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

Abstract

For a pair of maps ϕ : M → P and ψ : P → M 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.