向量场的积分曲线分类和链群与有向同伦不变性

The Classification of Integral Curves and Chain-groups in Vector Field and Homotopy Invariance

利用积分曲线的极限集,对紧流形M上向量场X所确定的积分曲线进行了分类,在分类的基础上定义了向量场X的链群、极限集闭链群和极限集边缘链群,以及两个同类群,最后还引入了向量场正向同伦和负向同伦的概念,并证明了极限集是正向同伦和负向同伦不变的,链群、极限集闭链群和极限集边缘链群以及两个同类群在向量场双向同伦的情况下是同构的.

By using limit sets of integral curves,the authors give a classification of integral curves which are determined by vector field on a compact manifold.Based on this,chain-groups, limit set closed chain-groups,limit set boundary chain-groups of vector field and two congener groups are defined respectively.Finally,the authors introduce concepts of positive homotopy and negative homotopy of vector field,and prove that limit sets possess positive and negative homotopy invariance,and chain-groups,limit set closed chain-groups and limit set boundary chain-groups and two congener groups are isomorphic on bidirectional homotopy of vector field.