曲面的边界-Wecken分类
The boundary-Wecken classification of surfaces
摘要：
设X是具有非空边界dX的紧致2-流形，f：（X，dX）->（X，dX）是保边界映射.用MF_d[f]表示通过到f的保边界映射同伦的所有保边界映射中的最小不动点个数。相对Nielsen数N_d（f）是限制f-杆的本质不动点类的数目：dX --> dX与不包含f-杆的本质不动点类的f的本质不动点类的数目之和。证明了若X是去掉一个（开）圆盘的Moebius带，则对所有映射f：（X，dX）->（X，dX），MF_d[f] - N_d（f）< 2.这个结果是曲面的边界韦肯分类的最后一步，如下所示。若X是圆盘、环带或Moebius带，则X是边界-Wecken的，即对所有保边界映射，MF_d[f] = N_d（f）.若X是去掉两个圆盘的圆盘或去掉一个圆盘的Moebius带，则X不是边界-Wecken，但MF_d[f] - N_d（f）< 2.所有其他曲面都是完全无边界的，即给定一个整数k > 0，存在一个映射$f_k：（X，dX）-->（X，dX）使得MF_d[f_k] - N_d（f_k）>= k。
英文摘要：
Let X be a compact 2-manifold with nonempty boundary dX and let f: (X, dX) --> (X, dX) be a boundary-preserving map. Denote by MF_d[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_d(f) is the sum of the number of essential fixed point classes of the restriction f-bar : dX --> dX and the number of essential fixed point classes of f that do not contain essential fixed point classes of f-bar. We prove that if X is the Moebius band with one (open) disc removed, then MF_d[f] - N_d(f) < 2 for all maps f : (X, dX) --> (X, dX). This result is the final step in the boundary-Wecken classification of surfaces, which is as follows. If X is the disc, annulus or Moebius band, then X is boundary-Wecken, that is, MF_d[f] = N_d(f) for all boundary-preserving maps. If X is the disc with two discs removed or the Moebius band with one disc removed, then X is not boundary-Wecken, but MF_d[f] - N_d(f) < 2. All other surfaces are totally non-boundary-Wecken, that is, given an integer k > 0, there is a map $f_k : (X, dX) --> (X, dX) such that MF_d[f_k] - N_d(f_k) >= k.
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