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The boundary-Wecken classification of surfaces. (English) Zbl 1053.55002
Let \(X\) be a compact \(n\)-manifold and \(f: X \to X\) be a map. The Nielsen number, \(N(f)\), of \(f\) is the number of essential fixed point classes; and \(MF[f]\) is the minimum number of fixed points for all maps homotopic to \(f\). Wecken proved that if \(n \geq 3\) then \(N(f)=MF[f]\); thus \(X\) is said to have the Wecken property if \(N(f)=MF[f]\) for every map \(f: X \to X\). On the other hand, a manifold \(X\) is said to be totally non-Wecken if, given an integer \(k \geq 1\), there is a map \(f_k: X \to X\) such that \(MF[f_k] - N(f_k) \geq k\). The Wecken classification of surfaces says that a compact surface \(X\) whose Euler characteristic is non-negative has the Wecken property; and that all other surfaces are totally non-Wecken.
Let \(f: (X, \partial X) \to (X, \partial X)\) be a boundary-preserving self-map of a compact \(n\)-manifold with boundary \(\partial X\), and let \(\overline f : \partial X \to \partial X\) be the restriction of \(f\) to the boundary. The minimum number of fixed points for all maps of pairs homotopic to \(f\) (through boundary-preserving maps) is denoted by \(M_\partial F[f]\); the relative Nielsen number, \(N_\partial (f)\), is the sum of the number of essential fixed point classes of \(\overline f\) and the number of essential fixed point classes of \(f\) that do not contain essential fixed point classes of \(\overline f\). The manifold \(X\) is said to have the boundary-Wecken property if \(N_\partial (f)=MF_\partial [f]\) for all maps of pairs \(f: (X, \partial X) \to (X, \partial X)\); \(X\) is said to be totally non-boundary-Wecken if, given \(k \geq 1\), there is a map \(f_k: (X, \partial X) \to (X, \partial X)\) such that \(MF_\partial [f] - N_\partial (f) \geq k\). \(X\) is said to have the almost boundary-Wecken property if \(X\) does not have the boundary-Wecken property, but there is an integer \(B \geq 1\) such that \(MF_\partial [f] - N_\partial (f) \leq B\) for all maps \(f: (X, \partial X) \to (X, \partial X)\). The following results are proved.
Theorem 2. If \(X\) is the Möbius band with one open disc removed then for any map \(f: (X, \partial X) \to (X, \partial X)\), \(MF_\partial [f] - N_\partial (f) \leq 1\).
Theorem 3 (Boundary-Wecken classification of surfaces). Let \(X\) be a compact surface with a non-empty boundary. (a) If \(X\) is a disc, an annulus or a Möbius band, then \(X\) has the boundary-Wecken property. (b) If \(X\) is a disc with two discs removed or the Möbius band with one disc removed, then \(X\) has the almost boundary-Wecken property, with \(B=1\). (c) All other surfaces with boundary are totally non-boundary-Wecken.

MSC:
55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
57N05 Topology of the Euclidean \(2\)-space, \(2\)-manifolds (MSC2010)
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