2019/10/11 by Andreas Knauf, Knauf, Andreas, Nikolay Martynchuk +1
Computer Science · Mathematics · Physics and Astronomy · #37N05 #55R25 #57N65 #57R65 #58E05 #70F10 #70H33 #Dynamical Systems (math.DS) #FOS: Mathematics #FOS: Physical sciences #Geometric Topology (math.GT) #Mathematical Physics (math-ph) #Symplectic Geometry (math.SG) #Topological and Geometric Data Analysis #math-ph #math.DS #math.GT #math.MP #math.SG #msc:37N05 #msc:55R25 #msc:57N65 #msc:57R65 #msc:58E05 #msc:70F10 #msc:70H33
paper · pdf · doi:10.48550/arxiv.1910.05294
arxiv created 2019/10/11 · openalex publication_date 2019/10/11 · arxiv updated 2019/10/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Classical Morse theory proceeds by considering sublevel sets f-1(-∞, a] of a Morse function f: M → R, where M is a smooth finite-dimensional manifold. In this paper, we study the topology of the level sets f-1(a) and give conditions under which the topology of f-1(a) changes when passing a critical value. We show that for a general class of functions, which includes all exhaustive Morse function, the topology of a regular level f-1(a) always changes when passing a single critical point, unless the index of the critical point is half the dimension of the manifold M. When f is a natural Hamiltonian on a cotangent bundle, we obtain more precise results in terms of the topology of the configuration space. (Counter-)examples and applications to celestial mechanics are also discussed.