The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds

Details The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds The equivariant Lefschetz fixed point theorem for proper cocompact G-manifolds

2002-08-22

arxiv.org/abs/math/0208162v1

High-dimensional manifold topology, 322-361, World Sci. Publ., River Edge, NJ, 2003

Mathematics

Algebraic Topology

(36 pages)

Scientific paper

Suppose one is given a discrete group G, a cocompact proper G-manifold M, and a G-self-map f of M. Then we introduce the equivariant Lefschetz class of f, which is globally defined in terms of cellular chain complexes, and the local equivariant Lefschetz class of f, which is locally defined in terms of fixed point data. We prove the equivariant Lefschetz fixed point theorem, which says that these two classes agree. As a special case, we prove an equivariant Poincare-Hopf Theorem, computing the universal equivariant Euler characteristic in terms of the zeros of an equivariant vector field, and also obtain an orbifold Lefschetz fixed point theorem. Finally, we prove a realization theorem for universal equivariant Euler characteristics.

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