The boundary-Wecken classification of surfaces

Brown, Robert F¹ ; Kelly, Michael R²

¹Department of Mathematics, University of California, Los Angeles CA 90095-1555, USA
²Department of Mathematics and Computer Science, Loyola University, 6363 St Charles Avenue, New Orleans LA 70118, USA

Algebraic and Geometric Topology, Tome 4 (2004) no. 1, pp. 49-71

Résumé

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.

Détail

DOI : 10.2140/agt.2004.4.49

Keywords: boundary-Wecken, relative Nielsen number, punctured Möbius band, boundary-preserving map

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