#### 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

and

Department of Mathematics and Computer Science, Loyola University

6363 St. Charles Avenue, New Orleans, LA 70118, USA

Email: rfb@math.ucla.edu, kelly@loyno.edu

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