2004/06/23 by David G. C. Handron, Handron, David G. C.
Computer Science · Mathematics · #57R25 #57R70 #FOS: Mathematics #Geometric Topology (math.GT) #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis #math.GT #msc:57R25 #msc:57R70
paper · pdf · doi:10.48550/arxiv.math/0406486
17 pages, 0 figures updated and clarified definitions, particularly in Section 1.1 (Setup and Definitions) and Definion 5 (in this version) of a Morse-Smale function
openalex publication_date 2004/06/23 · arxiv created 2004/08/12 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
A Morse function f on a manifold with corners M allows the characterization of the Morse data for a critical point by the Morse index. In fact, a modified gradient flow allows a proof of the Morse theorems in a manner similar to that of classical Morse theory. It follows that M is homotopy equivalent to a CW-complex with one cell of dimension λfor each essential critical point of index λ. The goal of this article is to determine the boundary maps of this CW-complex, in the case where M is compact and orientable. First, the boundary maps are defined in terms of the modified gradient flow. Then a transversality condition is imposed which insures that the attaching map is non-degenerate in a neighborhood of each critical point. The degree of this map is then interpreted as a sum of trajectories connecting two critical points each counted with a multiplicity determined by a choice of orientations on the tangent spaces of the unstable manifold at each critical point.