Abstract
AbstractWe construct a Floer type boundary operator for generalised Morse–Smale dynamical systems on compact smooth manifolds by counting the number of suitable flow lines between closed (both homoclinic and periodic) orbits and isolated critical points. The same principle works for the discrete situation of general combinatorial vector fields, defined by Forman, on CW complexes. We can thus recover the $$\mathbb {Z}_2$$
Z
2
homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.
Funder
Max Planck Institute for Mathematics in the Sciences
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Computational Mathematics,Statistics and Probability
Reference28 articles.
1. Banyaga, A., Hurtubise, D.: Morse-Bott homology. Trans. Am. Math. Soc. 362(8), 3997–4043 (2010)
2. Banyaga, A., Hurtubise, D., Ajayi, D.: Lectures on Morse Homology. Springer (2004)
3. Benedetti, B.: Discrete Morse theory is at least as perfect as Morse theory. arXiv preprintarXiv:1010.0548, (2010)
4. Bott, R.: Morse theory indomitable. Publ. Mathématiques de l’IHÉS 68, 99–114 (1988)
5. Conley, C., Zehnder, E.: Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math. 37(2), 207–253 (1984)