Plus constructions, assembly maps and group actions on manifolds

PLUS CONSTRUCTIONS, ASSEMBLY MAPS AND GROUP ACTIONS ON MANIFOLDS YE SHENGKUI (M.Sc. HIT) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY OF SINGAPORE 2012 Acknowledgements I would like to express my sincere appreciation to my supervisor Prof. A.J. Berrick for what he has done for me, from helping me get admitted to the PhD program, always being prepared to answer my questions, listening to my naive ideas, encouraging me to explore new areas, to correcting my English errors in both papers and this thesis. What I have learnt from him is not only the mathematics but also the way of life. I would also like to thank Prof. Wolfgang L¨ uck for supporting from his Leibniz-Preis a visit to Hausdorff Center of Mathematics in University of Bonn from April 2011 to July 2011, when parts of this thesis were written. I want to thank Professor T. Schick for noting a gap in a previous claim on G-dense rings. Thanks are also given to the members of graduate topology seminars at NUS, including Ji Feng, Yuan Zihong, Zhang Wenbin and so on, especially Ji Feng for his long-term weekly discussions. I am also greatly indebted to my wife Huang Chun for her constant encouragement and understanding with patience throughout the years of my PhD study. Finally, I would like to thank Ji Feng, Ma Jiajun, Wu Bin and Yuan Zihong for reading previous versions of this thesis and Sun Xiang for printing this thesis. I II Contents Acknowledgements I Summary VII Introduction 1.1 Generalized Quillen’s plus constructions . . . . . . . . . . . . . . . . . . 1.2 Assembly maps and isomorphism conjectures . . . . . . . . . . . . . . . 1.3 Matrix group actions on CAT(0) spaces and manifolds . . . . . . . . . 11 1.4 Organization of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Generalized plus constructions 2.1 2.2 2.3 21 A generalized Quillen’s plus construction for CW complexes . . . . . . . 21 2.1.1 G-dense rings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 2.1.2 Proof of Theorem A . . . . . . . . . . . . . . . . . . . . . . . . . 26 The manifold version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.1 Preliminary results and basic facts . . . . . . . . . . . . . . . . . 31 2.2.2 Proof of Theorem B . . . . . . . . . . . . . . . . . . . . . . . . . 34 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 2.3.1 Quillen’s plus-construction . . . . . . . . . . . . . . . . . . . . . 39 2.3.2 Bousfield’s integral localization . . . . . . . . . . . . . . . . . . . 40 2.3.3 Moore spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42 III IV CONTENTS 2.3.4 Partial k-completion of Bousfield and Kan . . . . . . . . . . . . . 43 2.3.5 Zero-in-the-spectrum conjecture . . . . . . . . . . . . . . . . . . 44 2.3.6 Surgery plus construction for manifolds . . . . . . . . . . . . . . 49 2.3.7 Surgery preserving integral homology groups . . . . . . . . . . . 51 2.3.8 The fundamental groups of homology manifolds . . . . . . . . . . 53 2.3.9 The fundamental groups of high-dimensional knots . . . . . . . . 54 Assembly maps and equivariant homology 3.1 57 Homology and cohomology theories over categories . . . . . . . . . . . . 58 3.1.1 Farrell-Jones conjecture and Baum-Connes conjecture . . . . . . 61 3.1.2 Bredon (co)homologies over categories . . . . . . . . . . . . . . . 63 Postnikov invariants and localization of a spectrum . . . . . . . . . . . . 65 3.2.1 Triangulated category and Postnikov invariants of spectra . . . . 65 3.2.2 Localization of a C-spectrum . . . . . . . . . . . . . . . . . . . . 70 Equivariant Chern characters . . . . . . . . . . . . . . . . . . . . . . . . 73 3.3.1 The case of equivariant homology theory . . . . . . . . . . . . . . 73 3.3.2 The case of equivariant cohomology theory . . . . . . . . . . . . 80 3.4 The ‘best’ possible bound in lower-degree cases . . . . . . . . . . . . . . 83 3.5 Applications to algebraic K-theory . . . . . . . . . . . . . . . . . . . . . 84 3.5.1 Algebraic K-theory of integral group rings . . . . . . . . . . . . . 84 3.5.2 Computation of Bredon homology . . . . . . . . . . . . . . . . . 89 3.5.3 Algebraic K-theory of rational group rings 94 3.2 3.3 . . . . . . . . . . . . Matrix group actions on CAT(0) spaces and manifolds 4.1 101 Basic notions and facts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 4.1.1 CAT(0) spaces and property FAn . . . . . . . . . . . . . . . . . . 102 4.1.2 Homology manifolds and Smith theory . . . . . . . . . . . . . . . 103 CONTENTS 4.2 4.3 V Elementary groups and K-theory . . . . . . . . . . . . . . . . . . . . . . 105 4.2.1 Elementary groups and Steinberg groups . . . . . . . . . . . . . 105 4.2.2 K-theory and stable range . . . . . . . . . . . . . . . . . . . . . . 109 Proof of theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112 4.3.1 Group actions on CAT(0) spaces . . . . . . . . . . . . . . . . . . 112 4.3.2 Group actions on spheres and acyclic manifolds . . . . . . . . . . 117 Bibliography 135 VI CONTENTS Summary The Farrell-Jones conjecture says that the algebraic K-groups of a group ring can be computed by the equivariant homology groups of a classifying space via an assembly map. Therefore, to address this conjecture it is important to understand the definition of K-theory, equivariant homology theory and group actions (for providing models of classifying spaces). The thesis consists of three parts. The first part aims to understand the definitions of algebraic K-theory. We introduce a construction adding low-dimensional cells (handles) to a CW complex (manifold) that satisfies certain low-dimensional conditions. It preserves high-dimensional homology with appropriate coefficients. This includes as special cases Quillen’s plus construction, Bousfield’s integral homology localization, Varadarajan’s existence of Moore spaces M (G; 1), Bousfield and Kan’s partial k-completion of spaces, the existences of high dimensional knot groups and homology spheres proved by Kervaire. We also use the construction to get some examples for the zero-in-the-spectrum conjecture, which give generalizations of the examples found by Farber-Weinberg and Higson-Roe-Schick. The second part investigates the equivariant homology theory. We give a computation of equivariant homology theories over categories. This generalizes both Arlettaz’s result for generalized homology theory and L¨ uck’s rational computation of Chern characters for equivariant K-theory. Some applications to algebraic K-theory are obtained as well. We prove that for a fixed group, there is an injection of the homology groups VII VIII CONTENTS of the group into the algebraic K-groups of the group ring, after tensoring with some subring of rationals. The third part studies matrix group actions on CAT(0) spaces and manifolds. It is shown that matrix groups can only act trivially on low-dimensional spheres and that matrix group actions on low-dimensional CAT(0) spaces always have a global fixed point. These results give generalizations of results obtained by Bridson-Vogtmann and Pawani concerning special linear group actions on spheres and of results obtained by Farb concerning Chevalley group actions on CAT(0) spaces. As applications to low-dimensional representations, we show that there are no non-trivial group homomorphisms from matrix groups to low-sized matrix groups for some rings. 4.3. PROOF OF THEOREMS 121 act effectively by homeomorphisms on a generalized d-sphere over Z3 . Without loss of generality, we may assume that the action of B1 is trivial. Note that [e32 (1), B1 ] = e31 (−1)e32 (2) and [e31 (−1)e32 (2), e12 (−1)] = e32 (1). The matrix e32 (1) normally generates the whole group En (R). This shows that the group action of En (R) is trivial. Now we prove (c). Suppose that the group EU2n (R, Λ) acts by homeomorphisms on a generalized d-homology sphere over Z2 or Z3 . There is a group homomorphism En (R) → EU2n (R, Λ) defined by the hyperbolic embedding A → diag(A, A∗−1 ) for any element A ∈ En (R). By the commutator formulas in Lemma 4.2.2, we see that EU2n (R, Λ) is normally generated by the image of En (R). Since the action of En (R) is trivial, the action of EU2n (R, Λ) is trivial as well. Remark 4.3.1. If the generalized spheres in Theorem H are smooth manifolds and the actions are smooth, the proof is much easier by noting the fact that Zk cannot act effectively by orientation-preserving diffeomorphisms on a d-sphere for d ≤ k − (cf. the proof of Theorem 2.1 in [28]). When we know that Theorem H is true for R = Z, the general-ring case can also be proved by using the normal generation of En (R) by the image of En (Z). Our intent here is to avoid the Margulis finiteness theorem. Moreover, the proof given here works for Steinberg groups as well. Proof of Theorem I. The strategy of the proof is similar to that of Theorem H. We construct an abelian subgroup Zn3 of EU2n (R, Λ) as follows. For i = 1, 2, . . . , n, let Ci = ρi,n+i (1)ρn+i,i (−1)ρi,n+i (1)ρn+i,i (−1) ∈ EU2n (R). 122CHAPTER 4. MATRIX GROUP ACTIONS ON CAT(0) SPACES AND MANIFOLDS It is obvious that the order of Ci is and the subgroup generated by Ci (i = 1, 2, . . . , n) is Zn3 . The remainder of the proof of (i) is the same as that of (b)(i) in Theorem H. Proof of Corollary 1.3.6. Let En−1 (A) act on the space Rn−1 by matrix multiplication. According to Theorem H a(ii), the image of En (R) in En−1 (A) acts trivially on Rn−1 . This implies that the image in (i) is the identity matrix. The second part can be proved similarly by using Theorem I and considering the group Sp2(n−1) (A) action on the space R2(n−1) . Proof of Theorem J. For the group E(R), the proof is similar to that of Lemma in [105]. The idea is as follows. For sufficiently large k, the abelian group Zk2 cannot act effectively on the manifolds in Theorem J. 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Zimmer, Lattices in semisimple groups and invariant geometric structures on compact manifolds, Discrete groups in geometry and analysis (New Haven, Conn.,1984), 152-210, Progr. Math., Vol. 67, Birkh¨auser Boston, Boston, MA, 1987. [109] B.P. Zimmermann, SL(n, Z) cannot act on small spheres, Topology and its Applications 156 (2009), 1167-1169. [110] B.P. Zimmermann, A note on actions of the symplectic group Sp(2g, Z) on homology spheres, arxiv.org/abs/0903.2946v1. 134 BIBLIOGRAPHY PLUS CONSTRUCTIONS, ASSEMBLY MAPS AND GROUP ACTIONS ON MANIFOLDS YE SHENGKUI NATIONAL UNIVERSITY OF SINGAPORE 2012 Plus constructions, assembly maps and group actions on manifolds Ye Shengkui 2012 [...]... Quillen’s plus construction Later on, there are also other approaches, such as Quillen’s Q-construction, Waldhausen’s s-construction and so on (see Weibel’s textbook [103] for 1 2 CHAPTER 1 INTRODUCTION more details on such constructions) In the first part of this thesis, we give a generalization of Quillen’s plus construction In the second part, the assembly map between equivariant homology and algebraic... by Chatterji and Kassabov (cf Corollary 4.5 in [32]) We now consider group actions on manifolds The following conjecture from Farb and Shalen [45], is related to Zimmer’s program which is trying to understand group actions on compact manifolds (see [107, 108] or the survey article [48]) This conjecture says that the special linear group SLn (Z) can only act on lower dimensional compact manifolds in... constrained way Conjecture 1.3.3 Any smooth action of a finite-index subgroup of SLn (Z), where n > 2, on an r-dimensional compact manifold M factors through a finite group action if r < n − 1 Parwani [86] considers this conjecture for the group SLn (Z) itself and M a sphere The idea is to use the theory of compact transformation groups to show that some 1.3 MATRIX GROUP ACTIONS ON CAT(0) SPACES AND MANIFOLDS. .. facts on CAT(0) spaces, homology manifolds (see Section 4.1) and matrix groups (see Section 4.2), we will prove that any matrix group action on low-dimensional CAT(0) spaces by isometries always has a global fixed point and that any matrix group action on low-dimensional spheres (or acyclic manifolds) by homeomorphisms is always trivial 20 CHAPTER 1 INTRODUCTION Chapter 2 Generalized plus constructions... trivial actions of SLn (Z) on small finite sets by Chatterji and Kassabov in [32] (Lemma 4.2) and so on Zimmermann [107] actually proves that any smooth action of SLn (Z) on small spheres is trivial It is natural to consider other kinds of group actions on compact manifolds Zimmermann [110] proves a similar trivial action of the symplectic group Sp2n (Z) The group action of Aut(Fn ), the automorphism group. .. Organization of the thesis The thesis consists of three parts As shown in the title, we will study a generalized plus construction, the equivariant homology in assembly map and matrix group actions on CAT(0) spaces and manifolds In Chapter 2, we introduce generalized plus constructions for both CW complexes and manifolds The results are stated in terms of G-dense rings, whose properties are studied in Section... an injection Hn (NG (C); Z[ S1n ]) → Kn (Q[G]) (C)∈(F Cyc) Z Z[ S1n ] Note that the second part of this theorem does not follow from the rational injection directly, since it is not obvious that the rational Chern character is induced from the one with Z[ S1n ]-coefficients 1.3 Matrix group actions on CAT(0) spaces and manifolds In this section, we study group actions on CAT(0) spaces and manifolds Recall... Theorem I are sharp 1.3 MATRIX GROUP ACTIONS ON CAT(0) SPACES AND MANIFOLDS 17 We consider some applications of the above results in this section As motivations, we make the following conjecture Conjecture 1.3.4 Let R be a ring and n > 2 an integer Then there is no nontrivial group homomorphism En (R) → En−1 (R) As the representations of groups with property FAn are quite constrained, we get that for... Quillen’s plusconstruction, Rodr´ ıguez and Scevenels’ work on Bousfield’s integral localization in [89], Varadarajan’s theorem on the existence of Moore spaces in [99], the partial k-completion of Bousfield and Kan in [20], and counterexamples to the zero-in-the-spectrum conjecture by Farber and Weinberger [47], and Higson, Roe and Schick [58] We introduce a construction to preserve high-dimensional homology... low-dimensional cells to a space satisfying certain low-dimensional conditions A manifold version of such results is also obtained This contains as special cases Quillen’s plus construction by handles obtained by Haussmann [55], the existence of homology spheres and high-dimensional knot groups obtained by Kervaire [66, 65] 2.1 A generalized Quillen’s plus construction for CW complexes In this section, a . matrix group actions on CAT(0) spaces and manifolds. It is shown that matrix groups can only act trivially on low-dimensional spheres and that ma- trix group actions on low-dimensional CAT(0) spaces. generalizations of results obtained by Bridson-Vogtmann and Pawani concerning special linear group actions on spheres and of results obtained by Farb con- cerning Chevalley group actions on CAT(0). PLUS CONSTRUCTIONS, ASSEMBLY MAPS AND GROUP ACTIONS ON MANIFOLDS YE SHENGKUI (M.Sc. HIT) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MATHEMATICS NATIONAL UNIVERSITY