Relative Morse Categorification Theory

• DOI: 10.48550/arxiv.1611.06471
• Publication Date: 2016-01-01
• Authors: Lu, Danning; Yan, Xiaohan
• Keywords: Mathematics - Differential Geometry; Differential Geometry (math.DG); Mathematics - Symplectic Geometry; 53D37 (Primary) 57R58, 57R70, 37D15 (Secondary); FOS: Mathematics; Symplectic Geometry (math.SG); Dynamical Systems (math.DS); Mathematics - Dynamical Systems; 0101 mathematics; 01 natural sciences

In this paper, we define a relative Morse complex for manifold with boundary using the handlebody decomposition of the manifold. We prove that the homology of the relative Morse complex is isomorphic to the relative singular homology. Furthermore, we construct $A_\infty$-category structure on the relative Morse complex by counting the trajectory trees among the critical points of different Morse functions. Our result generalizes Fukaya's construction on closed manifold and Manabu's construction of absolute homology on manifold with boundary.<br/>25 pages, 11 figures<br/>