after applying slam-dunks twice, we get q the same framing coefficient r just as we did after applying slam-dunk once to the canonical rational surgery diagram of L(p, −q). In this way, we see that deleting the first component of the link in Figure 2.6 and deleting the first two components in Figure 2.7 should result in the same framed link. From this computation, we also know, if the first component of the link in Figure 2.6 is n1, the second component of the link in Figure 2.7 must be n1 + 1, and the framing of the first component of the link in Figure 2.7 is always 1. This proves the claim. Secondly, we note Figure 2.7 can be obtained by sliding the first component of Figure 2.6 over a separate unknot with framing 1 as illustrated below:

Figure 2.8:

57 and the characteristic sublink of the diagram above is the union of the char- acteristic sublink of L p and that separated unknot. If the first component of q L p is not in the characteristic sublink. Sliding the first component of L p over q q this separated unknot does not change the color of this unknot, so we get the characteristic sublink of L p contains the first component of L p . Yet it is p+q p+q not possible because p + q is even, and Lemma 2.2.19 tells us the first compo- nent of L p must not in the characteristic sublink. We get a contradiction, p+q and therefore the first component of L p has to be in the characteristic sublink. q The proof is completed. Before returning to our 4-dimensional cobordism W , we recall that, given a framed link L and a framed knot K in S3, we have the 4-dimensional cobordism 2 2 WL∪K := ML × I ∪N(K) D × D where the attaching diffeomorphism from S1 × D2 to N(K) is determined by the framing of K. Now suppose a spin c structure, a characteristic sublink L of L, of ML is given. Then this spin structure of ML can be extended to WL∪K if and only if

lk(Lc,K) ≡ lk(K,K) (mod 2)

(see [GS99, p.189-190]). Now applying slam-dunk moves to the relative Kirby diagram (2.1), we ob- tain the following relative (integral) Kirby diagram:

Figure 2.9:

j where ai ∈ Z and which also represents the 4-dimensional cobordism W con- structed in (2.18). With Figure 2.9 and the knowledge we just obtain about the characteristic sublink of the integral surgery diagram for a lens space, we can now investigate the spinnability of W . Firstly, we assume ai is odd for every i ∈ {1, ..., n}. Hence, without loss of generality, we may assume bi is even for every i. In view of Lemma 2.2.19, we know the first component of L ai does bi not belong to the characteristic sublink, for every i. Let Lc be the union of the characteristic sublinks of L ai , for all i, and K be the knot with the framing bi coefficient b (see Figure 2.1). Then we have lk(Lc,K) ≡ 0(mod 2) (see Figure 2.9). On the other hand, the equality

n X bi a ...a (−b + ) = 1 1 n a i=1 i gives b ≡ 1(mod 2). Thus, in this case, the spin structure cannot be extended over W .

58 Suppose one of a1, ..., an is even, a1 say. Then there are two characteristic sublinks for the canonical integral surgery diagram of #L(ai, bi). We may as- i c c sume bi is even when i 6= 1 and let L1 and L2 be the characteristic sublinks corresponding to the two spin structures of #L(ai, bi). By Lemma 2.2.18, we i c c may assume lk(L1,K) ≡ 1(mod 2) and lk(L2,K) ≡ 0(mod 2) (See Figure 2.9 and note the only possible component of the characteristic sublink that has non-trivial linking number with K is the first component of L a1 ). So when b b1 c is odd, the spin structure on #L(ai, bi) corresponding to L1 can be extended, i c whereas when b is even, one can extend the spin structure corresponding to L2 over W . We summarize these properties of W in the following proposition: ` Proposition 2.2.21. 1. ∂W = #L(ai, bi) Σ(a1, ..., an) and 1≤i≤n

a1 an π1(W ) =< x1, x2, ..., xn | x1 = ... = xn = x1...xn = 1 > .

2. There is a short exact sequence

0 → Z → π1(Σ(a1, ..., an)) → π1(W ) → 0,

where the first homomorphism sends the generator of Z to h, and the second homomorphism is induced by the inclusion Σ(a1, ..., an) ,→ W .

3. When a1, ..., an are all odd, W is not spinnable.

4. When one of a1, ..., an is even, then W admits a unique spin structure. c 5. The spin structures on W are parameterized by Z. c 6. The spin structures on #L(ai, bi) are parameterized by 2Z/a1...an when i a1, ...an are odd, whereas when one of a1, ..., an is even they are parame- terized by Z/2 ⊕ 2Z/a1...an. c 7. When a1, ..., an are all odd, there is a spin structure on W such that it re- c stricts to the canonical spin structures on Σ(a1, ..., an) and #L(ai, −bi). i By canonical spinc structure of a spin manifold, we mean the spinc struc- ture induced by the spin structure and the trivial complex line bundle. Proof. Recall first the cohomology groups of the lens space L(p, q):  Z/p ∗ = 2 ∗  H (L(p, q); Z) = Z ∗ = 0, 3 . 0 otherwise

Hence, the cohomology groups of the connected sum #L(ai, bi) can be easily i deduced. They are:  Z/a1...an ∗ = 2 ∗  H (#L(ai, bi); Z) = Z ∗ = 0, 3 . i 0 otherwise

59 With the universal coefficient theorem, we can further obtain the cohomology groups of #L(ai, bi) with coefficients in Z/2: i ( ∗ Z/2 ∗ = 0, 3 H (#L(ai, bi); Z/2) = i 0 otherwise when the product a1....an is odd;  Z/2 ∗ = 0, 3 ∗  H (#L(ai, bi); Z/2) = Z/2 ∗ = 1, 2 i 0 otherwise when the product a1...an is even. Using the long exact sequence induced by the pair (W, Σ(a1, ..., an)) or (W, #L(ai, bi)), we can compute the cohomology groups of W with coefficients i in both Z and Z/2: ( ∗ Z ∗ = 2, 3 H (W ; Z) = ; 0 otherwise ( ∗ Z/2 ∗ = 2, 3 H (W ; Z/2) = . 0 otherwise In particular, we see W admits a spinc because the second Stiefel-Whitney class is the mod 2 reduction of an integral class, in view of the universal coefficient theorem. In fact, every 4-manifold admits a spinc structure (see [GS99, Remark 5.7.5]). Using the computation above we can also identify the parameter set for the sets of spinc and spin structures. Namely, the spinc structures are param- eterized by 2H2(W ; Z) ⊕ H1(W ; Z/2) (see [LM89, p.392]), while H1(W ; Z/2) parameterizes the set of spin structures when W is spin (see [LM89, p.82]). To see the last claim, we note to find such a spinc structure is equivalent to find a class x in H2(W ; Z) such that the following two conditions are fulfilled: i: Let H2(W ; Z) → H2(W ; Z/2) be the homomorphism induced by epimor- phism Z → Z/2. Then the image of x in H2(W ; Z/2) is the second Stiefel- Whitney class. 2 2 ii: Let H (W ; Z) → H (#L(ai, −bi), Z) be the homomorphism induced by i 2 the inclusion #L(ai, −bi) ,→ W . Then the image of x in H (#L(ai, −bi), Z) is i i trivial. Since H3(W ; Z) = Z, we have the homomorphism

2 2 H (W ; Z) → H (W ; Z/2) can be identified with the surjective homomorphism Z → Z/2, and thus every odd element in H2(W ; Z) satisfies the first condition. On the other hand, as 3 H (W, #L(ai, −bi); Z) = 0, the homomorphism i

2 2 Z ' H (W ; Z) → H (#L(ai, −bi); Z) ' Z/a1...an i

60 must be surjective. Hence every element that is divided by a1...an satisfies the second condition. Now because a1...an is odd, we can let x be the class corresponding to the element ka1...an ∈ Z with k odd. Then the resulting complex line bundle restricts to the trivial complex line bundle on the boundary of W and induces a spinc structure on W . Note there is no non-trivial complex line bundle over Σ(a1, ..., an).

Remark 2.2.22. Another construction of a 4-dimensional cobordism with bound- ary homeomorphic to #L(ai, bi) and Σ(a1, ..., an) is given in [JW95, p.951]. We 1≤i≤n follow their notation and name it W as well, yet it is not clear if their W is homeomorphic to our W . It is claimed in [JW95], their W is spin in all cases. However, the argument given there for this claim appears to contain some gaps. In effect, the homomorphism H2(W, Z/2) → H2(W \ A, Z/2) is not an isomor- phism as stated there. Hence, one cannot deduce W admits a spin structure from the fact that W \A is parallelizable. Actually, from their construction, this cobordism W should have H2(W, Z) = Z ⊕ Torsions, whereas H2(W \ A, Z) is actually trivial. This can be seen via the Mayer-Vietoris sequence as we have N(A), the thickened A, is homotopy equivalent to A; N(A) ∩ (W \ A) is homo- topy equivalent to T 2, a 2-torus; and W \ A a punctured 2-sphere with n-holes. By counting ranks, we see there is at least one free element in H2(W, Z). The torsion part can be seen from the fundamental group of W .

Given a representation ρ of π1(Σ(a1, a2, ...an)) in SLN (C) with ρ(h) a scalar multiple of the identity, then the tensor product ρ ⊗ ρ¯ has (ρ ⊗ ρ¯)(h) = id. That implies ρ can be extended to a representation % of π1(W ) (see Proposition 2.2.21) such that

% ◦ ι∗ = ρ ⊗ ρ¯; 0 M % ◦ ι∗ = ρsk−sl , 1≤k,l≤N

0 where ι and ι are the inclusions from Σ(a1, ..., an) and #L(ai, bi) into W , respec- 1≤i≤n tively, and ρsk−sl is the representation induced by the representations ρsk(i)−sl(i) ˜ on π1(L(ai, bi)), for 1 ≤ i ≤ n. Since ξ is a spin cobordism invariant (see [APS75, Theorem 3.3]), via the formula for the lens space with representation ρs

ds2 ds ξ˜(ρ ,L(p, q)) ≡ − − mod , s 2p 2 Z where dq ≡ 1(mod 2p), and Lemma 5.4 in [JW95], we can deduce the following lemma:

Lemma 2.2.23. Let ρ a representation of π1(Σ(a1, ..., an)) in SLN (C) of the type s = (sk(i))1≤i≤n;1≤k≤N with ρ(h) a scalar multiple of the identity matrix, we have

n N N 2 X X X a(sk(i) − sl(i)) 2NRe(e∗[Σ(a1, a2, ...an), ρ]) = . ai i=1 k=1 l=1

61 Proof. By Proposition 2.2.21, the cobordism W constructed earlier is spin when c a1...an is even and is spin when a1...an is odd; and by Lemma 5.4 in [JW95] there is e∗[Σ, ρ ⊗ ρ¯] = 2NRe(e∗[Σ, ρ]). Now, as ξ˜ is a spin (spinc) cobordism invariant, the following identities can be derived: n X X ˜ X ˜ ξ(ρsk(i)−sl(i),L(ai, −bi)) = ξ(ρsk−sl , #L(ai, −bi)) 1≤i≤n 1≤k,l≤N i=1 1≤k,l≤N ˜ = ξ(ρ ⊗ ρ,¯ Σ(a1, ..., an)). Plugging the formula (2.15) in, we get

n N N 0 2 0 ˜ X X X bi(sk(i) − sl(i)) bi(sk(i) − sl(i)) ξ(ρ ⊗ ρ,¯ Σ(a1, ..., an)) = ( + ) 2ai 2 i=1 k=1 l=1 n 0 2 X X b (sk(i) − sl(i)) = i , ai i=1 1≤k

bi a1 ··· an(−b + Σ( )) = 1 ai implies 2 0 a + ai m = aibi, for some m ∈ Z, and n 0 2 n 0 2 X X bi(sk(i) − sl(i)) X X aibi(sk(i) − sl(i)) = 2 , ai a i=1 1≤k

62 Chapter 3

Homotopy liftings and e∗

In this chapter, we discuss homotopy liftings of the canonical map from algebraic K-theory of complex numbers (resp. real numbers) to complex (resp. real) topological K-theory. In the first section, we recall some properties of Moore spaces (CW -spectra) and the rationalization of a simple space (CW -spectrum). It is to prepare the tools needed in the sections following. The existence of such homotopy liftings is proved in the second section. It is also shown there that any such homotopy lifting induces the same isomorphism on the torsion subgroup of [X,Ka], where X is a pointed topological space (see 3.2.1 for the notation Ka). In the second half of the section, we restrict our attention to the complex case, namely, homotopy liftings of the canonical map from algebraic K-theory of complex numbers to complex topological K-theory. Especially, we see, in this case, the induced homomorphism of any homotopy lifting restricts to 0 n n e∗, the e-invariant, on the torsion subgroup of [S ,KaC] =: K]aC (S ). Using the fact that, for every odd number n, the n-th algebraic K-group of complex numbers admits the following decomposition ([Jah99, (5.1),(5.2)],[Sus84, 4.9])

πn(KaC) = Q/Z ⊕ Dn, where Dn is a non-trivial divisible torsion-free abelian group, we conclude this section by showing there are infinite many different such homotopy liftings:

KaC → Ft,F/Z, where F = QR or C. In the third section, we consider only the homotopy liftings in the category of infinite loop spaces and extract further implications from Suslin’s theorem (3.1.5) and the proof of Lemma 3.2.2. Especially, we obtain some structure theorems for algebraic and relative K-theory of complex or real numbers. Meaning, there are homotopy equivalences of infinite loop spaces:

Ka ' Ka,Q × Ft,Q/Z rel K ' Ka,Q × ΩKt,Q

(The notations are explained in 3.2.1). With this insight, we identify a homotopy \ lifting eh, which is unique up to phantom maps, in the category of infinite loop

63 spaces, such that \ eh,∗ = e∗ : π∗(KaC) → π∗(Ft,C/Z). In other words, in the case of homology spheres, the ξ˜-invariant can be consid- \ ered as induced by an infinite loop map eh. We close our homotopy-theoretic investigation on “e-like invariants” with a comparison theorem for the Adams e-invariant and the e-invariant e∗, where we explain how the e-invariant generalizes the Adams e-invariant.

3.1 Moore prespectra and rationalization 3.1.1 Moore spaces and prespectra

Recall that a Moore space M(G, n) of an abelian group G of degree n ∈ N is a pointed topological space which is equipped with an isomorphism φ : ˜ ˜ Hn(M(G, n), Z) ' G and has H∗(M(G, n), Z) = 0, when ∗= 6 n. If n > 1, we further demand M(G, n) to be simply connected. Notice that the isomor- phism φ is part of the data, although it is usually omitted in the notation. Given an abelian group and n ∈ N, there is a standard construction of a model of M(G, n): Firstly, consider a free resolution of G,

q 0 → R −→i F −→ G, where F and R are the free abelian groups with generators {aα}α∈A and {bβ}β∈B, respectively. Secondly, assign every generator in {aα}α∈A and {bβ}β∈B with a n-sphere Sn, and construct a construct map f

n f n ∨ Sβ −→ ∨ Sα β∈B α∈A such that it realizes the homomorphism i. Then the mapping cone of f is a model for M(G, n). In particular, the cofiber of the degree m map

Sk → Sk gives us a model of M(Z/m, k). From this construction, the following lemma can be easily deduced.

Lemma 3.1.1. For n > 1, given any homomorphism φ : G → πn(X), one can find a continuous map f : M(G, n) → X such that f∗ = φ. This lemma implies the uniqueness of the homotopy type of the Moore space M(G, n), for n ≥ 2. On the contrary, when n = 1, the homotopy type of M(G, 1) is not unique. A fixed model is therefore needed. Moore spaces can be used to define homotopy groups with finite coefficients (see [Wei13, Chap.4] and [Sus84, Intro.]).

64 Definition 3.1.2. Given a topological space X, its n-th homotopy group with Z/m-coefficients is given by

πn(X)Z/m := [M(Z/m, n − 1),X], for n ≥ 3. If X is a H-space, one can further make the set of homotopy classes

π2(X)Z/m := [M(Z/m, 1),X] m into a group and define π1(X)Z/m to be the quotient of π1(X) −→ π1(X). When X is a connected H-space, these homotopy groups with Z/m-coefficients fit into the following long exact sequence of abelian groups

m ... → πn(X) −→ πn(X) → πn(X)Z/m → πn−1(X) m → ... → π1(X) −→ π1(X) → π1(X)Z/m → 0. Moore prespectra: For now, we are mainly working in the Adams category A, but most maps constructed here are, in effect, maps of prespectra. It is because we need the well-developed theory on the smash product and S-duality in A. In contrast, to the author’s knowledge, the theory on the smash product and S-duality in terms of prespectra in P is less well documented. Nevertheless, in the due course, we move back to the category P as we need to use the homotopy fiber construction and the functor Ω∞. The relation between A and P as well as the related notations are discussed in Appendix A. We use the term “CW -spectrum” for the objects in A, whereas, for those CW -objects in P, the term “CW -prespectrum” is reserved even though, by definition, they mean the same thing. Nonetheless, for those standard CW -spectra, like the sphere spectrum and the Eilenberg-Maclane spectra, we omit the prefixes CW and “pre-”. A Moore CW -spectrum MG of an abelian group G is a (−1)-connective CW -spectrum equipped with an isomorphism φ : π0(HZ∧MG) ' G and having π∗(HZ ∧ MG) = 0, for ∗ 6= 0. Via the Hurewicz isomorphism and an analog of Lemma 3.1.1, we see the homotopy type of a Moore CW -spectrum of an abelian group is uniquely determined. Similar to Moore spaces, there is also a standard construction for a Moore CW -spectrum of a given abelian group. In fact, it is basically the same construction as the one for Moore spaces. One only needs to replace the n-spheres by the sphere spectra. For example, Σ∞M(G, k)[−k] is a model for the Moore CW -spectrum MG, for every k ≥ 1. In view of this construction and Lemma 3.1.1, we have that, given a homo- morphism of abelian groups φ : H → G, there exists a map of CW -prespectra φ˜ : MH → MG such that ˜ φ = π0(id ∧ φ): π0(HZ ∧ MH) → π0(HZ ∧ MG). However, φ˜ is not unique in Ho(A) as Moore CW -spectra do not give us a functor from the category of abelian groups to the stable homotopy category. Like Moore spaces, Moore CW -spectra are useful for defining homotopy groups with finite coefficients. Given a CW -spectrum E, the n-the homotopy group with Z/m-coefficients πn(E)Z/m is defined to be

[S[n], E ∧ MZ/m]Ho(A),

65 where S[n] is the sphere spectrum with degree shift n (see 97). To see its rela- tion with the definition of homotopy groups with finite coefficients for pointed topological spaces, we recall the following lemma which is a corollary of those lemmas in [Swi02, 14.31-33].

Lemma 3.1.3. The CW -spectrum Σ∞M(Z/m, n)[−n − 1] is the S-dual of the Moore CW -spectrum MZ/m = Σ∞M(Z/m, n)[−n], and is denoted by M∗Z/m. Proof. Using the cofiber sequence

m ∞ S −→ S → Σ M(Z/m, n)[−n] → S[1] → S[1] and corollary 14.33 in [Swi02]. An immediate consequence of this lemma and the S-duality is that we have the following isomorphisms:

∗ ∞ [S[n], E∧MZ/m]Ho(A) ' [M Z/m[n], E]Ho(A) = [Σ M(Z/m, k)[n−k−1], E]Ho(A). When E is a 0-connective Ω-CW -spectrum and n ≥ 2, we can further deduce

∞ [Σ M(Z/m, k)[n − k − 1], E]Ho(A) ∞ ' [Σ M(Z/m, k)[n − k − 1], E]Ho(P) ' [M(Z/m, n − 1),E0] via the adjunction ∞ ∞ Σ : Ho(Top∗)  Ho(P):Ω (see A.2 and A.1.8). For the case n = 1, we observe the following isomorphisms

∞ [Σ M(Z/m, k)[n − k − 1], E]Ho(P) ' [M(Z/m, 1),E1] 2 m 2 ' Coker([S ,E1] −→ [S ,E1]) 1 m 1 ' Coker([S ,E0] −→ [S ,E0]) ' π1(E0)Z/m. Therefore one can conclude, for n ≥ 1, the definition given by Moore CW - spectra coincides with the definition given in 3.1.2, namely

[S[n], E ∧ MZ/m]Ho(A) = πn(E0)Z/m, where E is a 0-connective Ω-CW -spectrum. With this observation, Suslin’s theorem [Sus84, Cor.4.6] can be reformulated as follows: Corollary 3.1.4. The canonical map of CW -prespectra

Ka ∧ MQ/Z → Kt ∧ MQ/Z; Ka ∧ M(Q/Z, k)[−k] → Kt ∧ M(Q/Z, k)[−k]

are π∗-isomorphism, where Ka and Kt are the 0-connective cover of the algebraic K-prespectrum of complex numbers (resp. real numbers) and complex (resp, real ) topological K-prespectrum, respectively. We assume both Ka and Kt are Ω- CW -prespectra.

66 Proof. It is proved in [Sus84, Cor.4.6] that the canonical map

+ 0,δ 0 Ka = BGL (F ) → Kt = BGL(F ) induces an isomorphisms on the n-th homotopy group with finite coefficients Z/m, for every m, n ∈ N, and, with the observation above, it is equivalent to saying the following homomorphisms

[S[n], Ka ∧ MZ/m]Ho(A) → [S[n], Kt ∧ MZ/m]Ho(A); [S[n], Ka ∧ M(Z/m, k)[−k]]Ho(A) → [S[n], Kt ∧ M(Z/m, k)[−k]]Ho(A). are isomorphisms, for n, m ∈ N. Now because hocolim M( /m, k) = M( / , k), m Z Q Z we can further deduce

[S[n], Ka ∧ M(Q/Z, k)[−k]]Ho(A) → [S[n], Kt ∧ M(Q/Z, k)[−k]]Ho(A) is an isomorphism, for all n, and hence the canonical map of prespectra

Ka ∧ M(Q/Z, k)[−k] → Kt ∧ M(Q/Z, k)[−k] is a π∗-isomorphism. Note that Ka → Kt can be chosen to be a map of pre- spectra by Theorem A.2.5 and the definition of the stable homotopy group for CW -spectra in A and CW -prespectra in P are the same. The first π∗-isomorphism follows from the homotopy equivalences between the naive smash product and the smash product [Swi02, p.258-259]. Now given a map φ˜ : M(H, k) → M(G, k) that realizes a homomorphism φ : H → G, the homotopy cofiber of φ˜ is a model for the Moore space M(coker(φ), k). Similarly, using the construction given earlier, the cofiber of the induced map of CW -prespectra

MH → MG that realizes φ is a model for the Moore CW -spectrum M(coker(φ)). Especially, given a homomorphism φ : H → G, there is a homotopy cofiber sequence in Ho(P)

E ∧ M(H, k)[−k] → E ∧ M(G, k)[−k] → E ∧ M(coker(φ), k)[−k].

Now the Serre theorem gives us the following homotopy equivalence:

MQ ' S ∧ MQ ' HQ which implies MQ is rational, and therefore, by Lemma 3.1.12 and Lemma 3.1.10, there is a unique map of CW -spectra in Ho(A)

MZ = S → MQ = HQ that realizes the canonical inclusion

π0(MZ) ' Z → Q ' π0(MQ).

67 Hence the following associated maps of CW -prespectra can be defined without ambiguity

E ∧ MZ → E ∧ MQ, E ∧ M(Z, k)[−k] → E ∧ M(Q, k)[−k].

Letting E = Ka or Kt, we obtain the following homotopy cofiber sequences:

f Ka ∧ M(Z, k)[−k] −→ Ka ∧ M(Q, k)[−k] → Ka ∧ M(Q/Z, k)[−k]; g Kt ∧ M(Z, k)[−k] −→ Kt ∧ M(Q, k)[−k] → Kt ∧ M(Q/Z, k)[−k], and Corollary 3.1.4 says

Ka ∧ M(Q/Z, k)[−k] → Kt ∧ M(Q/Z, k)[−k], or equivalently, Cofib(f) → Cofib(g) is a π∗-isomorphism. Therefore, by Lemma A.1.4 and Corollary 3.1.4, we can obtain another reformulation of Suslin’s theorem [Sus84, Cor.4.6]: Theorem 3.1.5. The induced map of prespectra

Fib(f) → Fib(g) is a π∗-isomorphism. Proof. The assertion follows quickly from Lemma A.1.4 as we have the following commutative diagram:

Fib(f) Fib(g)

o o

ΩCofib(f) ΩCofib(g)

3.1.2 Rationalization of spaces and CW -spectra In this subsection, we recall some properties and definitions of rational spaces (CW -spectra) and the rationalization of a simple spaces (CW -spectrum).

Rational spaces: A rational equivalence is a map f : X → Y ∈ Top∗ such that the induced homomorphism

f∗ : H∗(X, Q) → H∗(Y, Q) is an isomorphism, for all ∗ ∈ N ∪ {0}. Since we are mainly concerned with connected H-spaces or rather connected infinite loop spaces, we assume all spaces here are simple spaces—connected spaces whose fundamental group acts trivially on its homotopy group, even though most of the theorems here can be applied to more general spaces as well (see [MP12, Chap.5-6]).

68 Lemma 3.1.6. Given a simple space Z, the following statements are equivalent: 1. For any rational equivalence f : X → Y , the induced map of sets

f ∗ :[Y,Z] → [X,Z]

is a bijection.

2. π∗(Z) is rational, for ∗ ≥ 1 ˜ 3. H∗(Z, Z) is rational, for ∗ ≥ 0. Proof. see [MP12, Theorem 6.1.1] A space that satisfies any of the three statements in the lemma above is called a rational space. Definition 3.1.7. Given a map of simple spaces f : X → Y , we call (Y, f) a rationalization of X if and only if Y is rational and f is a rational equivalence. Now the existence of the rationalization of a simple space is proved in [MP12, Theorem 5.3.2], and, with the definition, we see it must be unique up to homo- topy equivalence. We usually denote the rationalization of X simply by XQ and omit the map X → XQ. The following statements characterize the rationalization of a simple space. Lemma 3.1.8. The following statements are equivalent: 1. f : X → Y is a rationalization of X. 2. For any rational space Z, we have

f ∗ :[Y,Z] → [X,Z]

is a bijection. 3. f∗ : π∗(X) → π∗(Y )

is the rationalization of π∗(X), for ∗ ≥ 1. 4. ˜ ˜ f∗ : H∗(X, Z) → H∗(Y, Z) ˜ is the rationalization of H∗(X, Z), for ∗ ≥ 1. Proof. See [MP12, Theorem 6.1.2].

One simple observation is that, given a simple space X, if πn(X) is a torsion group, for every n ≥ 1, then XQ is contractible. Rationalization of CW -spectra

Definition 3.1.9. Let MQ be the Moore CW -spectrum for rational numbers and f a map of CW -spectra E → F, then (F, f) is a rationalization of E if and only if there is a homotopy equivalence

h : E ∧ MQ → F such that the following commutes

69 E ∧ MQ

E h f

F

A CW -spectrum E is called rational if and only if the canonical map E → E ∧ MQ is a homotopy equivalence in A. If thinking of a 0-connective rational CW -spectrum E as an object in P, then Ω∞E is a rational space (see A.1.8 for the functor Ω∞), and suppose E is already an Ω-prespectrum (fibrant object in P), then E0, the zero component of E, is a rational space. The following statements are equivalent and they all characterize a rational- ization of a CW -spectrum.

Lemma 3.1.10. 1. f : E → F is a rationalization of E. 2. For any CW -spectrum G, the induced homomorphism

f∗ :[S[∗], G ∧ E]Ho(A) → [S[∗], G ∧ F]Ho(A)

is a rationalization of the abelian group [S[∗], G ∧ E]Ho(A) (or localization at empty set). 3. Given any rational CW -spectrum G, we have

∗ f :[F, G]Ho(A) → [E, G]Ho(A)

is an isomorphism. Proof. See [Rud08, 5.4-5;5.8-9]. Actually any rational CW -spectrum is equivalent to a graded Eilenberg- Maclane spectrum.

Lemma 3.1.11. Let E be a rational CW -spectrum, then there is a homotopy equivalence ∼ E −→ ∨ Hπn(E)[n]. −∞

An important feature of the rational CW -spectra is that their maps are determined by the induced homomorphisms on the homotopy groups. Lemma 3.1.12. Given two CW -spectra E and F, and assume F is rational, then the canonical homomorphism

0 [E, F]Ho(A) → Hom (π∗(E) ⊗ Q, π∗(F)) is an isomorphism, where 0 Hom (A∗.B∗) is the abelian group of graded homomorphisms of degree 0.

70 Proof. See [Rud08, Theorem 7.11 (iii)]. Finally, we note, if F is an Ω-CW -spectrum, the homomorphism

∼ [E, F]Ho(P) ' [E, F]l −→ [E, F]Ho(A)

is an isomorphism (see Theorem A.2.5 and A.2.6), and hence the lemma above can be rephrased in terms of prespectra. Corollary 3.1.13. Given two CW -prespectra E and F and assume F is a rational Ω-prespectrum, then the canonical homomorphism

0 [E, F]Ho(P) → Hom (π∗(E) ⊗ Q, π∗(F)) is an isomorphism.

3.2 Homotopy liftings

For convenience, we introduce the following notations and assume that all the prespectra are 0-connective:

Convention 3.2.1. F0 = R or C, and F = Q or F0. 0 0 KaF : The infinite loop space of the Ω-CW -prespectrum KaF which represents algebraic K-theory of complex numbers (resp. real numbers)—When there is no risk of confusion, we drop F0 from the notation. 0 0 KtF : The infinite loop space of the Ω-CW -prespectrum KtF which represents complex (resp. real) topological K-theory. We usually drop F0 from the notation as the context will make it clear.

XF: The infinite loop space of the prespectrum X ∧ MF, or equivalently the zero component of its fibrant replacement. 0 0 Ft,F/Z: The homotopy fiber of KtF → KtF , or equivalently the infinite loop space 0 F of the prespectra Ω(KtF ∧ MF/Z) (see A.1.4). 0 0 Fa,Q/Z: The homotopy fiber of KaF → KaF , or equivalently the infinite loop 0 Q space of Ω(KaF ∧ MQ/Z) (see A.1.4). rel 0 0 0 K F : The homotopy fiber of KaF → KtF . It is the infinite loop space repre- senting relative K-theory. We usually drop F0 as the context should make it clear. Following the discussion in the last section, there is a commutative diagram of prespectra in Ho(P).

71 Ω(Ka ∧na MZ) Ω(Kt ∧na MZ) Ω(Kt ∧na MZ)

Ωf Ωg Ωh

0 Ω(Ka ∧na MQ) Ω(Kt ∧na MQ) Ω(Kt ∧na MF )

Su, ∼ Fib(f) Fib(g) Fib(h)

id Ka ∧na MZ Kt ∧na MZ Kt ∧na MZ

f g h

0 Ka ∧na MQ Kt ∧na MQ Kt ∧na MF where Su stands for the π∗-isomorphism in Theorem 3.1.5, and E∧na MH is the naive smash product E∧M(H, k)[−k]. Using the model structure of P, one can replace each CW -prespectrum in the diagram above by an equivalent fibrant- cofibrant prespectrum, then, applying Ω∞, we obtain the following diagram of homotopy fiber sequences.

ΩKa ΩKt ΩKt

Ωf Ωg Ωh

0 ΩKa,Q ΩKt,Q ΩKt,F

Su, ∼ 0 Fa,Q/Z Ft,Q/Z Ft,F /Z

id Ka Kt Kt

f g h

0 Ka,Q Kt,Q Kt,F

72 Now, since Kt ∧na MQ is rational, the following composition

Ka ∧ MZ → Kt ∧ MZ → Kt ∧ MQ is determined by their induced homomorphisms (see Lemma 3.1.12 and Lemma 3.1.10):

π∗(Ka ∧na MZ) → π∗(Kt ∧na MZ) → π∗(Kt ∧na MQ) = π∗(Kt) ⊗ Q. It is a zero homomorphism as the algebraic K-groups of real or complex num- bers are isomorphic to a direct sums of its torsion subgroup and a uniquely divisible group [Wei84, VI.Theorem 1.6; Theorem 3.1], and there is no non- trivial homomorphism from a divisible group to Z. Therefore, we have shown the composition Ka → Kt → Kt,Q is homotopic to zero as an infinite loop map.

Lemma 3.2.2. The homotopy liftings of the canonical map Ka → Kt exist, called e1 and e2 in the following, and they fit into the commutative diagram below.

ΩKa ΩKt ΩKt

Ωf Ωg Ωh

0 ΩKa,Q ΩKt,Q ΩKt,F

Su, ∼ j 0 Fa,Q/Z Ft,Q/Z Ft,F /Z

e2 e1

id Ka Kt Kt

f g h

0 Ka,Q Kt,Q Kt,F

Proof. The existence of the map has been shown in the discussion preceding the lemma. To see the diagram is commutative, we note Fa,Q/Z has all its homotopy groups are torsion groups. That is because both coker(πn(Ωf)) and ker(πn(f)) are torsion groups, for n ≥ 1, and πn(Fa,Q/Z) fits into the short exact sequence

0 → coker(πn(Ωf)) → πn(Fa,Q/Z) → ker(πn(f)) → 0.

Therefore the rationalization of Fa,Q/Z is contractible and hence the abelian group [Fa,Q/Z, ΩKt,F] is trivial. This gives us the commutativity of the following two triangles:

73 Su j ◦ Su 0 Fa,Q/Z Ft,Q/Z Fa,Q/Z Ft,F /Z

e2 e1

Ka Ka

We thus proved the lemma.

Remark 3.2.3. Because [Fib(f), Ω(Kt ∧MF)]Ho(A) is also trivial, when e1 and e2 are infinite loop maps, the diagram in Lemma 3.2.2 is commutative in the category of infinite loop spaces. The next theorem shows that any such homotopy lifting restricts to an iso- 0 morphism on the torsion subgroup [X,KaF ].

Theorem 3.2.4. Given a pointed topological space X, the homotopy liftings e2 and e1 induce the following isomorphisms:

0 ∼ e2,∗|Tor : Tor[X,KaF ] −→ [X,Ft,Q/Z] 0 ∼ 0 e1,∗|Tor : Tor[X,KaF ] −→ Tor[X,Ft,F /Z].

Proof. 1. Our first step is to show that j∗ induces an isomorphism

coker((Ωg)∗) → Tor((coker(Ωh)∗).

In order to see this, we recall some simple facts from homological algebra: Given an abelian group A, there is a short exact sequence

0 → AT → A → A/AT → 0, where AT is the torsion subgroup of A. Now, by the right exactness of the tensor product, and the fact that AT ⊗ F/Z = 0, we see the homomorphism

A ⊗ F/Z → A/AT ⊗ F/Z (3.1) is an isomorphism. Moreover, since A/AT is torsion free and hence flat, there is another short exact sequence

0 0 0 → A/AT ⊗ Q/Z → A/AT ⊗ F /Z → A/AT ⊗ F /Q → 0

0 where the tensor product A/AT ⊗ F /Q is flat and hence torsion free as A/AT and F0/Q both are flat. Now, applying the left exactness of Tor, we can further obtain the following isomorphism

0 A/AT ⊗ Q/Z → Tor(A/AT ⊗ F /Z). (3.2) Now we return to the proof of the theorem and let A be the abelian group [X, ΩKt]. Then the isomorphisms (3.1) and (3.2) give us the isomorphisms

∼ ∼ 0 ∼ coker((Ωg)∗) = A/AT ⊗ Q/Z −→ Tor(A/AT ⊗ F /Z) = Tor(coker((Ωh)∗)

74 in the claim at the beginning of the proof. 2. The second step is to show the homomorphism

[X,Ft,Q/Z] → Tor[X,Ft,F/Z] is an isomorphism, yet this follows quickly from the following two diagrams of exact sequences: The first is

0 coker((Ωg)∗) [X,Ft,Q/Z] ker(g∗) 0

0 0 coker((Ωh)∗) [X,Ft,F /Z] ker(h∗) 0, and, applying the functor Tor, we obtain the second one:

0 coker((Ωg)∗) [X,Ft,Q/Z] ker(g∗) 0

l 0 0 Tor(coker((Ωh)∗) Tor([X,Ft,F /Z]) ker(h∗) 0

Note that the functor Tor does not always preserve the short exact sequence, but, in this case, it does. The only thing to check is the surjectivity of l, yet it follows from the surjectivity of the homomorphism

[X,Ft,Q/Z] → Tor([X,Ft,F/Z]) → ker(h∗) = ker(g∗). Therefore, by the short five lemma, we have shown the homomorphism

[X,Ft,Q/Z] → Tor[X,Ft,F/Z] is an isomorphism. 3. Once we have these, the first statement follows quickly from the homotopy commutative diagram below:

Su∗, ∼ [X,Fa,Q/Z] [X,Ft,Q/Z] l

e2,∗|Tor Tor[X,Ka]

e2,∗

[X,Ka]

In more details, we have Su∗ is an isomorphism by Suslin’s theorem (see The- orem 3.1.5), and therefore l is injective. On the other hand, by definition, l is also surjective and hence an isomorphism. In this way, we see e2,∗|Tor is an isomorphism. Similarly, one can deduce the second assertion from the following homotopy commutative diagram:

75 Su∗ j∗ [X,F ] [X,F ] Tor([X,F 0 ]) [X,F 0 ] a,Q/Z ∼ t,Q/Z ∼ t,F /Z F /Z l

e1,∗|Tor Tor[X,Ka]

e1,∗ [X,Ka] as we have shown, in the step 2, that j induces an isomorphism

∼ 0 [X,Ft,Q/Z] −→ Tor[X,Fa,F /Z], and hence e1,∗|Tor has to be an isomorphism.

Now one can further observe e1,∗|Tor (resp. e2,∗|Tor) is identical to the com- −1 −1 position j∗ ◦ Su∗ ◦l (resp. Su∗ ◦l ). That means any such homotopy lifting induces the same isomorphism on the torsion subgroup Tor[X,Ka]. Further- more, in the case of complex numbers (Kt represents complex topological K- theory) and X = Sn, Lemma 2.2.10 shows the e-invariant also satisfies the same commutative diagram, namely

n Su∗ n n [S ,Fa,Q/Z] [S ,Ft,Q/Z] [S ,Ft,C/Z]

e

n [S ,KaC]

Hence, in the case of complex numbers, the induced homomorphism of every ho- n motopy lifting restricts to the e-invariant on the torsion subgroup of [S ,KaC]. We summarize this in the following:

Theorem 3.2.5. Every homotopy lifting Ka → Ft,F/Z of Ka → Kt induces the same isomorphism on the torsion subgroup of [X,Ka]. n Assume Kt represents complex topological K-theory, and X = S , we further have every homotopy lifting KaC → Kt,C/Z of the canonical map KaC → Kt n induces the e-invariant on the torsion subgroup of [S ,KaC].

From now on till the end of the section, Kt only stands for complex topo- logical K-theory. As we have seen, in the first part of the section, the existence of the homotopy lifting

KaC → Ft,F/Z. Now we want to show there are infinite many different homotopy liftings in both category of topological spaces and infinite loop spaces. Recall that the number of different homotopy liftings is measured by the size of the subgroup

im([KaC, ΩKt,F]) ⊂ [KaC,Ft,F/Z].

76 This is a consequence of the following long exact sequence

... → [KaC, ΩKt,F] → [KaC,Ft,F/Z] → [KaC.Kt] → ....

Since Ω(Kt ∧ MF) is rational, by Lemma 3.1.13 and Lemma 3.1.10 (see also [Rud08, Theorem 5.8 and 7.11]), we have the following commutative diagram of abelian groups

[K , Ω(K ∧ M )] 0 aC t F Ho(P) Hom (π∗(KaC) ⊗ Q, π∗(ΩKt,F))

Ω∞,inj. surj.

[Ka , ΩKt, ] C F Top∗ (3.3) On the other hand, It is known that ( F ∗ = odd π∗(ΩKt, ) = F 0 ∗ = even.

However, to the author’s knowledge, the precise size of the abelian group π∗(KaC)⊗ Q is not determined yet. Nevertheless, according to [Jah99, Sec.4-5], we have π∗(KaC)⊗Q is a non-trivial Q-vector space, when ∗ is odd. In fact, he constructs a homomorphism from π∗(KaC) → R, for ∗ is odd and proves that this homo- morphism reduces to the Borel classes after precomposing the homomorphisms induced by the conjugate embeddings of number fields in C and tensoring R. This shows π∗(KaC) ⊗ Q cannot be trivial. With this observation, if one can construct infinite many different homomor- phisms

π∗(KaC) → π∗(ΩKt,F) such that after composing with the following homomorphism

π∗(ΩKt,F) → π∗(Ft,F/Z), they remain different, they we show there are infinite many different homotopy liftings. Here we provide one possible construction: Pick up a non-trivial element 1 x ∈ π∗(KaC) ⊗ Q, where ∗ is odd, and assign to it the number n ∈ F, where n ∈ N \{1}. Then extend this assignment to a homomorphism

π∗(KaC) ⊗ Q → F = π∗(ΩFt,F). It is not to difficult to construct the extension. One can, for instance, let < x >⊥ go to zero. Then, in view of diagram 3.3, we see these subgroups

im([KaC, ΩKt,F]) ⊂ [KaC,Ft,F/Z] im([KaC, Ω(Kt ∧ MQ)]Ho(P)) ⊂ [KaC, Fib(g)]Ho(P) 0 im([KaC, Ω(Kt ∧ MF )]Ho(P)) ⊂ [KaC, Fib(h)]Ho(P) all contain at least countably infinite many different elements.

77 Theorem 3.2.6. There are infinite many different homotopy liftings of the canonical map from algebraic K-theory of complex numbers to complex topo- logical K-theory. Equivalently, there are infinite many dashed arrows in the following diagram making the whole diagram commutative:

Fa,Q/Z Ft,F/Z

KaC Kt

Proof. This follows from the discussion preceding the theorem. The upper tri- angle is always commutative as the rationalization of Fa,Q/Z is contractible and hence the abelian groups

[Fa,Q/Z, ΩKt,F] and [Fib(f), Ω(Kt ∧ MF)]Ho(P) both are trivial (see 3.1.8).

\ 3.3 Homotopy liftings, e∗, t∗, and eh Here we summarize some implications from the previous two sections and ex- plain how these homotopy liftings are related to the homomorphisms t∗ and e∗ defined in 2.2.1 and 2.1.30. In this section we consider only the homotopy liftings in the category of infinite loop spaces. Corollary 3.3.1. There exists a homotopy equivalence of infinite loop spaces:

∼ Ka −→ Ka,Q × Ft,Q/Z.

Q Proof. By Lemma 3.2.2, there exists an infinite loop map eh : Ka → Ft,Q/Z as a homotopy lifting of Ka → Kt with respect to the homotopy fiber sequence

Ft,Q/Z → Kt → Kt,Q.

Combining with the rationalization uQ : Ka → Ka,Q, we obtain a homotopy equivalence of infinite loop spaces

(u ,eQ ) Q h Ka −−−−−→ Ka,Q × Ft,Q/Z.

F Corollary 3.3.2. Assume eh is a homotopy lifting

F eh Ka −→ Ft,F/Z of Ka → Kt with respect to the homotopy fiber sequence

Ft,F/Z → Kt → Kt,F.

78 Then the composition

F i2 eh Ft,Q/Z −→ Ka,Q × Ft,Q/Z ' Ka −→ Ft,F/Z is homotopic to the canonical map

j : Ft,Q/Z → Ft,F/Z, where i2 is the inclusion into the second component and Ka ' Ka,Q × Ft,Q/Z is Q the homotopy equivalence induced by eh . In other words, what really determines a homotopy lifting is its restriction on the divisible part Ka,Q. Proof. This follows from the commutative diagram below

j Su ∼ Ft,Q/Z Fa,Q/Z

i2

(π , eQ ) ∼ F Q h eh Ka,Q × Ft,Q/Z Ka Ft,F/Z oo

Let Krel be the infinite loop space of relative K-theory of complex or real numbers, namely the homotopy fiber of Ka → Kt, and we fix also a homotopy Q lifting eh as in the proof above. Corollary 3.3.3. There is a homotopy equivalence of infinite loop spaces:

rel ∼ K −→ Ka,Q × ΩKt,Q. Proof. This can be deduced from the following maps of homotopy fiber se- quences:

79 ΩKt ΩKt

Q (u ◦ π, th) rel Q K Ka,Q × ΩKt,Q

π

(u , eQ ) Q h Ka Ka,Q × Ft,Q/Z

p

Kt Kt where p is the composition

π2 Ka,Q × Ft,Q/Z −→ Ft,Q/Z → Kt,

Q π2 is the projection onto the second component and th is the infinite loop map Q induced by eh and a filler (homotopy) of the following triangle:

Kt

Q eh K F a t,Q/Z (3.4)

F The following lemma shows that the map th is in effect independent of the choice of fillings of diagram 3.4.

F Corollary 3.3.4. Given a homotopy lifting eh, there is a unique map

F rel th : K → ΩKt,C

80 making the following diagram commute:

ΩKt ΩKt

i

F th rel K ΩKt,F

π (3.5)

F eh Ka Ft,F/Z

ι

Kt Kt

F,0 Proof. Suppose there is another map th that also fits into the commutative F F,0 diagram, then the difference of th and th is measured by the image of an rel rel element of [K , ΩKt] in [K , ΩKt,F]. Since ΩKt,F is rational and any map

rel K → ΩKt induces the trivial homomorphism between homotopy groups, so we have the homomorphism rel rel [K , ΩKt] → [K , ΩKt,F]

F F,0 is trivial. Hence, th and th are homotopic.

F Now if we do it the other way around, meaning fixing th in diagram 3.4, we have a slightly weaker result.

F F Lemma 3.3.5. The map th determines the map eh up to phantom maps. F,0 F F,0 Namely, if there is another homotopy lifting eh such that the pair (th, eh ) also F,0 F satisfies commutative diagram (3.5), then eh and eh differ only by a phantom map.

Q Proof. Firstly, recall that eh induces an identification

Ka ' Ka,Q × Ft,Q/Z,

F F,0 and, by Corollary 3.3.2, we know eh and eh restrict to the homotopic maps (as infinite loop maps) on Ft,Q/Z. Secondly, via the Serre class theory ([Rud08, Proposition 4.23, 4.25]), one can deduce [F, Kt]Ho(A) is a finitely generated abelian group, for every finite CW -spectrum F. On the other hand, since Ka ∧ MQ is rational, the abelian group

[E, Ka ∧ MQ]Ho(A)

81 is always divisible, for any CW -spectrum E. Therefore the homomorphism

[F, Ka ∧ MQ]Ho(A) → [F, Kt]Ho(A) is trivial, for any finite CW -spectrum F. In particular, this implies all divisible rel elements of [F, Ka]Ho(A) are in the image of [F, K ]Ho(A) → [F, Ka]Ho(A), and F hence, via diagram (3.5) and th, we have shown

F F,0 eh,∗ = eh,∗ :[F, Ka]Ho(A) → [F, Fib(ch)]Ho(A), for every finite CW -spectrum F, where Fib(ch) is the homotopy fiber of

Kt → Kt ∧ MF.

F F,0 Therefore eh and eh differ only by a phantom map.

The homomorphisms t∗ and e∗ constructed in 2.1.30 and 2.2.1 also satisfy a similar diagram:

Lemma 3.3.6. The homomorphisms t∗ and e∗ fit into the following commuta- tive diagram:

π∗(ΩKt) π∗(ΩKt)

i∗

t rel ∗ π∗(K ) π∗(ΩKt,C)

π∗

e∗ π∗(KaC) π∗(Ft,C/Z)

ι∗

π∗(Kt) π∗(Kt)

Proof. Note first when n is even, the diagram commute trivially as we have Y ΩHS (BU) = ΩHS( K( , 2i − 1)) = ΩHS (BU)0 = 0. n+1 n C n+1 C/Z i In the case when n is odd, the diagram can be realized by the homology sphere models as follows:

82 HS HS Ωn+1(BU) Ωn+1(BU)

R i∗ ◦ch

t¯ HS rel (A),0 ∗ HS Q Ωn (BGL (C)) Ωn ( i K(C, 2i − 1))

π∗ p

e0 HS δ (A) ∗ ΩHS (BU)0 Ωn (BGL(C )) n+1 C/Z

ι∗ b

HS HS Ωn+1(BU) Ωn (BU) where R ◦ch is defined in the remark following 2.1.22 (see also 2.1.31)

p([ω]Σ) := (ωΣ, [(Σ, c; W, c)1]), where ∂W = −Σ and c is the constant map ;

b(ω, [(Σ, f; W, F )1]) := (Σ, f); n i∗(Σˆ,F ) := (S , c; W, F ), where W := Σˆ \ intDn+1,

F |Dn+1 = c, Sn := ∂W,

π∗(Σ, f; W, F ) := (Σ, f);

ι∗(Σ, f) := (Σ, ι ◦ f).

Note for n = 1, the geometric model still works as we still have

HS + Ω1 (X) = π1(X) → π1(X ) is surjective, where X = BGLrel(C) or BGL(Cδ). The commutativity of the diagram can be easily deduced from this geometric realization.

\ Theorem 3.3.7. There is a homotopy lifting eh, which is unique up to phantom maps, such that \ eh,∗ = e∗ : π∗(KaC) → π∗(Ft,C/Z).

Proof. Since Kt,Q is rational (see 3.1.12), we can find a unique infinite loop map \ th such that \ rel th,∗ = t∗ : π∗(K ) → π∗(ΩKt,C). As the homotopy cofiber and fiber sequences are isomorphic in the stable ho- motopy category, we can, by choosing a filler of the following triangle

83 ΩKt

Q th rel K ΩKt,C obtain an infinite loop map

\ eh : KaC → Ft,C/Z which makes diagram 3.5 commute. Applying commutative diagram 3.5, we can easily deduce the following

\ eh,∗ = e∗ : π∗(KaC) → π∗(Ft,C/Z)

rel as π∗(Kt) = 0 when ∗ is odd, and hence π∗(K ) → π∗(KaC) is onto. Note, for ∗ is even, π∗(Ft,C/Z) = 0. [ To see the uniqueness, we assume there is another lifting eh such that

[ \ eh,∗ = e∗ = eh,∗.

[ In view of Lemma 3.3.4, we may assume th is the induced infinite loop map rel from K to ΩKt,C, and we also let p∗ be the homomorphism

π∗(ΩKt,C) → π∗(Ft,C/Z). Then diagram 3.5 tells us

[ [ \ \ p∗ ◦ th,∗ = e∗ ◦ π∗ = e∗ ◦ π∗ = p∗ ◦ th,∗. (3.6) Now observe there is an exact sequence

rel rel rel 0 → Hom(π∗(K ), π∗(ΩKt)) → Hom(π∗(K ), π∗(ΩKt,C)) → Hom(π∗(K ), π∗(Ft,C/Z)), which we get from the short exact sequence

0 → π∗(ΩKt) → π∗(ΩKt,C) → π∗(Ft,C/Z) → 0. Since there is no non-trivial homomorphism from a divisible abelian groups to rel a finite generated abelian group, Hom(π∗(K ), π∗(ΩKt)) = 0. Consequently, the surjective homomorphism

rel rel Hom(π∗(K ), π∗(ΩKt,C)) → Hom(π∗(K ), π∗(Ft,C/Z))

[ \ is actually bijective and hence th,∗ = th,∗ because of (3.6). As a result, we see [ \ th and th are homotopic as infinite loop maps (see Lemma 3.1.12). Applying [ \ Lemma 3.3.5, we obtain eh and eh indeed differ only by a phantom map.

84 3.4 The Adams e-invariant

In this section, we identify the Adams e-invariant with the e-invariant con- structed in 2.2.1. Recall first the Adams e-invariant can be obtained from the following homotopy lifting:

Ft,Q/Z Ft,C/Z

e¯Adams

BΣ∞ Kt

Applying the universal property of the plus construction, we obtain a map

+ eAdams : BΣ∞ → Ft,Q/Z. Then its induced homomorphism

+ eAdams,∗ : π∗(BΣ∞) → π∗(Ft,Q/Z) gives us the Adams e-invariant (see [Qui76]). On the other hand, from Construction 2.2.1, we also know that the mape ¯ fits into the following homotopy commutative diagram:

Ft,C/Z

e¯ W

δ BΣ∞ BGL(C ) Kt where W means the triangle commutes only up to weak homotopy. Then, apply- ing the universal property of the plus construction, we also obtain a well-defined homomorphism + e∗ : π∗(BΣ∞) → π∗(Ft,C/Z). The following theorem then says that they are actually identical. Theorem 3.4.1.

+ e∗ = eAdams,∗ : π∗(BΣ∞) → π∗(Ft,C/Z).

Proof. 1. To see this, we recall that, given any map f : BΣ∞ → Ft,C/Z, there is a commutative diagram (see Theorem 2.1.9):

f HS ∗ HS Ωn (BΣ∞) Ωn (Ft,C/Z)

surj. o

f + + ∗ πn(BΣ∞) πn(Ft,C/Z)

85 + + for n = 1, 3 or n ≥ 5, where f : BΣ∞ → Ft,C/Z is the map induced by the universal property of the plus construction. 2. With this observation, it suffices to show

HS HS e¯Adams,∗ =e ¯∗ :Ωn (BΣ∞) → Ωn (Ft,C/Z).

Now because of the compactness, any map from a homology sphere to BΣ∞ fac- (N) (N) tors through BΣk , for some N ∈ N, where, by (−) means the N-skeleton. (N) We can further assume N is even, and therefore [BΣk , ΩKt,Q] is trivial as the cohomology groups of finite group are always finite (see [Web, Corollary 4.3]). This can be summed up in the following diagram:

Ft,C/Z

e¯Adams e¯

(N) ι BΣk BΣ∞ Kt

(N) Since BΣk is compact, bothe ¯Adams ◦ ι ande ¯ ◦ ι are homotopy liftings of the composition (N) BΣk ,→ BΣ∞ → Kt. (N) On the other hand, because the abelian group [BΣk , ΩKt,Q] is trivial, there is only one homotopy lifting of

(N) BΣk ,→ BΣ∞ → Kt in Ho(Top∗), and hence these two maps are homotopic:

e¯Adams ◦ ι ' e¯ ◦ ι.

n Now, given any map α from a homology n-sphere Σ to BΣ∞, there always (N) exists an even number k, N ∈ N such that α factors through BΣk , and hence the following two maps

e¯ ◦ ι α n (N) F Σ BΣk t,C/Z

e¯Adams ◦ ι

HS represent the same element in Ωn (Ft,C/Z), for n = 1, 3 or n ≥ 5. For n is even, they both induce the trivial homomorphism. Thus we have proved the theorem.

86 Chapter 4

Summary and future works

We conclude the thesis by summarizing what we have achieved so far and the limitation of our method. Some other possible approaches for further investi- gation of the relation between the e-invariant and the Becker-Gottlieb transfer are also presented here.

4.1 An index theorem

As explained in the introduction, the primary motivation behind the thesis is to construct the following commutative diagram:

ξ˜(−,E,A)

e¯ ind δ ∗ A [E, BGL(C )] [E,Ft,C/Z] C/Z

π! ? ?

e¯ ind δ ∗ A [B, BGL(C )] [B,Ft,C/Z] C/Z

ξ˜(−,B,A0)

δ Now, instead of usinge ¯, we know there are infinite many mapse ¯h : BGL(C ) → Ft,C/Z making the following commute e¯ δ h,∗ [E, BGL(C )] [E,Ft,C/Z]

! ∗ π trBG

e¯ δ h,∗ [B, BGL(C )] [B,Ft,C/Z] wheree ¯h is the composition of the canonical map δ + δ BGL(C ) → BGL (C ) = Ka

87 and a homotopy lifting in the category of infinite loop spaces

eh : Ka → Ft,C/Z. In addition, we also realize, in the category of infinite loop spaces, there is a \ \ homotopy lifting eh unique up to phantom maps such that e∗ = eh,∗. Actually, Theorem 3.3.7 implies, for every flat vector bundle over a homology sphere ˜ ∗ \ (Σ, α), the ξ-invariant of (Σ, α) is identical to the value of c ◦ TD ◦ eh,∗(Σ, α). Notice, in this case, the index map indD can be identified with the composition ∗ c ◦TD (see 2.2.11 and [APS76, p.87]), where D is the Dirac operator associated \ to the spin manifold. However, we still do not know ife ¯h, the associated map \ of eh, satisfies

ξ˜(−,M,A)

\ e¯h,∗ ind δ A [M, BGL(C )] [M,Ft,C/Z] C/Z,

\ for every compact smooth manifold M. Nevertheless, we still have eh refines the Borel regulator, in view of Theorem 3.1 in [JW95]. Especially, we obtain a refined Bismut-Lott index theorem (see [BL95, Theorem 3.17]).

4.2 Delooping and algebraic K-theory machines

Inspired by Dwyer, Weiss and Williams’ approach in [DWW03], we start think- ing about using algebraic K-theory and infinite loop space machines to approach our problem. The idea is to construct directly an e-like map on the level of pre- spectra. More precisely, we are asking the following questions: Question 4.2.1. Is there an algebraic K-theory or an infinite loop space ma- chine that can produce both prespectra KaC and Fib(ch)? If yes, could one find an “exact” functor between the corresponding “structured” categories such that it realizes an e-like map ? Actually, one could make use of coassembly and ask instead: Question 4.2.2. Is there an algebraic K-theory or an infinite loop space ma- chine that can produce both prespectra

Ka(X, C) and Fib(X, ch), where Ka(X, C) is the presepctrum induced by the Waldhausen category of flat vector bundles over X, and Fib(−, ch) is a contravariant homotopy functor from the category of ENR’s to the category of prespectra such that

Fib(∗, ch) ' Fib(ch)

(∗ is a point).

88 If yes, coassembly gives us (functorially) the following (zig-zag) maps of pre- spectra

Ka(X, C) → map(X, KaC) Fib(X, ch) → map(X, Fib(ch)).

Then one could further search for an “exact” functor between the categories that generate Ka(X, C) and Fib(X, ch) such that, after applying coassembly, one obtains an e-like map. For either case, it seems necessary to have an algebraic K-theory machine that can deal with categories internal in Top.

4.3 Realizing the Becker-Gottlieb transfer

Another approach worth considering is, instead of provinge ¯ is induced from an infinite loop map e, one tries to realize the Becker-Gottlieb transfer

∗ trBG :[E,Ft,C/Z] → [B,Ft,C/Z] with some geometric models for topological K-theory with C/Z-coefficients. In this way, all homomorphisms in the diagram

e¯ δ ∗ [E, BGL(C )] [E,Ft,C/Z]

∗ π! trBG

e¯ δ ∗ [B, BGL(C )] [B,Ft,C/Z] are realized by the geometric models. In fact, Lott, in his article [Lot94], has used his geometric model for topological K-theory with R/Z-coefficients, which resembles Jones and Westbury’s, to realize an Umkehr map, and it is likely his method can be carried over to this case as the Umkehr map considered there ∗ only differs from trBG by an Eular class ([BG74]), provided E → B is a smooth fiber bundle with the vertical tangent bundle admitting a spin structure.

89 Appendices

90 Appendix A

The stable homotopy category

In this thesis, we have made use of both the category of prespectra P and the category of CW -spectra A—the Adams category. The reason for considering both categories is, although in the Adams category A, there is well-developed theory on the smash product, the S-duality and rationalization, A is not large enough to contain the homotopy fiber construction needed in our work. On the other hand, though the category P has the homotopy fiber construction and admits a model structure, which enables us to consider the following Quillen adjunction: ∞ ∞ Σ : Top∗  P :Ω , its theory on the smash product, the S-duality and rationalization is not very well documented. As there are many reference books for the Adams category A, for example, [Swi02] and [Rud08], which cover most theorems used in the thesis, we only review some properties of the category P. In order to shuffle between them comfortably, we also prove a comparison theorem for A and P.

A.1 The category of prespectra

In this section, we recall the definition of the category of prespectra P and discuss its model structure. Especially, in this model structure, a prespectrum is fibrant if and only if it is an Ω-prespectrum, and all CW -prespectra are cofibrant objects. Next, we describe how it stabilizes the category of pointed topological spaces Top∗. Most of the material presented here is taken from [Sch] and [MMSS01, Part II]. The simplicial analogue of the category P can be found in the article [BF78, Sec.2]. We provide proofs for those statements that is not found in the references given above.

A.1.1 The model structure

Note first the index set used here is N ∪ {0} unless otherwise specified, and secondly, the convenient category of topological spaces considered in this thesis is the symmetric monoidal category of pointed k-spaces Top∗. It is endowed with

91 the Quillen model structure whose fibrations, cofibrations and weak equivalences are given by Serre fibrations, the retracts of relative cell complexes and weak homotopy equivalences, respectively (see [Hov99, p.58-60]).

Definition A.1.1 (Prespectra).

1. A prespectrum is a sequence of spaces with structure maps

1 {Ei, σi : S ∧ Ei → Ei+1}i∈N∪{0}. When the adjoint σ˜i : Ei → ΩEi+1 is a weak homotopy equivalence, it is called an Ω-prespectrum. If each component Ei is a CW -complex and the structure maps are inclusions of subcomplexes, then it is called a CW -prespectrum. Throughout the thesis, the bold letters E, F... are reserved for prespectra, and we usually omit the structure maps when there is no risk of confusion.

0 2. A map of prespectra f : E = {Ei, σi} → F = {Fi, σi} is a sequence of continuous maps fi : Ei → Fi such that

0 σi ◦ fi = fi+1 ◦ σi,

for i ∈ N ∪ {0}.

3. A map of prespectra f : E → F is called a π∗-isomorphism or weakly stable equivalence if and only if the induced homomorphism

f∗ : π∗(E) → π∗(F)

is an isomorphism, for ∗ ∈ Z, where we define

π∗(E) := colim πi+∗(Ei). i;i+∗≥0

We denote the category of prespectra with maps of prespectra by P. Theorem A.1.2. ([MMSS01] and [Sch, Theorem 0.70]) There is a model struc- ture on the category of prespectra such that

1. A map is a weak equivalence if and only if it is a π∗-isomorphism.

2. A map p : E → B is a fibration if and only if pi : Ei → Bi is a fibration

and the following commutative diagram is a homotopy pullback in Top∗:

Ei ΩEi+1

pi Ωpi+1

Bi ΩBi+1

92 In particular, a prespectrum is fibrant if and only if it is an Ω-prespectrum.

3. A map of prespectra A → E is a cofibration if and only if A0 → E0 and

1 1 An+1 ∪S ∧An S ∧ En → En+1

are cofibrations in Top∗. Especially, if a prespectrum E = {Ei, σi} is cofibrant then Ei has the homotopy type of CW -complex and σi is a cofi- bration in Top. Thus every CW -prespectrum is a cofibrant object in this model structure on P. 4. With this model structure on P, one has

Ho(A) ' Ho(P).

Proof. For the detailed proof, see [MMSS01, Sec.6-11] and [Sch]. The latter contains more details specializing in this particular case, following the general proof given in the former. Here we describe briefly the construction of this model structure. First we recall, given a right proper model category C and a Quillen idempotent monad (Q, φ), which consists of an endofunctor Q : C → C and a natural transformation φ : id 7→ Q (see [Sch, Definition 0.62]), one can consider a new model structure, denoted by CQ, by demanding a morphism f is a weak equivalence in CQ if and only if Q(f) is, a cofibration if and only if it is in C, and a fibration if and only if it satisfies the right lifting property against trivial cofibrations in CQ. This is the Bousfield-Friedlander theorem, see [Sch, Proposition 0.66]. Now observe there is a strict model structure on P given by level-wise weak homotopy equivalences and level-wise Serre fibrations (see [Sch, Defi- nition 0.38]). We also have a Quillen idempotent monad induced by the Ω- prespectrification functor Q : P → P and a natural transformation φ : id 7→ Q, where we have QE is an Ω-prespectrum and φE : E → QE a π∗-isomorphism (see [Sch, Definition 0.19]). Then, applying the Bousfield- Friedlander theorem, we obtain the model structure required in the theorem. For the description of the cofibrations and fibrations in this model structure, we refer to [Sch, Definition 0.38, Definition 0.60] or [MMSS01, Proposition 9.5, Lemma 11.4]. From now on, P denotes the model category of prespectra described in the theorem above.

A.1.2 Homotopy fiber and cofiber

Homotopy fiber: Given a map of prespectra f : E → F, the homotopy fiber Fib(f) is the prespectra whose n-component is given by

Fib(f)n := En ×fn PFn,

93 where PFn := Top∗(I,Fn) with 0 ∈ I the base point. This gives us a sequence of prespectra f Fib(f) → E −→ F. Furthermore, if E and F are Ω-prespectra, then so is Fib(f). Homotopy cofiber: Given a map of prespectra f : E → F, the homotopy cofiber Cofib(f) is the prespectra whose n-component is the space En ∪fn CFn where CFn = Fn ∧ I+. This also yields a sequence of prespectra f E −→ F → Cofib(f), and if E and F are CW -prespectra, then Cofib(f) is also a CW -prespectrum. These two sequences of prespectra are related by the following lemma proved in [LMS86, p.128-130]:

Lemma A.1.3. There is a π∗-isomorphism, for every map of prespectra f, ζ Fib(f) −→f ΩstdCofib(f) std such that ζf and its adjoint ζ˜f :Σ Fib → Cofib(f) fit into the following diagram:

ΣstdΩstdE ΣstdΩstdF ΣstdFib(f)

o o o, ζ˜f

Fib(f) E F Cofib(f)

, o, ζf o o

ΩstdCofib(f) ΩstdΣstdE ΩstdΣstdF where Σstd and Ωstd are the standard suspension and loop functors (see (A.1) for the definition). In fact, the map Fib(f) → ΩstdCofib(f) is natural with respect to f. Meaning we have the following: Lemma A.1.4. Let Ar P be the category of maps of prespectra, then we have the homotopy fiber and cofiber constructions induce functors from Ar P to P, denoted by Fib and Cofib, respectively. Then the map ζf gives a natural trans- formation ζ : Fib 7−→ ΩstdCofib. Proof. Recall the map std ζf : Fib(f) → Ω (Cofib(f)) is constructed level-wisely. That is given f : X → Y a map of spaces, ζf is defined as follows (I,0) Fib(f) = X ×f (Y, ∗) → Ω Cofib(f) = Ω(Y ∪f X ∧ I+) (x, γ(t)) 7→ λ(t), ( 1 γ(2t) t ≤ 2 , where λ(t) = 1 (x, 2t − 1) t ≥ 2 . With this construction, it is not difficult to see ζ is natural.

94 The next lemma verify these two constructions indeed yields homotopy (co)fiber sequences in the model category P. First recall the definition of ho- motopy fiber sequences in a model category M.

Definition A.1.5. Given a commutative diagram in Mf , the subcategory of fibrant objects in M:

X Y

f

g W Z

We say it is a homotopy cartesian square if and only if there is a factorization f = p ◦ i with p : Y˜ → Z a fibration and i : Y → Y˜ a weak equivalence such that the canonical morphism from X to a pullback of W → Z ← Y˜ (dashed arrow) is a weak equivalence. It is illustrated as follows:

X Y i

W ×Z Y˜ Y˜ f p

g W Z

If W → ∗ is a weak equivalence, where ∗ is the terminal object, then X → Y → Z is called a homotopy fiber sequence. The dual notion gives us the definitions of homotopy cocartesian squares and homotopy cofiber sequences.

Lemma A.1.6. Suppose F and E are Ω-prespectra, then the sequences of pre- spectra f Fib(f) → E −→ F is a homotopy fiber sequence in P. Suppose E and F are CW -prespectra, then the sequence of prespectra

f E −→ F → Cofib(f) is a homotopy cofiber sequence in P. Proof. By Theorem A.1.2 and the construction of homotopy fiber Fib(f), we know the map P F → F is a fibration in P and Fib(f) is a pullback of

P F → F ← E.

In particular, we have the commutative diagram

95 Fib(f) E

P F F is a homotopy cartesian. Since ∗ → P F is a π∗-isomorphism, the sequence

Fib(f) → E → F is a homotopy fiber sequence. Similarly, we have E → E ∧ I+ is a cofibration in P by A.1.2 and Cofib(f) is a pushout of the cospan

F ← E → E ∧ I+.

Therefore the commutative diagram

E E ∧ I+

F Cofib(f) is a homotopy cocartesian. Since E ∧ I+ is π∗-isomorphic to ∗, the sequence

E → F → Cofib(f) is a homotopy cofiber sequence

Remark A.1.7. It seems the assumption E and F are Ω-prespectra in the first assertion (resp. E and F are CW -prespectra in the second), is redundant as P might actually be proper or even simplicial like its simplicial analogue [BF78, Sec.2]. It is customary that people consider homotopy fiber (resp. cofiber) sequences only in the category of fibrant objects (resp. cofibrant objects) or in a proper model category. The reason is that one has to ensure it is independent of the choice of fibration (resp. cofibration) replacements of E → F. The gluing lemma then is used to achieve this purpose, and, to prove the gluing lemma, one makes use of the property that weak equivalences are stable under pullback (resp. pushout) along fibration (resp. cofibration) which is one of the main properties of the category of fibrant objects (resp. cofibrant objects) or a proper model category.

96 A.1.3 Stability of P Firstly, we consider the infinite loop functor

∞ Ω : P → Top∗ which associates a prespectra E with its zero component E0 ∈ Top∗ and the suspension functor ∞ Σ : Top∗ → P. They together constitute a Quillen adjunction

∞ ∞ Σ : Top∗  P :Ω (see [Sch] for more details). Secondly, given a pointed model category M, there are the (canonically) induced (reduced) suspension functor

Σ : Ho(M) → Ho(M), and the (canonically) induced loop functor

Ω : Ho(M) → Ho(M).

They are a homotopy pushout of ∗ ← C → ∗, and a homotopy pullback of ∗ → C ← ∗, respectively, and hence these functors are uniquely determined up to isomorphisms in Ho(M). A model category M is a stable model category if and only if the induced suspension functor Σ and loop functor Ω are inverse equivalences on Ho(P). Standard suspension and loop functors: Now in the model category of prespectra, the induced suspension functor and loop functor can be realized by the standard suspension Σstd and loop functor Ωstd constructed as below. Let 1 E := {Ek, σk} andσ ˜k : S ∧ Ek → ΩEk+1 be the adjoint of σk. Then

std 1 1 Σ E := {Ek ∧ S , σk ∧ S } std 1 0 Ω E := {Top∗(S ,Ek), σk}, (A.1) where

0 1 1 (const.,id) 1 1 σk := S ∧ Top∗(S ,Ek) −−−−−−−→ Top∗(S ,S ∧ Ek) (S1,σ ) Top∗ k 1 −−−−−−−→ Top∗(S ,Ek+1) These constitute a Quillen adjunction

std std Σ : P  P :Ω . (One can deduce this adjunction by the approach described in [Sch, Lemma 0.72], although the lemma there is for the “alternative” suspension which is different from the standard suspension. The same approach, however, can be applied to this case as well. On the other hand, we have the k-fold shifting functor which associates a prespectrum E with another prespectrum ,denoted by E[k], whose n-component is En+k, when n + k ≥ 0; and ∗, otherwise. These readily give us the following Quillen equivalence

(−)[−1] : P  P :(−)[1]

97 Alternative suspension and loop functors: Although theoretically the construction of the standard suspension functor Σstd is the natural one, it suffers some disadvantages. For example, it is hard to compare Σstd with the shifting functor (−)[1]. For this reason, we introduce the alternative suspension and loop functors, denoted by Σalt and Ωalt. They are given as follows:

alt 1 1 Σ E := {S ∧ Ek,S ∧ σk} alt 1 1 Ω E := {Top∗(S ,Ek), Top∗(S , σ˜k)} 1 where E := {S ∧ Ek, σk : Ek → Ek+1} is a prespectrum andσ ˜k is the adjoint of σk. One can check they constitute a Quillen adjunction

alt alt Σ : P  P :Ω , and there are natural transformations

Σalt(−) 7→ (−)[1] (−)[−1] 7→ Ωalt(−).

alt alt such that Σ E → E[1] and E[−1] → Ω E are π∗-isomorphisms, for every E ∈ P. Therefore, Σalt and Ωalt actually constitute a Quillen equivalence

alt alt Σ : P  P :Ω . The relation between Σstd and Σalt: Although there seems no direct std alt 1 1 way to compare Σ and Σ —The most natural map Ek ∧ S → S ∧ Ek does not give us a map in P as it is not compatible with the structure maps of the prespectra (see the explanation in [Sch, Remark 0.34]), it is possible to compare them in Ho(P). In effect, it is proved in [Sch, Lemma 0.81] that there is a natural transformation from Σstd to Σalt in Ho(P). Notice both of them, as Quillen left adjoints, descend to functors of the homotopy category Ho(P). In this way, we see P is a stable model category. The discussion above is summarized in the following proposition: Proposition A.1.8. 1. There is a diagram of Quillen’s adjunctions (below), and the functors at the bottom constitute a Quillen equivalence.

Σ Top∗ ⊥ Top∗ Ω

Σ∞ a Ω∞ Σ∞ a Ω∞ Σalt/std, ∼ P ⊥ P Ωalt/std, ∼

2. The model category P is a stable model category as we have

Σ : Ho(P)  Ho(P):Ω are inverse equivalences. Especially, every homotopy fiber sequence is a homotopy cofiber sequence and vice verse. The distinguished triangles in

98 Ho(P) are the closure of homotopy cofiber sequences under isomorphisms, and there is the long exact sequence

... → [X, ΩB]Ho(P) → [X, F]Ho(P) → [X, E]Ho(P)

→ [X, B]Ho(P) → [X, ΣF]Ho(P) → ...,

for every homotopy (co)fiber sequence

F → E → B

and any X ∈ Ho(P).

Proof. The discussion preceding the theorem is just an outline. The detailed proof for the first assertion can be found in [Sch, Theorem 0.84] whereas the proof for the second assertion has been given in [Hov99, Chap.6-7], [Sch, Prop.0.110,0.116].

A.2 Relation with the Adams category

Here we describe how the Adams category A is related to the category of pre- spectra P. To distinguish, we reserve the term “CW -spectra” for objects in A while using the term “CW -prespectra” for the CW -objects in P. We also use [−, −]Ho(A) and [−, −]Ho(P) to distinguish the homotopy classes of maps in the corresponding homotopy categories. Recall the maps of CW -spectra are the eventually-defined maps and a map is a π∗-isomorphism if and only if it is a homotopy equivalence (see [Swi02, Chap.8] for the definitions). Note also, without loss of generality, we can assume CW -spectra are indexed over N ∪ {0}. Lemma A.2.1. For every CW -spectrum E in A, one can construct an Ω-CW - spectrum E¯ such that there is a homotopy equivalence of CW -spectra

φE : E → E¯.

Proof. When E is already an Ω-CW -spectrum, we let E¯ be E and φ identity. Otherwise, we use the construction described in [Rud08, Chap II, Prop.1.21]. Alternatively, one can first use the functor Q in [Sch, Definition 0.19], and then apply the CW -approximation as did in [Rud08, II, Prop. 1.21]. With this lemma, we obtain Proposition A.2.2. The inclusion

Ho(ΩA) ,→ Ho(A) is an equivalence of the categories, where ΩA is the full subcategory of Ω-CW - spectra. Proof. The functor in the other direction can be constructed by the lemma above. In more details, it sends a CW -spectrum E to E¯ and a map from E to F to a map from E¯ to F¯ via the isomorphism ∼ ¯ ¯ [E, F]Ho(A) −→ [E, F]Ho(A),

99 which is induced by the following cospan of isomorphisms ¯ ¯ ∼ ¯ ∼ [E, F]Ho(A) −→ [E, F]Ho(A) ←− [E, F]Ho(A). In this way, we see the homotopy equivalence

φE : E → E¯ in the preceding lemma is natural and induces the natural isomorphism required.

On the other hand, via the model structure of P, we have the following observation Proposition A.2.3. The inclusion

Ho(Pcf ) ,→ Ho(P) induces an equivalence of categories, where Pcf is the full subcategory of cofibrant- fibrant objects. Proof. This follows from the definition of the homotopy category of a model category. The next lemma about CW -approximation is proved in [Sch, Prop.0.53]. Lemma A.2.4. For any prespectrum E, one can construct a CW -prespectrum Eˆ and a map of prespectra νE : Eˆ → E, such that it is a level-wise weak homotopy equivalence. Moreover, if E is already a CW -prespectrum, one can choose Eˆ = E and νE = idE. Proof. By induction and the CW -approximation theorem (see [Sch, Prop.0.95]).

Theorem A.2.5. There are inverse equivalences of categories

F : Ho(ΩA)  Ho(Pcf ): G. Proof. Step 1: Construct the functor F: Recall a map of CW -spectra E → F is a pair (E0, f 0) where E0 is a cofinal subspectrum and f 0 : E0 → F a map of prespectra. Because E0 is cofinal subspectrum of E and both of them are CW -prespectra, we have the following induced homomorphism

θ 0 [E, F]Ho(P) −→ [E , F]Ho(P) is an isomorphism (see Remark A.2.6 or [MP12, Theorem 14.4.8]). Thus we can define F((E0, f 0)) to be θ−1(E0, f 0). To see the well-definedness, we recall two maps of CW -spectra (E0, f 0); (E00, f 00): E → F are homotopic if and only 000 000 if there exists a pair (E ∧ I+,H) such that E is a cofinal subspectrum of E0 ∩ E00 and 000 H : E ∧ I+ → F 0 00 is a map of presepctra that restricts to f |E000×{0} and f |E000×{1}. Then the following commutative diagram of isomorphisms

100 0 [E , F]Ho(P)

000 [E, F]Ho(P) [E , F]Ho(P)

00 [E , F]Ho(P) implies F((E0, f 0)) = F((E00, f 00)). Now let (E0, f 0): E → F and (F0, g0): F → G, we want to check the following equality

F((F0, g0) ◦ (E0, f 0)) = F((F0, g0)) ◦ F((E0, f 0)).

We first note, without loss of generality, one can assume f 0 factors through F0, the cofinal subspectrum of F. Then the identity above is the consequence of the following commutative diagram:

f 0 g0 E0 F0 G

F(E0, f 0) F(F0, g0) E F G

Step 2: Construct the functor G: The functor G sends a cofibrant Ω- prespectrum E to Eˆ, the Ω-CW -prespectrum constructed in the lemma pre- ceding the theorem. Then, by the following cospan of isomorphisms

ν∗ ν E ˆ F,∗ ˆ ˆ [E, F]Ho(P) −−→ [E, F]Ho(P) ←−−− [E, F]Ho(P), we obtain an isomorphism ˆ ˆ [E, F]Ho(P) → [E, F]Ho(P).

−1 ∗ Then, given a map of prespectra f : E → F, we define G(f) to be νF,∗ ◦ νE(f) which depends only on the homotopy class of f. The following identity is clear

G(g ◦ f) = G(g) ◦ G(f) as we have the commutative diagram

f g E F G

νE νF νG

G(f) G(g) Eˆ Fˆ Gˆ

101 Step 3: The natural isomorphism from F ◦G and id is given by the level-wise homotopy equivalence νE : Eˆ → E, as, from the cospan of isomorphisms:

ν∗ ν E ˆ F,∗ ˆ ˆ [E, F]Ho(P) −−→ [E, F]Ho(P) ←−−− [E, F]Ho(P), we know, given a map f : E → F, the following diagram commutes

f E F

νE νF

F ◦ G(f) Eˆ Fˆ

Lastly, the identity assigning map provides the natural isomorphism between G ◦ F and id. Namely, one assigns a CW -spectrum E the identity map idE ∈ 0 0 [E, E]Ho(A). To see how it works, we recall the definition of F((E , f )) which is the map of prespectra from E to F such that its image under the map

0 [E, F]Ho(P) → [E , F]Ho(P) is f 0, but that implies the following commutative diagram

f 0 E0 F

idF

G ◦ F(f) E F

Since the inclusion of a cofinal subspectrum represents the identity map in Ho(A), we see the identity-assigning map yields the natural isomorphism re- quired.

Remark A.2.6. Here we recall some notions about homotopy in a model cate- gory M: Let HomM(X,Y ) be the set of morphisms from X to Y in M, then one can use cylinder objects of X to define left homotopy and path objects of Y right homotopy. They are, however, not equivalence relations in general. However, if X is cofibrant, left homotopy is an equivalence relation on HomM(X,Y ); and, if Y is fibrant, right homotopy is an equivalence relation on HomM(X,Y ) (see [MP12, Proposition 14.3.9]). We denote the left and right homotopy classes of morphisms from X to Y by [X,Y ]l and [X,Y ]r, respectively. Then we have the following variant of Whitehead theorem (see [MP12, Proposition 14.3.14]): 1. Given p : Y → Z an acyclic fibration and X a cofibrant object, then p∗ :[X,Y ]l → [X,Z]l is a bijection. 2. Given i : X → Y an acyclic cofibration and Z a fibrant object, then ∗ i :[Y,Z]r → [X,Z]r is a bijection.

102 The notions of left and right homotopy are not the same in general, but when X is cofibrant, we have f, g : X → Y are left homotopic implies f, g are right homotopic; and when Y is fibrant, f, g are right homotopic implies f, g are left homotopic (see [MP12, Proposition 14.3.11]). Especially, if X is cofibrant and Y is fibrant, then left homotopy and right homotopy induce the same equivalence relation on HomM(X,Y ), and, furthermore, combing with the variant of Whitehead theorem above, one can obtain the following bijections

∼ [R(X),Q(Y )]r = [R(X),Q(Y )]l =: [X,Y ]Ho(M) −→ [X,Y ]l = [X,Y ]r, (A.2) where R(X) and Q(Y ) are the fibrant and cofibrant replacements of X and Y , respectively. Now apply this to the model category P, and assume E is a CW -prespectrum and F an Ω-CW -prespectrum, then Theorem A.2.5 shows the canonical homomorphism

[E, F]l → [E, F]Ho(A) is an isomorphism.

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