The Pathwise Connected Components of Path Space on Manifold Mn

Let Ωbe the path space consists of all piecewise C-path joining point pair A≠Aon a n-dimensional connected differential manifold M. We know that the connectivity numbers of Ωare invariants of M. In other words,these quantities are independent of the point pair A≠Aon M. Ωis not pathwise connected in general. So we can divide Ωto several pathwise connected components. Let C（h） be the pathwise connected component involving the piecewise C-path h.