Equivariant Novikov conjecture for groups acting on Euclidean buildings

Gong, Donggeng

doi:10.1090/S0002-9947-98-01990-4

Equivariant Novikov conjecture for groups acting on Euclidean buildings

Author: Donggeng Gong
Journal: Trans. Amer. Math. Soc. 350 (1998), 2141-2183
MSC (1991): Primary 46L80; Secondary 55N15, 19K56, 58G12
DOI: https://doi.org/10.1090/S0002-9947-98-01990-4
MathSciNet review: 1433118
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Abstract: We prove the equivariant Novikov conjecture for groups acting on Euclidean buildings by using an equivariant Hilsum-Skandalis method. We also obtain an equivariant version of the Connes-Gromov-Moscovici theorem for almost flat $C^{*}$ -algebra bundles.

[AtS] M. F. Atiyah and I. Singer, Index theorem of elliptic operators, I, III, Ann. Math., 87 (1968), 484-530, 546-609. MR 38:5243, MR 38:5245
[BaJ] S. Baaj and P. Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les $C^{*}$ -modules hilbertiens, C. R. Acad. Sci. Paris Ser. I, 296 (1983), 875-878. MR 84m:46091
[BCH] P. Baum, A. Connes and N. Higson, Classifying space for proper action and $K$ -theory of group $C^{*}$ -algebras, $C^{*}$ -Algebras: -, A Fifty Year Celebration, R. S. Doran, ed., Contemp. Math., Vol. 167, AMS., Providence, 1994, 241-291. MR 96c:46070
[Bla] B. Blackadar, $K$ -Theory for Operator Algebras, MSRI Publ. Vol. 5, Springer, NY, 1986.MR 88g:46082
[BHM] M. B $\ddot{o}$ kstedt, W. C. Hsiang and I. Madsen, The cyclotomic trace and algebraic $K$ -theory of spaces, Invent. Math., 111 (1993), 465-539. MR 94g:55011
[Bre] G. E. Bredon: Introduction to compact transformation groups, Academic Press, NY, 1980. MR 54:1265
[Bro] K. S. Brown, Buildings, Springer, NY, 1989.MR 90e:20001
[BuS] K. Burns and R. Spatzier, Manifolds of nonpositive curvature and their buildings, Publ. Math. IHES, No. 65, 35-59, 1987.MR 88g:53050
[Cap] S. E. Cappell, On homotopy invariance of higher signatures, Invent. Math., 33 (1976), 171-179.MR 55:13264
[CaP] G. Carlsson and E.K. Pedersen, Controlled algebra and the Novikov conjectures for $K$ and $L$ -theory, Topology, 34 (1995), 731-758.MR 96f:19006
[CoJ] R. L. Cohen and J. D. S. Jones, Algebraic $K$ -theory of spaces and the Novikov conjecture, Topology, 29 (1990), 317-344.MR 91i:19001
[CGM] A. Connes, M. Gromov and H. Moscovici, Group cohomology with Lipschitz control and higher signature, Geom. Funct. Anal., 3 (1993), 1-78.MR 93m:19011
[CoM] A. Connes and H. Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology, 29 (1990), 345-388.MR 92a:58137
[CMW] R. Curto, P. Muhly and D. Williams, Cross products of strongly Morita equivalent $C^{*}$ -algebras, Proc. Amer. Math. Soc., 90 (1984), 528-530. MR 85i:46083
[FaJ] F. T. Farrell and L. E. Jones, Rigidity in geometry and topology, Proc. ICM, Kyoto 1990, Springer, 1991, 653-663.MR 93g:57041
[FeW] S. C. Ferry and S. Weinberger, Curvature, tagentiality and controlled topology, Invent. Math., 105 (1991), 401-414.MR 94c:57043
[Gong1] D. Gong, Bivariant twisted cyclic theory and spectral sequences of crossed products, J. Pure. Appl. Alg., 79 (1992), 225-254.MR 93f:16013
[Gong2] D. Gong, Excision of equivariant cyclic cohomology of topological algebras, Michigan Math. J. 39 (1992), 455-473.MR 94f:19005
[Gong3] D. Gong, $L^{2}$ -Analytic Torsions, Equivariant Cyclic Cohomology and the Novikov Conjecture, thesis, SUNY at Stony Brook, 1992.
[Gong4] D. Gong, The Chern characters in equivariant cyclic theory, preprint, 1994.
[Gong5] D. Gong, A relation between the Novikov and Thurston conjectures, Applied Functional Anal., Vol. 2, Sci-Tech Information Services, Hong Kong, 1995, 52-62.
[GoR] D. Gong and M. Rothenberg, Analytic torsion forms for noncompact fiber bundles, preprint, 1995.
[GrS] M. Gromov and R. Schoen, Harmonic maps into singular spaces and $p$ -adic superrigidity for lattices in groups of rank one, Publ. Math. IHES, No.76, 1993, 165-246.MR 94e:58032
[Hil] M. Hilsum, Fonctorialité en $k$ -theórie bivariante pour les variétés lipschitziennes, $K$ -Theory, 3 (1989), 401-440. MR 91j:19012
[HiS] M. Hilsum and G. Skandalis, Invariance par homotopie de la signature a coefficients dans un fibré presque plat, J. reine angew. Math., 423 (1992), 73-99.MR 93b:46137
[Hus] D. Husemoller, Fibre Bundles, McGraw-Hill, NY, 1966. MR 37:4821
[KaM] J. Kaminker and J. G. Miller, Homotopy invariance of the analytic signature operators over $C^{*}$ -algebras, J. Oper. Theory, 14 (1985), 113-127. MR 87b:58082
[Kas] G. G. Kasparov, Equivariant -theory and the Novikov conjecture, Invent. Math., 91 (1988), 147-201.MR 88j:58123
[KaS] G. G. Kasparov and G. Skandalis, Groups acting on buildings, operator $K$ -theory and Novikov's conjecture, $K$ -theory, 4 (1991), 303-338. MR 92h:19009
[KoS] N.J. Korevaar and R. M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Commun. Anal. Geom., 1 (1993), 1-99.MR 95b:58043
[Lus] G. Lusztig, Novikov's higher signature and families of elliptic operators, J. Diff. Geom., 7 (1971), 229-256.MR 48:1250
[Mis1] A. S. Mishchenko, Homotopy invariants of non-simply connected manifolds I, Math. USSR Izv. 4 (1970), 506-519.MR 42:3795
[Mis2] A. S. Mishchenko, Infinite dimensional representations of discrete groups, and higher signatures, Math. USSR Izv., 8 (1974), 85-111.MR 50:14848
[Roe] J. Roe, Hyperbolic metric spaces and the exotic cohomology Novikov conjecture, $K$ -Theory, 4 (1991), 501-512; 5 (1991), 189. MR 93e:58180a, b
[Ros] J. Rosenberg, $C^{*}$ -algebras, positive scalar curvature, and the Novikov conjecture, Publ. Math. IHES, No. 58, 1983, 409-424.MR 85g:58083
[RoW1] J. Rosenberg and S. Weinberger, Higher $G$ -indices and applications, Ann. Sci. École Norm. Sup., 21 (1988), 479-495.MR 90f:58170
[RoW2] J. Rosenberg and S. Weinberger, An equivariant Novikov conjecture, $K$ -theory, 4 (1990), 28-53.MR 91k:57039
[Ska] G. Skandalis, Approche de la conjecture de Novikov par la cohomologie cyclique, Astérisque, 201-203 (1991), 299-320.MR 93i:57035
[Tits] J. Tits, Buildings of Spherical Type and Finite BN-Pairs, Lecture Notes in Math., No. 386, Springer, NY, 1974.MR 57:9866
[Wein] S. Weinberger, The Topological Classification of Stratified Spaces, Univ. of Chicago Press, Chicago, 1994.MR 96b:57024