2004/11/30 by Joa Weber
Computer Science · Mathematics · Medicine · #Advanced Neuroimaging Techniques and Applications #Algebra over a field #Balanced flow #Cellular homology #Circle-valued Morse theory #Closed manifold #Compact space #Discrete Morse theory #Dynamical systems theory #Homology (biology) #Homotopy and Cohomology in Algebraic Topology #Invariant manifold #Mathematical analysis #Mathematics #Moduli space #Morse code #Morse homology #Morse theory #Physics #Pure mathematics #Riemannian manifold #Topological and Geometric Data Analysis #math.DS #math.GT #math.SG #msc:57R19 #msc:58-02
paper · pdf · doi:10.1016/j.exmath.2005.09.001
published as Expo. Math. 24 (2006), 127--159 · 38 pages, 17 figures, minor modifications and corrections
openalex publication_date 2005/10/12 · arxiv created 2005/10/26 · arxiv updated 2014/02/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Given a smooth closed manifold M, the Morse-Witten complex associated to a Morse function f and a Riemannian metric g on M consists of chain groups generated by the critical points of f and a boundary operator counting isolated flow lines of the negative gradient flow. Its homology reproduces singular homology of M. The geometric approach presented here was developed in [We-93] and is based on tools from hyperbolic dynamical systems. For instance, we apply the Grobman-Hartman theorem and the Lambda-Lemma (Inclination Lemma) to analyze compactness and define gluing for the moduli space of flow lines.