The relative Nielsen number and boundary-preserving surface maps

Michael R. Kelly

Vol. 161 (1993), No. 1, 139–153

DOI: 10.2140/pjm.1993.161.139

Abstract

Let X be a compact 2-manifold with nonempty boundary ∂X. Given a boundary-preserving map f : (X,∂X) → (X,∂X) the relative Nielsen number N_∂(f) gives a lower bound for the number of fixed points of f. Let MF_∂[f] denote the minimum number of fixed points of all boundary-preserving maps homotopic to f as maps of pairs. This paper continues the study of the difference MF_∂[f] − N_∂(f) for surface maps begun by Brown and Sanderson [BS]. Their results are extended by (i) adding to their list of surfaces for which this difference can be arbitrarily large, and (ii) producing an example of a boundary-preserving map of the pants surface for which the difference is equal to one. This answers a conjecture raised by the authors.

Mathematical Subject Classification 2000

Primary: 55M20

Secondary: 57N05

Milestones

Received: 3 November 1991

Revised: 18 July 1992

Published: 1 November 1993