Algebraic and Geometric Topology 4 (2004), paper no. 4, pages 49-71.

The boundary-Wecken classification of surfaces

Robert F. Brown, Michael R. Kelly

Abstract. Let X be a compact 2-manifold with nonempty boundary dX and let f: (X, dX) --> (X, dX) be a boundary-preserving map. Denote by MF_d[f] the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to f. The relative Nielsen number N_d(f) is the sum of the number of essential fixed point classes of the restriction f-bar : dX --> dX and the number of essential fixed point classes of f that do not contain essential fixed point classes of f-bar. We prove that if X is the Moebius band with one (open) disc removed, then MF_d[f] - N_d(f) < 2 for all maps f : (X, dX) --> (X, dX). This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If X is the disc, annulus or Moebius band, then X is boundary-Wecken, that is, MF_d[f] = N_d(f) for all boundary-preserving maps. If X is the disc with two discs removed or the Moebius band with one disc removed, then X is not boundary-Wecken, but MF_d[f] - N_d(f) < 2. All other surfaces are totally non-boundary-Wecken, that is, given an integer k > 0, there is a map $f_k : (X, dX) --> (X, dX) such that MF_d[f_k] - N_d(f_k) >= k.

Keywords. Boundary-Wecken, relative Nielsen number, punctured Moebius band, boundary-preserving map

AMS subject classification. Primary: 55M20. Secondary: 54H25, 57N05.

DOI: 10.2140/agt.2004.4.49

E-print: arXiv:math.AT/0402334

Submitted: 21 November 2002. (Revised: 15 October 2003.) Accepted: 26 November 2003. Published: 7 February 2004.

Notes on file formats

Robert F. Brown, Michael R. Kelly

Department of Mathematics, University of California
Los Angeles, CA 90095-1555, USA
Department of Mathematics and Computer Science, Loyola University
6363 St. Charles Avenue, New Orleans, LA 70118, USA


AGT home page

EMIS/ELibM Electronic Journals

Outdated Archival Version

These pages are not updated anymore. They reflect the state of 21 Apr 2006. For the current production of this journal, please refer to