Homeomorphisms of 3-manifolds and the realization of Nielsen Number
Homeomorphisms of 3-manifolds and the realization of Nielsen Number
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DOI：
10.4310/cag.2001.v9.n4.a6
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发表时间：
1996-10
期刊：
影响因子：
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通讯作者：
Boju Jiang;Shicheng Wang;Yingga Wu
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文献类型：
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作者：
Boju Jiang;Shicheng Wang;Yingga Wu
同胚的Nielsen猜想断言，闭流形的任何同胚$f$与实现$f$的Nielsen数的映射同位，这是与$f$同伦的所有映射中不动点个数的下界。本文的主要定理证明了几何或Haken三维流形上的所有保向映射的这一猜想。还将证明，在许多流形上，所有映射都是不动点自由映射的同位映射。本文的证明是基于对2维流形和3维流形上同胚的理解。将瑟斯顿对曲面同胚的分类推广到二维orbilold，用来研究Seifert纤维空间的保纤维映射。除了透镜空间上的四个流形和方向反转映射或$S^3$以外，大多数Seifert纤维空间上的映射确实是保纤维的同位素映射。它还将准确地确定哪些流形具有独特的Seifert纤维，直至同位素。这些信息将被用来在JSJ分解的每一块上以及在分解环面的邻域上将映射变形为特定的标准映射，这将使将每个不动点类收缩到单个点成为可能，并删除不必要的不动点类。
The Nielsen Conjecture for Homeomorphisms asserts that any homeomorphism $f$ of a closed manifold is isotopic to a map realizing the Nielsen number of $f$, which is a lower bound for the number of fixed points among all maps homotopic to $f$. The main theorem of this paper proves this conjecture for all orientation preserving maps on geometric or Haken 3-manifolds. It will also be shown that on many manifolds all maps are isotopic to fixed point free maps. The proof is based on the understanding of homeomorphisms on 2-orbifolds and 3-manifolds. Thurston's classification of surface homeomorphisms will be generalized to 2-dimensional orbifolds, which is used to study fiber preserving maps of Seifert fiber spaces. Maps on most Seifert fiber spaces are indeed isotopic to fiber preserving maps, with the exception of four manifolds and orientation reversing maps on lens spaces or $S^3$. It will also be determined exactly which manifolds have a unique Seifert fibration up to isotopy. These informations will be used to deform a map to certain standard map on each piece of the JSJ decomposition, as well as on the neighborhood of the decomposition tori, which will make it possible to shrink each fixed point class to a single point, and remove inessential fixed point classes.