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. M. Singer, The index of elliptic operators. I, Ann. of Math. (2) 87 (1968), 484–530. MR 236950, https://doi.org/10.2307/1970715
[BaJ] Saad Baaj and Pierre Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les 𝐶*-modules hilbertiens, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983), no. 21, 875–878 (French, with English summary). MR 715325
[BCH] Paul Baum, Alain Connes, and Nigel Higson, Classifying space for proper actions and 𝐾-theory of group 𝐶*-algebras, 𝐶*-algebras: 1943–1993 (San Antonio, TX, 1993) Contemp. Math., vol. 167, Amer. Math. Soc., Providence, RI, 1994, pp. 240–291. MR 1292018, https://doi.org/10.1090/conm/167/1292018
[Bla] Bruce Blackadar, 𝐾-theory for operator algebras, Mathematical Sciences Research Institute Publications, vol. 5, Springer-Verlag, New York, 1986. MR 859867
[BHM] M. Bökstedt, W. C. Hsiang, and I. Madsen, The cyclotomic trace and algebraic 𝐾-theory of spaces, Invent. Math. 111 (1993), no. 3, 465–539. MR 1202133, https://doi.org/10.1007/BF01231296
[Bre] Glen E. Bredon, Introduction to compact transformation groups, Academic Press, New York-London, 1972. Pure and Applied Mathematics, Vol. 46. MR 0413144
[Bro] Kenneth S. Brown, Buildings, Springer-Verlag, New York, 1989. MR 969123
[BuS] Keith Burns and Ralf Spatzier, Manifolds of nonpositive curvature and their buildings, Inst. Hautes Études Sci. Publ. Math. 65 (1987), 35–59. MR 908215
[Cap] Allen L. Shields, A note on invariant subspaces, Michigan Math. J. 17 (1970), 231–233. MR 440389
[CaP] Gunnar Carlsson and Erik Kjær Pedersen, Controlled algebra and the Novikov conjectures for 𝐾- and 𝐿-theory, Topology 34 (1995), no. 3, 731–758. MR 1341817, https://doi.org/10.1016/0040-9383(94)00033-H
[CoJ] Ralph L. Cohen and John D. S. Jones, Algebraic 𝐾-theory of spaces and the Novikov conjecture, Topology 29 (1990), no. 3, 317–344. MR 1066175, https://doi.org/10.1016/0040-9383(90)90002-2
[CGM] A. Connes, M. Gromov, and H. Moscovici, Group cohomology with Lipschitz control and higher signatures, Geom. Funct. Anal. 3 (1993), no. 1, 1–78. MR 1204787, https://doi.org/10.1007/BF01895513
[CoM] Alain Connes and Henri Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990), no. 3, 345–388. MR 1066176, https://doi.org/10.1016/0040-9383(90)90003-3
[CMW] Raúl E. Curto, Paul S. Muhly, and Dana P. Williams, Cross products of strongly Morita equivalent 𝐶*-algebras, Proc. Amer. Math. Soc. 90 (1984), no. 4, 528–530. MR 733400, https://doi.org/10.1090/S0002-9939-1984-0733400-1
[FaJ] F. Thomas Farrell and Lowell E. Jones, Rigidity in geometry and topology, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 653–663. MR 1159252
[FeW] Steven C. Ferry and Shmuel Weinberger, Curvature, tangentiality, and controlled topology, Invent. Math. 105 (1991), no. 2, 401–414. MR 1115548, https://doi.org/10.1007/BF01232272
[Gong1] Dong Geng Gong, Bivariant twisted cyclic theory and spectral sequences of crossed products, J. Pure Appl. Algebra 79 (1992), no. 3, 225–254. MR 1167574, https://doi.org/10.1016/0022-4049(92)90052-H
[Gong2] Dong Geng Gong, Excision of equivariant cyclic cohomology of topological algebras, Michigan Math. J. 39 (1992), no. 3, 455–473. MR 1182501, https://doi.org/10.1307/mmj/1029004600
[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] Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and 𝑝-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246. MR 1215595
[Hil] Michel Hilsum, Fonctorialité en 𝐾-théorie bivariante pour les variétés lipschitziennes, 𝐾-Theory 3 (1989), no. 5, 401–440 (French, with English summary). MR 1050489, https://doi.org/10.1007/BF00534136
[HiS] Michel Hilsum and Georges Skandalis, Invariance par homotopie de la signature à coefficients dans un fibré presque plat, J. Reine Angew. Math. 423 (1992), 73–99 (French). MR 1142484, https://doi.org/10.1515/crll.1992.423.73
[Hus] Dale Husemoller, Fibre bundles, McGraw-Hill Book Co., New York-London-Sydney, 1966. MR 0229247
[KaM] Jerome Kaminker and John G. Miller, Homotopy invariance of the analytic index of signature operators over 𝐶*-algebras, J. Operator Theory 14 (1985), no. 1, 113–127. MR 789380
[Kas] G. G. Kasparov, Equivariant 𝐾𝐾-theory and the Novikov conjecture, Invent. Math. 91 (1988), no. 1, 147–201. MR 918241, https://doi.org/10.1007/BF01404917
[KaS] G. G. Kasparov and G. Skandalis, Groups acting on buildings, operator 𝐾-theory, and Novikov’s conjecture, 𝐾-Theory 4 (1991), no. 4, 303–337. MR 1115824, https://doi.org/10.1007/BF00533989
[KoS] Nicholas J. Korevaar and Richard M. Schoen, Sobolev spaces and harmonic maps for metric space targets, Comm. Anal. Geom. 1 (1993), no. 3-4, 561–659. MR 1266480, https://doi.org/10.4310/CAG.1993.v1.n4.a4
[Lus] Gheorghe Lusztig, Novikov’s higher signature and families of elliptic operators, J. Differential Geometry 7 (1972), 229–256. MR 322889
[Mis1] A. S. Miščenko, Homotopy invariants of multiply connected manifolds. I. Rational invariants, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 501–514 (Russian). MR 0268898
[Mis2] A. S. Miščenko, Infinite-dimensional representations of discrete groups, and higher signatures, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974), 81–106 (Russian). MR 0362407
[Roe] J. Roe, Hyperbolic metric spaces and the exotic cohomology Novikov conjecture, $K$ -Theory, 4 (1991), 501-512; 5 (1991), 189. MR 1123175
[Ros] Jonathan Rosenberg, 𝐶*-algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. 58 (1983), 197–212 (1984). MR 720934
[RoW1] Jonathan Rosenberg and Shmuel Weinberger, Higher 𝐺-indices and applications, Ann. Sci. École Norm. Sup. (4) 21 (1988), no. 4, 479–495. MR 982331
[RoW2] Jonathan Rosenberg and Shmuel Weinberger, An equivariant Novikov conjecture, 𝐾-Theory 4 (1990), no. 1, 29–53. With an appendix by J. P. May. MR 1076524, https://doi.org/10.1007/BF00534192
[Ska] Georges Skandalis, Approche de la conjecture de Novikov par la cohomologie cyclique (d’après A. Connes, M. Gromov et H. Moscovici), Astérisque 201-203 (1991), Exp. No. 739, 299–320 (1992) (French). Séminaire Bourbaki, Vol. 1990/91. MR 1157846
[Tits] Jacques Tits, Buildings of spherical type and finite BN-pairs, Lecture Notes in Mathematics, Vol. 386, Springer-Verlag, Berlin-New York, 1974. MR 0470099
[Wein] Shmuel Weinberger, The topological classification of stratified spaces, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1994. MR 1308714