Abstract
Let X be a locally compact space with a continuous proper action of a locally compact group G. Assuming that X satisfies a certain kind of duality in equivariant bivariant Kasparov theory, we can enrich the classical construction of Lefschetz numbers for self-maps to an equivariant K-homology class. We compute the Lefschetz invariants for self-maps of finite-dimensional simplicial complexes and smooth manifolds. The resulting invariants are independent of the extra structure used to compute them. Since smooth manifolds can be triangulated, we get two formulas for the same Lefschetz invariant in this case. The resulting identity is closely related to the equivariant Lefschetz Fixed Point Theorem of Lück and Rosenberg.
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References
Baum P., Block J.: Equivariant bicycles on singular spaces. C. R. Acad. Sci. Paris Sér. I Math. 311(2), 115–120 (1990) (English, with French summary). MR 1065441
Baum, P., Douglas, R.G.: K-homology and index theory. Operator algebras and applications, Part I (Kingston, Ont., 1980). In: Proc. Sympos. Pure Math., vol. 38, pp. 117–173. American Mathematical Society, Providence (1982). MR 679698
Connes A., Skandalis G.: The longitudinal index theorem for foliations. Publ. Res. Inst. Math. Sci. 20(6), 1139–1183 (1984) MR 775126
Echterhoff, S., Emerson, H., Kim, H.-J.: Fixed point formulas for proper actions (2007, to appear). arXiv: 0708.4279
Emerson H., Meyer R.: Euler characteristics and Gysin sequences for group actions on boundaries. Math. Ann. 334(4), 853–904 (2006) MR 2209260
Emerson, H., Meyer, R.: Dualities in equivariant Kasparov theory (2007). eprint. arXiv: 0711.0025
Illman S.: Existence and uniqueness of equivariant triangulations of smooth proper G -manifolds with some applications to equivariant Whitehead torsion. J. Reine Angew. Math. 524, 129–183 (2000) MR 1770606
Kasparov G.G.: Equivariant KK-theory and the Novikov conjecture. Invent. Math. 91(1), 147–201 (1988) MR 918241
Kasparov G.G., Skandalis G.: Groups acting on buildings, operator K -theory, and Novikov’s conjecture, K -Theory 4(4), 303–337 (1991) MR 1115824
Lück, W., Rosenberg, J.: The equivariant Lefschetz fixed point theorem for proper cocompact G -manifolds. High-dimensional manifold topology, pp. 322–361. World Scientific Publication, River Edge (2003). MR 2048727
Le Gall, P.-Y.: Théorie de Kasparov équivariante et groupoï des. I. K -Theory, 16(4), 361–390 (1999) (French, with English and French summaries). MR 1686846
Weber J.: The universal functorial equivariant Lefschetz invariant. K -Theory 36(1–2), 169–207 (2006) MR 2274162
Acknowledgments
This research was supported by the National Science and Engineering Research Council of Canada Discovery Grant program, the Marie Curie Action Noncommutative Geometry and Quantum Groups (Contract MKTD-CT-2004-509794), and by the German Research Foundation [Deutsche Forschungsgemeinschaft (DFG)] through the Institutional Strategy of the University of Göttingen.
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Open Access This is an open access article distributed under the terms of the Creative Commons Attribution Noncommercial License (https://creativecommons.org/licenses/by-nc/2.0), which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.
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Emerson, H., Meyer, R. Equivariant Lefschetz maps for simplicial complexes and smooth manifolds. Math. Ann. 345, 599–630 (2009). https://doi.org/10.1007/s00208-009-0367-z
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DOI: https://doi.org/10.1007/s00208-009-0367-z