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On equivariant Euler characteristics. (English) Zbl 0708.19004
Let G be a finite group of symmetries of a compact manifold M. In string theory one uses an Euler characteristic based on simultaneous fixed-point sets of commuting pairs of elements of G. The authors show that this is the same as the Euler characteristic of the equivariant K-theory \(K^*_ G(M)\). The method depends on expressing \(K^*_ G(M)\) tensored with the complex numbers in terms of the non-equivariant K- theory of the fixed-point sets of single elements of G.
Reviewer: R.J.Steiner

19L47 Equivariant \(K\)-theory
55N91 Equivariant homology and cohomology in algebraic topology
81T30 String and superstring theories; other extended objects (e.g., branes) in quantum field theory
Full Text: DOI
[1] Atiyah, M.F., K -theory, (1967), Benjamin New York · Zbl 0735.57001
[2] Atiyah, M.F.; Bott, R., The moment map and equivariant cohomology, Topology, 23, 1-28, (1984) · Zbl 0521.58025
[3] Atiyah, M.F.; Segal, G.B., The index of elliptic operators II, Ann. of math., 87, 531-545, (1968) · Zbl 0164.24201
[4] Baum, P.; Brylinski, J.L.; MacPherson, R., Cohomologieéquivariante délocalisée, C.R. acad. sci. Paris, 300, 605-608, (1985) · Zbl 0589.55003
[5] Baum, P.; Connes, A., Chern character for discrete groups, () · Zbl 0656.55005
[6] Brieskorn, E., Singular elements of semi-simple groups, Proc. international congress of mathematical, Vol. 2, 279-284, (1970), Nice
[7] Dixon, L.; Harvey, J.A.; Vafa, C.; Witten, E., Strings on orbifolds, Nucl. phys. B, 261, 678, (1985)
[8] M.J. {\scHopkins}, N.J. {\scKuhn} and D.C. {\scRavenel}: Generalized group characters and complex oriented cohomology theories, (to appear).
[9] Lusztig, G., Leading coefficients of character values of Hecke algebras, A.M.S. proc. symp. in pure maths., 47, 235-262, (1987)
[10] McKay, J., Graphs, singularities and finite groups, American math. soc. proc. symp. pure math., 37, 235-262, (1980)
[11] Segal, G.B., Equivariant K -theory, Publ. math. inst. hautes etudes sci. Paris, (1968) · Zbl 0199.26202
[12] Witten, E., Supersymmetry and Morse theory, J. diff. geom., 17, 661-692, (1982) · Zbl 0499.53056
[13] Dijkgraaf, R.; Vafa, C.; Verlinde, E.; Verlinde, H., The operator algebra of orbifold models, Commun. math. phys., 123, 485-526, (1989) · Zbl 0674.46051
[14] R. {\scDijkgraaf} and E. {\scWitten}: Topological gauge theories and group cohomology, (to appear).
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