Algebraic & Geometric Topology

The boundary-Wecken classification of surfaces

Robert F Brown and Michael R Kelly

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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:XX 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 k1, there is a map fk:(X,X)(X,X) such that MF[fk]N(fk)k.

Article information

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

Received: 21 November 2002
Revised: 15 October 2003
Accepted: 26 November 2003
First available in Project Euclid: 21 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55M20: Fixed points and coincidences [See also 54H25]
Secondary: 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20] 57N05: Topology of $E^2$ , 2-manifolds

boundary-Wecken relative Nielsen number punctured Möbius band boundary-preserving map


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.

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