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. When the characteristic of R is 2, we take such Zk2 as the subgroup in E(R) generated by e1j (1) for ≤ j ≤ k + 1. By commutator formulas (cf. Lemma 4.2.1), any nontrivial element in Zk2 normally generates E(R). This shows that the action of E(R) is trivial. When the characteristic of R is not 2, we take such Zk2 as the subgroup generated by Ai,i+1 defined in Lemma 4.2.3 for ≤ i ≤ k. Any nontrivial element in such Zk2 is noncentral in E(R). By Lemma 4.2.3, any noncentral element in such Zk2 generates a normal subgroup containing E(R, 2R). Therefore the action of E(R) factors through that of E(R/2R), which is already proved since the characteristic of R/2R is 2. For the group EU (R, Λ), note that there is a hyperbolic embedding E(R) → EU (R, Λ) defined by A → diag(A, A∗−1 ). The action of EU (R, Λ) is trivial since E(R) normally generates EU (R, Λ). Bibliography [1] J.F. Adams, Infinite Loop Spaces. Ann. of Math. Studies no. 90, Princeton Univ. Press, Princeton, N. J., 1978. [2] R. Alperin, Locally compact groups acting on trees and property T, Monatsh. Math. 93 (1982), 261-265. [3] D. Arlettaz, On the homology of the special linear group over a number field, Comment. Math. Helv. 61 (1986), 556-564. [4] D. Arlettaz, On the k-invariants of iterated loop spaces, Proc. Roy. Soc. Edinburgh Sect. A 110 (1988), 343-350. [5] D. Arlettaz, Exponents for extraordinary homology groups, Comment Math Helvetici 68(1993) 653-672. [6] P. Balmer and M. Matthey, Codescent theory I: Foundations, Topology and its Applications 145 (2004), 11-59. [7] A. Bak, K-theory of Forms. Ann. of Math. Stud., vol. 98. Princeton Univ. Press, Princeton, 1981. [8] A. Bak and G.P. Tang, Stability for Hermitian K1 , J. Pure Appl. Algebra 150 (2000), 107-121. 123 124 BIBLIOGRAPHY [9] A. Bak, V. Petrov and G.P. Tang, Stability for quadratic K1 , K-theory 30 (2003), 1-11. [10] H. Bass, Algebraic K-Theory, Benjamin, New York, 1968. [11] H. Bass, Groups of integral representation type, Pacific Journal of Math., 86 (1980), 15-51. [12] A. Barnhill, The F An conjecture for Coxeter groups, Algebraic & Geometric Topology (2006), 2117-2150. [13] A.J. Berrick, An Approach to Algebraic K-Theory, Pitman Research Notes in Math 56, London,1982. [14] A.J. Berrick and C. Casacuberta, A universal space for plus-constructions, Topology 38 (1999), 467-477. [15] A.J. Berrick, I. Chatterji and G. Mislin, Homotopy idempotents on manifolds and Bass’ conjectures, Proc Nishida Fest (Kinosaki 2003), Geometry & Topology Monographs 10 (2007), 41-62. [16] D. Blanc, Algebraic invariants for homotopy types, Math. Proc. Camb. Philos. Soc. 127 (1999), 497-523. [17] A. K. Bousfield, The localization of spaces with respect to homology, Topology, 14 (1975) 133-150. [18] A. K. Bousfield, Homological Localization Towers for Groups and π-modules, Memoirs of the American Mathematical Society 10, no. 186,1977. [19] A. K. Bousfield, Homotopical localizations of spaces, American Journal of Mathematics 119(1997), 1321-1354. BIBLIOGRAPHY 125 [20] A. K. Bousfield and D. M. Kan, Homotopy Limits, Completions and Localizations, Lecture Notes in Mathematics 304. Berlin, Heidelberg, New York: Springer, 1972. [21] G. Bredon, Equivariant Cohomology Theories, Springer Lect. Notes Math. 34 (1967). [22] G.E. Bredon, Sheaf Theory, second edition. Graduate Texts in Mathematics, 170. Springer-Verlag, New York, 1997. xii+502 pp. [23] M.R. Bridson, Semisimple actions of mapping class groups on CAT(0) spaces, to appear in “The Geometry of Riemann Surfaces” (F. P. Gardiner, G. GonzalezDiez and C. Kourouniotis, eds.), LMS Lecture Notes 368, Cambridge Univ. Press, Cambridge, 2010, pp. 1–14. [24] M.R. Bridson, On the dimension of CAT(0) spaces where mapping class groups act, Journal f¨ ur reine und angewandte Mathematik, to appear. arXiv:0908.0690. [25] M.R. Bridson, Helly’s theorem, CAT(0) spaces, and actions of automorphism groups of free groups, preprint 2007. [26] M.R. Bridson, The rhombic dodecahedron and semisimple actions of Aut(Fn ) on CAT(0) spaces, arXiv:1102.5664v1. [27] M.R. Bridson and A. Haefliger, Metric Spaces of Nonpositive Curvature, Grundlehren der Math. Wiss. 319, Springer-Verlag, Berlin, 1999. [28] M.R. Bridson, K. Vogtmann, Actions of automorphism groups of free groups on homology spheres and acyclic manifolds, Commentarii Mathematici Helvetici 86 (2011), 73-90. [29] R. Brooks, The fundamental group and the spectrum of the Laplacian, Comment. Math. Helv. 56 (1981), 581–598. 126 BIBLIOGRAPHY [30] K. Brown, Cohomology of Groups, volume 87 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1994. [31] H. Cartan, Alg`ebres d’Eilenberg–MacLane et Homotopie, S´eminaire H. Cartan Ecole Norm. Sup. (1954/1955), exp. 11; see also: Oeuvres, Vol. III, Springer, 1979, 1374-1394. [32] I. Chatterji and M. Kassabov, New examples of finitely presented groups with strong fixed point properties, Journal of Topology and Analysis (2009), 1-12. [33] B. S. Chwe and J. Neggers, On the extension of linearly independent subsets of free modules to bases. Proc. Amer. Math. Soc. 24(1970), 466-470. [34] B. S. Chwe and J. Neggers, Local rings with left vanishing radical. J. London Math. Soc. 4(1971), 374-378. [35] C.W. Curtis and I. Reiner, Methods of Representation Theory with Applications to Finite Groups and Orders (Volume 2), Wiley-Interscience Publication, J. Wiley & Sons, Inc., New York, 1987. [36] J. Davis and W. L¨ uck, Spaces over a category and assembly maps in isomorphism conjectures in K- and L- theory, K-theory 15(1998) 201-252. [37] A. Dold, Relations between ordinary and extraordinary homology, Colloq. algebr. Topology, Aarhus(1962), 2-9. [38] E. Dror Farjoun, Cellular Spaces, Null Spaces and Homotopy Localization, Lecture Notes in Math. vol. 1622, Springer-Verlag, Berlin, Heidelberg, New York, 1996. [39] E. Dror Farjoun, K. Orr and S. Shelah, Bousfield localization as an algebraic closure of groups, Israel Journal of Mathematics 66 (1989), 143-153. BIBLIOGRAPHY 127 [40] S. Eilenberg and S. MacLane, Relations between homology and homotopy groups of spaces, Ann. of Math., 46(1945), 480-509. [41] A. D. Elmendorf, Systems of fixed point sets, Trans. Amer. Math. Soc. 277 (1983), 275-284. [42] M. Ershov and A. Jaikin-Zapirain, Property ( T ) for noncommutative universal lattices, Inventiones Mathematicae 179 (2010), 303-347. [43] M. Ershov, Andrei Jaikin-Zapirain and M. Kassabov, Property ( T ) for groups graded by root systems, arXiv:1102.0031. [44] B. Farb, Group actions and Helly’s theorem, Advances in Mathematics 222 (2009), 1574-1588. [45] B. Farb and P. Shalen, Real-analytic actions of lattices, Inventiones Mathematicae 135 (1999), 273-296. [46] M. Farber, Homological algebra of Novikov-Shubin invariants and Morse inequalities, GAFA 6(1996), 628–665. [47] M. Farber and S. Weinberger, On the zero-in-the-spectrum conjecture, Ann. of Math. (2) 154 (2001),139-154. [48] D. Fisher, Groups acting on manifolds: around the Zimmer program, To appear in the Zimmer Festschrift, arXiv:0809.4849. [49] M. Fukunaga, Fixed points of elementary subgroups of Chevalley groups acting on trees, Tsukuba J. Math., (1979), 7-16. [50] C.R. Guilbault and F.C. Tinsley, Manifolds with non-stable fundamental groups at infinity III, Geometry and Topology, 10(2006) 541-556. 128 BIBLIOGRAPHY [51] C.R. Guilbault and F.C. Tinsley, Spherical alterations of handles: embedding the manifold plus construction, arXiv:1108.5166. [52] A.J. Hahn, O.T. O’Meara, The Classical Groups and K-Theory, Springer-Verlag, Berlin, 1989. [53] I. Hambleton and E.K. Pedersen, Identifying assembly maps in K- and L-theory, Math. Ann. 328 (2004), 27-57. [54] A. Hatcher, Algebraic Topology, Cambridge, New York, NY: Cambridge University Press, 2002. [55] J-C. Hausmann, Homological surgery, Annals of Math. 104(1976), 573-584. [56] J-C. Hausmann, Manifolds with a Given Homology and Fundamental Group, Commentarii Mathematici Helvetici 53(1978) (1): 113–134. [57] J.-C. Hausmann and S. Weinberger, Caract´eristiques d’Euler et groupes fondamentaux des vari´et´es de dimension 4, Comment. Math. Helv. 60 (1985), 139–144. [58] N. Higson, J. Roe and T. Schick, Spaces with vanishing l2 -homology and their fundamental groups (after Farber and Weinberger), Geometriae Dedicata, 87 (2001), 335-343. [59] P. Hilton and U. Stammbach, A Course in Homological Algebra, 2nd ed Graduate Texts in Math., Springer, New York, 1997. [60] P. Hirschhorn, Model Categories and their Localizations, American Mathematical Society, 2002. [61] M. V. Horosevskii, On automorphisms of finite groups, Mat. Sbornik Tom 93 (135) (1974), 584-594. [62] M. Hovey, Model Categories, American Mathematical Society, 1999. BIBLIOGRAPHY 129 [63] M. Hovey, Spectra and symmetric spectra in general model categories, Journal of Pure and Applied Algebra 165(2001), 63-127. [64] M. Joachim, K-homology of C ∗ -categories and symmetric spectra representing Khomology, Math. Ann. 327 (2003) 641-670. [65] M.A. Kervaire, On higher dimensional knots, in Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse) (edited by S.S.Cairns), Princeton University Press, Princeton (1965), 105-119. [66] M. Kervaire, Smooth homology spheres and their fundamental groups, Transactions of the American Mathematical Society 144 (1969) 67–72. [67] H. Lenzing, A homological characterization of Steinitz rings, Proc. Amer. Math. Soc., 29(1971), 269-271. [68] W. L¨ uck. Chern characters for proper equivariant homology theories and applications to K- and L-theory, J. reine angew. Math. 543 (2002), 193-234. [69] W. L¨ uck, L2 -Invariants: Theory and Applications to Geometry and K-Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer, 2002. [70] W. L¨ uck and H. Reich, Detecting K-theory by cyclic homology, Proceedings of the LMS 93(2006) 593–634. [71] W. L¨ uck and H. Reich, The Baum-Connes and the Farrell-Jones Conjectures in K- and L-theory, Handbook of K-theory Volume 2, editors: E.M. Friedlander, D.R. Grayson, Springer, 2005. [72] W. L¨ uck, H. Reich and M. Varisco, Commuting homotopy limits and smash products. K-Theory, 30(2):137–165, 2003. Special issue in honor of Hyman Bass on his seventieth birthday. Part II. 130 BIBLIOGRAPHY [73] B.A. Magurn. An Algebraic Introduction to K -Theory, Cambridge University Press, 2002. [74] M. A. Mandell, J. P. May, S. Schwede and B. Shipley, Model Categories of Diagram Spectra, Proc. London Math. Soc. 82(2001), 441-512. [75] G.A. Margulis, Discrete subgroups of semisimple Lie groups, Ergeb. Math. Grenzgeb. (3) 17, Springer-Verlag, Berlin, 1991. [76] M. Matthey, A delocalization property for assembly maps and an application in algebraic K-theory of group rings, K-Theory 24(2001) 87-107. [77] J. May et al., Equivariant Homotopy and Cohomology Theory dedicated to the memory of Robert J. Piacenza, the Conference Board of the Mathematical Sciences, American Mathematical Society, 1996. [78] W. Meier, Acyclic maps and knots complements, Math. Ann. 243(1979), 247-259. [79] J.W. Milnor, A procedure for killing the homotopy groups of differentiable manifolds, Differential geometry, Proc. Symp. Pure Math. , , Amer. Math. Soc. (1961) pp. 39-55. [80] G. Mislin and G. Peschke, Central extensions and generalized plus-constructions, Trans. Amer. Math. Soc. 353 No. 2(2001), 585-608. [81] G. Mislin and A. Valette, Proper Group Actions and the Baum-Connes Conjecture, Basel: Birk¨ auser, 2003. [82] B. Nashier and W. Nichols, On Steinitz properties, Arch. Math., 57(1991), 247-253. [83] B. T. Noe, Spaces over a category and Brown Representability, preprint. [84] K. Ohshika, Discrete Groups, Translations of mathematical monographs 207, Amer. Math. Soc., 2002. BIBLIOGRAPHY 131 [85] R. Oliver, Whitehead Groups of Finite Groups, London Mathematical Society Lecture Note Series .132, Cambridge University Press, 1988. [86] K. Parwani, Actions of SL(n, Z) on homology spheres, Geom. Dedicata 112 (2005), 215-223. [87] D. Quillen, Cohomology of groups, Actes Congr`es intern. math., 1970, Tome 2, 47-51. [88] A Ranicki, Algebraic and Geometric Surgery, Oxford Mathematical Monograph (OUP), 2002. [89] J. Rodr´ıguez and D. Scevenels, Homology equivalences inducing an epimorphism on the fundamental group and Quillen’s plus-construction, Proc. Amer. Math. Soc. 132(2004), no. 3, 891–898. [90] J. Rosenberg, Algebraic K-theory and its Applications, Springer, 1994. [91] J. Rosenberg, A Minicourse on Applications of Non-Commutative Geometry to Topology, Surveys in Noncommutative Geometry, Clay Mathematics Proceedings 6, 2006. [92] Y.B. Rudyak, On Thom Spectra, Orientability, and Cobordism, Springer (1st ed.), 1998. [93] J.P. Serre. Linear representations of finite groups. Springer-Verlag, New York, 1977. [94] P.A. Smith, Transformations of finite period, Ann. Math. 39 (1938), 137-164. [95] P.A. Smith, Transformations of finite period II, Ann. Math. 40 (1939), 690-711. [96] P.A. Smith, Permutable periodic transformations, Proc. Nat. Acad. Sci. U.S.A. 30 (1944), 105-108. 132 BIBLIOGRAPHY [97] Tammo tom Dieck, Transformation groups, Walter de Gruyter & Co., Berlin, 1987. [98] R. Swan, Induced representations and projective modules, Ann. of Math. 71 (1960), 552-578. [99] K. Varadarajan, Groups for which Moore spaces M (π, 1) exist, Ann. of Math. 84 (2)(1966), 368-371. [100] L.N. Vaserstein and H. You, Normal subgroups of classical groups over rings, J. Pure Appl. Algebra 105 (1995), 93-105. [101] C.T.C. Wall, Surgery on compact manifolds,. Academic Press, London, 1970. London Mathematical Society Monographs, No. 1. [102] C. Weibel, An Introduction to Homological Algebra, Cambridge Univ. Press, 1994. [103] C. Weibel, An introduction to algebraic K-theory, preprint. [104] S. Weinberger, SL(n, Z) cannot act on small tori. Geometric topology (Athens, GA, 1993), 406–408, AMS/IP Stud. Adv. Math., Amer. Math. Soc., Providence, RI, 1997. [105] S. Weinberger. Some remarks inspired by the C Zimmer program. in Rigidity and Group actions, Chicago lecture notes, 2011. [106] S. Ye, A unifid approach to the plus-construction, Bousfield localization, Moore spaces and zero-in-the-spectrum examples. to appear in Israel Journal of Mathematics. arXiv:1107.3392. [107] R. Zimmer, Actions of semisimple groups and discrete subgroups, Proc. I.C.M., Berkeley 1986, 1247-1258 BIBLIOGRAPHY 133 [108] R. 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