Elsevier

Topology and its Applications

Volume 53, Issue 2, 26 November 1993, Pages 221-228
Topology and its Applications

Commutativity and Wecken properties for fixed points on surfaces and 3-manifolds

https://doi.org/10.1016/0166-8641(93)90138-4Get rights and content
Under an Elsevier user license
open archive

Abstract

For a pair of maps ϕ : MP and ψ : PM between compact surfaces, the minimum number of fixed points in the homotopy class of ϕ ∘ ψ may differ from that of ψ ∘ ϕ. We give a sufficient condition for them to be the same, improving a recent result of M.R. Kelly. It is then applied to show that for every surface of negative Euler characteristic, the difference between the minimum number of fixed points and the Nielsen number can be arbitrarily large. The corresponding question for boundary-preserving self-maps of orientable 3-manifolds is also discussed.

Keywords

Surface maps
Fixed points
Minimum number of fixed points
Commutativity property
Wecken property
Boundary-Wecken property
Nielsen fixed point theory

MSC

55M20
57N05, 57N10

Cited by (0)

Partially supported by NSERC Grant A7579 and an NSFC grant.