Floer Homology: From Generalized Morse–Smale Dynamical Systems to Forman’s Combinatorial Vector Fields

Abstract

We 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 $\mathbb {Z}_2$ homology of both smooth and discrete structures directly from the flow lines (V-paths) of our vector field.

References

[1] Banyaga A, Hurtubise D. Morse-Bott homology. Trans. Am. Math. Soc., 2010, 362(8): 3997-4043

[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 preprint arXiv:1010.0548, (2010)

[4] Bott R. Morse theory indomitable. Publ. Mathématiques de l’IHÉS, 1988, 68: 99-114

[5] Conley C, Zehnder E. Morse-type index theory for flows and periodic solutions for Hamiltonian equations. Commun. Pure Appl. Math., 1984, 37(2): 207-253

[6] Conley, C.C.: Isolated Invariant Sets and the Morse Index, vol. 38. American Mathematical Society (1978)

[7] Floer A. An instanton-invariant for 3-manifolds. Commun. Math. Phys., 1988, 118(2): 215-240

[8] Floer A. Morse theory for lagrangian intersections. J. Differ. Geom., 1988, 28(3): 513-547

[9] Floer A. A relative Morse index for the symplectic action. Commun. Pure Appl. Math., 1988, 41(4): 393-407

[10] Floer A. The unregularized gradient flow of the symplectic action. Commun. Pure Appl. Math., 1988, 41(6): 775-813

[11] Floer A. Witten’s complex and infinite-dimensional Morse theory. J. Differ. Geom., 1989, 30(1): 207-221

[12] Floer A, Hofer H. Coherent orientations for periodic orbit problems in symplectic geometry. Mathematische Zeitschrift, 1993, 212(1): 13-38

[13] Forman R. Combinatorial vector fields and dynamical systems. Mathematische Zeitschrift, 1998, 228(4): 629-681

[14] Forman R. Morse theory for cell complexes. Adv. Math., 1998, 134(1): 90-145

[15] Forman R. Witten–Morse theory for cell complexes. Topology, 1998, 37(5): 945-980

[16] Franks JM. Morse–Smale flows and homotopy theory. Topology, 1979, 18(3): 199-215

[17] Franks, J.M.: Homology and Dynamical Systems, vol. 49. American Mathematical Society (1982)

[18] Hurtubise DE. Three approaches to Morse–Bott homology. Afr. Diaspora J. Math. New Ser., 2012, 14(2): 145-177

[19] Jost, J.: Dynamical Systems: Examples of Complex Behaviour. Springer (2005)

[20] Jost, J., Jost, J.: Riemannian Geometry and Geometric Analysis, vol. 42005. Springer (2008)

[21] Knudson, K.P.: Morse Theory: Smooth and Discrete. World Scientific Publishing Company (2015)

[22] Meyer KR. Energy functions for Morse Smale systems. Am. J. Math., 1968, 90(4): 1031-1040

[23] Morse M. Relations between the critical points of a real function of independent variables. Trans. Am. Math. Soc., 1925, 27(3): 345-396

[24] Schwarz, M.: Morse Homology, vol. 111. Springer (1993)

[25] Scoville, N.A.: Discrete Morse Theory, vol. 90. American Mathematical Society (2019)

[26] Smale S. Morse inequalities for a dynamical system. Bull. Am. Math. Soc., 1960, 66(1): 43-49

[27] Smale S. On gradient dynamical systems. Ann. Math., 1961, 74: 199-206

[28] Smoller, J.: Shock Waves and Reaction–Diffusion Equations, vol. 258. Springer (2012)

Funding

Max Planck Institute for Mathematics in the Sciences (2)