Equivariant Nielsen invariants for discrete groups

Julia Weber

Vol. 231 (2007), No. 1, 239–256

DOI: 10.2140/pjm.2007.231.239

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 2000

Primary: 55M20, 57R91

Secondary: 54H25, 57S99

Milestones

Received: 16 November 2005

Revised: 21 June 2006

Accepted: 17 August 2006

Published: 1 May 2007