arXiv subjects
Publications and source records attributed to Julia Weber.
We introduce the universal functorial equivariant Lefschetz invariant for endomorphisms of finite proper G-CW-complexes, where G is a discrete group. We use K_0 of the category of "phi-endomorphisms of finitely generated free RPi(G,X)-modules". We derive results about fixed points of equivariant endomorphisms of cocompact proper smooth G-manifolds.
We geometrically construct a homology theory that generalizes the Euler characteristic mod 2 to objects in the unoriented cobordism ring N_*(X) of a topological space X. This homology theory Eh_* has coefficients Z/2 in every nonnegative dimension. There exists a natural transformation N_*(X)->Eh_*(X) that for X=pt assigns to each smooth manifold its Euler characteristic mod 2. The homology theory is constructed using cobordism of stratifolds. For discrete groups G, we also define an equivariant version of the homology theory Eh_*, generalizing the equivariant Euler characteristic.
We define equivariant homology theories using bordism of stratifolds with a G-action, where G is a discrete group. Stratifolds are a generalization of smooth manifolds which were introduced by Kreck. He defines homology theories using bordism of suitable stratifolds. We develop the equivariant generalization of these ideas.
For discrete groups G, we introduce equivariant Nielsen invariants. They are equivariant analogs of the Nielsen number and give lower bounds for the number of fixed point orbits in the G-homotopy class of an equivariant endomorphism f:X->X. Under mild hypotheses, these lower bounds are sharp. We use the equivariant Nielsen invariants to show that a G-equivariant endomorphism f is G-homotopic to a fixed point free G-map if the generalized equivariant Lefschetz invariant of f is zero. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem.