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2004 The boundary-Wecken classification of surfaces
Robert F Brown, Michael R Kelly
Algebr. Geom. Topol. 4(1): 49-71 (2004). DOI: 10.2140/agt.2004.4.49

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: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.

Citation

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Robert F Brown. Michael R Kelly. "The boundary-Wecken classification of surfaces." Algebr. Geom. Topol. 4 (1) 49 - 71, 2004. https://doi.org/10.2140/agt.2004.4.49

Information

Received: 21 November 2002; Revised: 15 October 2003; Accepted: 26 November 2003; Published: 2004
First available in Project Euclid: 21 December 2017

zbMATH: 1053.55002
MathSciNet: MR2031912
Digital Object Identifier: 10.2140/agt.2004.4.49

Subjects:
Primary: ‎55M20
Secondary: 54H25, 57N05

Rights: Copyright © 2004 Mathematical Sciences Publishers

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