Geometry & Topology

Equivariant Euler characteristics and $K$–homology Euler classes for proper cocompact $G$–manifolds

Wolfgang Lueck and Jonathan Rosenberg

Full-text: Open access

Abstract

Let G be a countable discrete group and let M be a smooth proper cocompact G-manifold without boundary. The Euler operator defines via Kasparov theory an element, called the equivariant Euler class, in the equivariant KO–homology of M. The universal equivariant Euler characteristic of M, which lives in a group UG(M), counts the equivariant cells of M, taking the component structure of the various fixed point sets into account. We construct a natural homomorphism from UG(M) to the equivariant KO-homology of M. The main result of this paper says that this map sends the universal equivariant Euler characteristic to the equivariant Euler class. In particular this shows that there are no “higher” equivariant Euler characteristics. We show that, rationally, the equivariant Euler class carries the same information as the collection of the orbifold Euler characteristics of the components of the L–fixed point sets ML, where L runs through the finite cyclic subgroups of G. However, we give an example of an action of the symmetric group S3 on the 3–sphere for which the equivariant Euler class has order 2, so there is also some torsion information.

Article information

Source
Geom. Topol., Volume 7, Number 2 (2003), 569-613.

Dates
Received: 2 August 2002
Accepted: 9 October 2003
First available in Project Euclid: 21 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.gt/1513883316

Digital Object Identifier
doi:10.2140/gt.2003.7.569

Mathematical Reviews number (MathSciNet)
MR2026542

Zentralblatt MATH identifier
1034.19003

Subjects
Primary: 19K33: EXT and $K$-homology [See also 55N22]
Secondary: 19K35: Kasparov theory ($KK$-theory) [See also 58J22] 19K56: Index theory [See also 58J20, 58J22] 19L47: Equivariant $K$-theory [See also 55N91, 55P91, 55Q91, 55R91, 55S91] 58J22: Exotic index theories [See also 19K56, 46L05, 46L10, 46L80, 46M20] 57R91: Equivariant algebraic topology of manifolds 57S30: Discontinuous groups of transformations 55P91: Equivariant homotopy theory [See also 19L47]

Keywords
equivariant $K$–homology de Rham operator signature operator Kasparov theory equivariant Euler characteristic fixed sets cyclic subgroups Burnside ring Euler operator equivariant Euler class universal equivariant Euler characteristic

Citation

Lueck, Wolfgang; Rosenberg, Jonathan. Equivariant Euler characteristics and $K$–homology Euler classes for proper cocompact $G$–manifolds. Geom. Topol. 7 (2003), no. 2, 569--613. doi:10.2140/gt.2003.7.569. https://projecteuclid.org/euclid.gt/1513883316


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References

  • H Abels, A universal proper $G$-space, Math. Z. 159 (1978) 143–158
  • M F Atiyah, I M Singer, The index of elliptic operators, III, Ann. of Math. 87 (1968) 546–604
  • S Baaj, P Julg, Théorie bivariante de Kasparov et opérateurs non bornés dans les $C^*$-modules hilbertiens, C. R. Acad. Sci. Paris Sér. I Math. 296 (1983) 875–878
  • B Blackadar, $K$-Theory for Operator Algebras, Math. Sci. Res. Inst. Publ. 5, Springer-Verlag, New York, Berlin (1986)
  • J Davis, W Lück, Spaces over a Category, Assembly Maps in Isomorphism Conjecture in $K$-and $L$-Theory, $K$-Theory 15 (1998) 201–252
  • T tom Dieck, Transformation groups, Studies in Math. 8, de Gruyter (1987)
  • J Dixmier, $C^*$-Algebras, North-Holland Mathematical Library, 15, North-Holland, Amsterdam and New York (1977)
  • M P Gaffney, A special Stokes's theorem for complete Riemannian manifolds, Ann. of Math. 60 (1954) 140–145
  • N Higson, A primer on $KK$-theory, Operator theory: operator algebras and applications, Part 1 (Durham, NH, 1988), W. Arveson and R. Douglas, eds., Proc. Sympos. Pure Math. 51, part 1, Amer. Math. Soc. Providence, RI (1990) 239–283
  • D S Kahn, J Kaminker, C Schochet, Generalized homology theories on compact metric spaces, Michigan Math. J. 24 (1977) 203–224
  • J Kaminker, J G Miller, Homotopy invariance of the analytic index of signature operators over $C\sp *$-algebras, J. Operator Theory 14 (1985) 113–127
  • G Kasparov, The operator $K$-functor and extensions of $C^*$-algebras, Izv. Akad. Nauk SSSR, Ser. Mat. 44 (1980) 571–636; English transl. in Math. USSR–Izv. 16 (1981) 513–572
  • G Kasparov, Equivariant $KK$-theory and the Novikov conjecture, Invent. Math. 91 (1988) 147–201
  • T Kato, Perturbation Theory for Linear Operators, 2nd ed., corrected printing, Grundlehren der Mathematischen Wissenschaften, 132, Springer-Verlag, Berlin (1980); reprinted in Classics in Mathematics, Springer-Verlag, Berlin (1995)
  • H B Lawson, Jr., M-L Michelsohn, Spin Geometry, Princeton Mathematical Ser., 38, Princeton Univ. Press, Princeton, NJ (1989)
  • W Lück, Transformation groups and algebraic $K$-theory, Lecture Notes in Math. 1408, Springer (1989)
  • W Lück, $L^2$-Invariants: Theory and Applications to Geometry and $K$-Theory, Ergebnisse der Mathematik und ihrer Grenzgebiete 44, Springer (2002)
  • W Lück, Chern characters for proper equivariant homology theories and applications to $K$- and $L$-theory, Journal für reine und angewandte Mathematik 543 (2002) 193–234
  • W Lück, The relation between the Baum-Connes Conjecture and the Trace Conjecture, Inventiones Math. 149 (2002) 123–152
  • W Lück, J Rosenberg, The equivariant Lefschetz fixed point theorem for proper cocompact $G$-manifolds, Proc. 2001 Trieste Conf. on High-Dimensional Manifolds, T. Farrell, L. Göttsche, and W. Lück, eds., World Scientific, to appear. Available at http://www.math.umd.edu/\raisebox-.6ex \symbol"7Ejmr/jmr\char'137pub.html
  • D Meintrup, On the Type of the Universal Space for a Family of Subgroups, Ph. D. thesis, Münster (2000)
  • J Rosenberg, Analytic Novikov for topologists, Novikov Conjectures, Index Theorems and Rigidity, vol. 1, S. Ferry, A. Ranicki, and J. Rosenberg, eds., London Math. Soc. Lecture Notes, 226, Cambridge Univ. Press, Cambridge (1995), 338–372
  • J Rosenberg, The $K$-homology class of the Euler characteristic operator is trivial, Proc. Amer. Math. Soc. 127 (1999) 3467–3474
  • J Rosenberg, The $G$-signature theorem revisited, Tel Aviv Topology Conference: Rothenberg Festschrift, international conference on topology, June 1-5, 1998, Tel Aviv, Contemporary Mathematics 231 (1999) 251–264
  • J Rosenberg, The $K$-homology class of the equivariant Euler characteristic operator, unpublished preprint, available at \phantomxxxxxxxxxxxxxxxxx http://www.math.umd.edu/\raisebox-.6ex \symbol"7Ejmr/jmr\char'137pub.html
  • J Rosenberg, S Weinberger, Higher $G$-signatures for Lipschitz manifolds, $K$-Theory 7 (1993) 101–132
  • J Rosenberg, S Weinberger, The signature operator at 2, in preparation
  • J-P Serre, Linear representations of finite groups, Springer-Verlag (1977)
  • S Waner, Y Wu, The local structure of tangent $G$-vector fields, Topology and its Applications 23 (1986) 129–143
  • S Waner, Y Wu, Equivariant $SKK$ and vector field bordism, Topology and its Appl. 28 (1988) 29–44