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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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