Algebraic & Geometric Topology

A finite-dimensional approach to the strong Novikov conjecture

Daniel Ramras, Rufus Willett, and Guoliang Yu

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The aim of this paper is to describe an approach to the (strong) Novikov conjecture based on continuous families of finite-dimensional representations: this is partly inspired by ideas of Lusztig related to the Atiyah–Singer families index theorem, and partly by Carlsson’s deformation K–theory. Using this approach, we give new proofs of the strong Novikov conjecture in several interesting cases, including crystallographic groups and surface groups. The method presented here is relatively accessible compared with other proofs of the Novikov conjecture, and also yields some information about the K–theory and cohomology of representation spaces.

Article information

Algebr. Geom. Topol., Volume 13, Number 4 (2013), 2283-2316.

Received: 18 October 2012
Revised: 29 January 2013
Accepted: 19 March 2013
First available in Project Euclid: 19 December 2017

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Zentralblatt MATH identifier

Primary: 19K56: Index theory [See also 58J20, 58J22] 19L99: None of the above, but in this section 55N15: $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19- XX} 57R20: Characteristic classes and numbers
Secondary: 20C99: None of the above, but in this section 46L80: $K$-theory and operator algebras (including cyclic theory) [See also 18F25, 19Kxx, 46M20, 55Rxx, 58J22] 46L85: Noncommutative topology [See also 58B32, 58B34, 58J22]

Baum–Connes conjecture $K$–homology deformation $K$–theory index theory


Ramras, Daniel; Willett, Rufus; Yu, Guoliang. A finite-dimensional approach to the strong Novikov conjecture. Algebr. Geom. Topol. 13 (2013), no. 4, 2283--2316. doi:10.2140/agt.2013.13.2283.

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  • A Adem, J M Gómez, On the structure of spaces of commuting elements in compact Lie groups, from: “Configuration Spaces: Geometry, Combinatorics and Topology”, (A Björner, F Cohen, C De Concini, C Procesi, M Salvetti, editors), Publ. Scuola Normale Superiore 14, Edizioni Della Normaleinger, Pisa, Italy (2012) 1–26
  • M F Atiyah, Characters and cohomology of finite groups, Inst. Hautes Études Sci. Publ. Math. (1961) 23–64
  • M F Atiyah, $K$–theory, W A Benjamin, New York (1967)
  • M F Atiyah, I M Singer, The index of elliptic operators. IV, Ann. of Math. 93 (1971) 119–138
  • T J Baird, Cohomology of the space of commuting $n$–tuples in a compact Lie group, Algebr. Geom. Topol. 7 (2007) 737–754
  • T Baird, D A Ramras, Smooth approximation in algebraic sets and the topological Atiyah–Segal map
  • P Baum, A Connes, N Higson, Classifying space for proper actions and $K$–theory of group $C^*$–algebras, from: “$C^*$-algebras: 1943–1993”, (R S Doran, editor), Contemp. Math. 167, Amer. Math. Soc. (1994) 240–291
  • B Bekka, P de la Harpe, A Valette, Kazhdan's property (T), New Mathematical Monographs 11, Cambridge Univ. Press (2008)
  • B Blackadar, $K$–theory for operator algebras, 2nd edition, MSRI Publications 5, Cambridge Univ. Press (1998)
  • K S Brown, Cohomology of groups, Graduate Texts in Mathematics 87, Springer, New York (1982)
  • M Burger, S Mozes, Lattices in product of trees, Inst. Hautes Études Sci. Publ. Math. (2000) 151–194
  • A Connes, M Gromov, H Moscovici, Group cohomology with Lipschitz control and higher signatures, Geom. Funct. Anal. 3 (1993) 1–78
  • A Connes, H Moscovici, Cyclic cohomology, the Novikov conjecture and hyperbolic groups, Topology 29 (1990) 345–388
  • M Dadarlat, Group quasi-representations and index theory, J. Topol. Anal. 4 (2012) 297–319
  • M Dadarlat, Group quasi-representations and almost flat bundles, to appear in J. Noncommut. Geom. (2013)
  • N Higson, $C^*$–algebra extension theory and duality, J. Funct. Anal. 129 (1995) 349–363
  • N Higson, J Roe, Analytic $K$–homology, Oxford University Press (2000)
  • J Kaminker, J G Miller, Homotopy invariance of the analytic index of signature operators over $C^*$–algebras, J. Operator Theory 14 (1985) 113–127
  • G G Kasparov, Equivariant $KK$–theory and the Novikov conjecture, Invent. Math. 91 (1988) 147–201
  • G Lusztig, Novikov's higher signature and families of elliptic operators, J. Differential Geometry 7 (1972) 229–256
  • J Milnor, Construction of universal bundles. II, Ann. of Math. 63 (1956) 430–436
  • J W Milnor, J D Stasheff, Characteristic classes, Annals of Mathematics Studies 76, Princeton Univ. Press (1974)
  • A S Miščenko, Infinite-dimensional representations of discrete groups, and higher signatures, Izv. Akad. Nauk SSSR Ser. Mat. 38 (1974) 81–106 In Russian; translated in Math. USSR-Izv. 8 (1974) 85–111
  • S Mukai, Duality between $D(X)$ and $D(\hat X)$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981) 153–175
  • D A Ramras, Excision for deformation $K$–theory of free products, Algebr. Geom. Topol. 7 (2007) 2239–2270
  • D A Ramras, Yang–Mills theory over surfaces and the Atiyah–Segal theorem, Algebr. Geom. Topol. 8 (2008) 2209–2251
  • D A Ramras, On the Yang–Mills stratification for surfaces, Proc. Amer. Math. Soc. 139 (2011) 1851–1863
  • D A Ramras, Periodicity in the stable representation theory of crystallographic groups, to appear in Forum Math. (2011)
  • D A Ramras, The stable moduli space of flat connections over a surface, Trans. Amer. Math. Soc. 363 (2011) 1061–1100
  • J Roe, Index theory, coarse geometry, and topology of manifolds, CBMS Regional Conference Series in Mathematics 90, Amer. Math. Soc. (1996)
  • J Rosenberg, $C^*$–algebras, positive scalar curvature, and the Novikov conjecture, Inst. Hautes Études Sci. Publ. Math. (1983) 197–212
  • F W Roush, Transfer in generalized cohomology theories, Akadémiai Kiadó, Budapest (1999)
  • G Segal, Classifying spaces and spectral sequences, Inst. Hautes Études Sci. Publ. Math. (1968) 105–112
  • D T Wise, Non-positively curved squared complexes: Aperiodic tilings and non-residually finite groups, PhD thesis, Princeton (1996) Available at \setbox0\makeatletter\@url {\unhbox0
  • G Yu, Localization algebras and the coarse Baum–Connes conjecture, $K$–Theory 11 (1997) 307–318
  • G Yu, The Novikov conjecture for groups with finite asymptotic dimension, Ann. of Math. 147 (1998) 325–355