Abstract

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 λ_G(f) is zero. Finally, we prove a converse of the equivariant Lefschetz fixed point theorem.

Keywords

Nielsen number, discrete groups, equivariant, Lefschetz fixed point theorem

Mathematical Subject Classification

Primary: 55M20, 57R91

Secondary: 54H25, 57S99

Milestones

Received: 16 November 2005
Revised: 21 June 2006
Accepted: 17 August 2006

Authors