A Generalization of Winding Number Functions on Surfaces
A Generalization of Winding Number Functions on Surfaces
复制标题
曲面上绕数函数的推广
DOI：
10.1112/plms/s3-58.2.366
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发表时间：
1989
影响因子：
1.8
通讯作者：
Dennis Johnson
中科院分区：
文献类型：
--
作者：
Stephen P. Humphries;Dennis Johnson
设M是可定向曲面。本文给出了M上缠绕数函数的一个新的组合定义，并刻画了所有具有这类缠绕数函数的三个共同性质的函数。这些函数与M的单位切丛的第一上同调群的元素相同。为了更精确地表述我们的结果，我们首先定义了一些概念，设f =[0，1]。如果切线映射Ta：77-· TM是连续的和单射的，则弧a：77-·M被称为正则的。如果a没有零同伦环，那么我们说a是直接的。设A '是M上的向量场，且只有一个奇点。令Wx表示X\的缠绕数函数，即对于M上不与X的奇点相交的定向直正则闭曲线c，Wx（c）是c的切线相对于c被X框定旋转的次数。我们开始描述Wx的一些性质。设L是M中的弧的集合（可能为空）。弧a的L-直同伦ht，其中tel是a在M中的同伦，使得对于M\L的所有分支E，h的每个分支，（a）D E是直正则弧。显然，如果L< zj，则L-直接同伦也是l-直接同伦。一条闭曲线c是L-直的，如果对于M\L的每个分量E，每个弧c n E都是直的。如果c是一条具有最小自相交数的闭曲线，z是一个横截自交点，那么我们说（a，B）是c在z处的分裂，如果将z视为c的基点，我们有c= ba，其中a，B是c的子弧。参见图1。
Let M be an orientable surface. In this paper we give a new combinatorial definition of winding number functions on M and characterize all functions having three specific properties common with such winding number functions. These functions turn out to be identified with the elements of the first cohomology group of the unit tangent bundle of M. We first make some definitions in order to state our results more precisely.Let/=[0, 1]. An arc a:/—» M is called regular if the tangent map Ta: 77—• TM is continuous and injective. If a has no null-homotopic loops, then we say that a is direct. Throughout this paper we let A'be a vector field on M with only one singular point. Let Wx denote the winding number function of X\that is, for an oriented direct regular closed curve c on M not meeting the singular points of X, Wx (c) is the number of times the tangents to c rotate with respect to the framing of c by X. We begin by describing some properties of Wx. Let L be a set of arcs in M (possibly empty). An L-direct homotopy ht, with tel, of an arc a is a homotopy of a in M such that for all f€/and all components E of M\L, each component of h,(a) D E is a direct regular arc. Clearly if L< zj, then an L-direct homotopy is also a/-direct homotopy. A closed curve c is L-direct if for each component E of M\L, each arc c n E is direct. If c is a closed curve with minimal self-intersection number and z is a transverse self-intersection point, then we say that (a, b) is a splitting of c at z if, upon thinking of z as a base point for c, we have c= ba, where a, b are subarcs of c. See Fig. 1.