## Algebraic & Geometric Topology

### The boundary-Wecken classification of surfaces

#### Abstract

Let $X$ be a compact $2$-manifold with nonempty boundary $∂X$ and let $f:(X,∂X)→(X,∂X)$ be a boundary-preserving map. Denote by $MF∂[f]$ the minimum number of fixed point among all boundary-preserving maps that are homotopic through boundary-preserving maps to $f$. The relative Nielsen number $N∂(f)$ is the sum of the number of essential fixed point classes of the restriction $f̄:∂X→∂X$ and the number of essential fixed point classes of $f$ that do not contain essential fixed point classes of $f̄$. We prove that if $X$ is the Möbius band with one (open) disc removed, then $MF∂[f]−N∂(f)≤1$ for all maps $f:(X,∂X)→(X,∂X)$. This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If $X$ is the disc, annulus or Möbius band, then $X$ is boundary-Wecken, that is, $MF∂[f]=N∂(f)$ for all boundary-preserving maps. If $X$ is the disc with two discs removed or the Möbius band with one disc removed, then $X$ is not boundary-Wecken, but $MF∂[f]−N∂(f)≤1$. All other surfaces are totally non-boundary-Wecken, that is, given an integer $k≥1$, there is a map $fk:(X,∂X)→(X,∂X)$ such that $MF∂[fk]−N∂(fk)≥k$.

#### Article information

Source
Algebr. Geom. Topol., Volume 4, Number 1 (2004), 49-71.

Dates
Revised: 15 October 2003
Accepted: 26 November 2003
First available in Project Euclid: 21 December 2017

https://projecteuclid.org/euclid.agt/1513882466

Digital Object Identifier
doi:10.2140/agt.2004.4.49

Mathematical Reviews number (MathSciNet)
MR2031912

Zentralblatt MATH identifier
1053.55002

#### Citation

Brown, Robert F; Kelly, Michael R. The boundary-Wecken classification of surfaces. Algebr. Geom. Topol. 4 (2004), no. 1, 49--71. doi:10.2140/agt.2004.4.49. https://projecteuclid.org/euclid.agt/1513882466

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