LisMath Seminar 

29/04/2020, 17:00 — 18:00 — Online
Miguel Santos, LisMath, Instituto Superior Técnico, Universidade de Lisboa

Morse Homology and Floer Homology

Morse theory relates the topology of a smooth manifold with the critical points of Morse functions. Under Morse-Smale transversality one can define a chain complex generated by critical points which com- putes the singular homology of closed manifolds; in particular this implies the Morse inequalities on the number of critical points of Morse functions. Floer homology originated as a version of Morse homology for the symplectic action on the (infinite dimensional) free loop space on a symplectic manifold, which culminated in proving a conjecture by Arnol’d on the number of 1-periodic orbits of non-degenerate Hamiltonians on closed symplectic manifolds. Other examples of applications and generalizations are Viterbo’s theorem on the Floer homology of cotangent bundles, or $S^1$-equivariant Floer homology.

Bibiliography:

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

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

[3] Audin, M. and Damian, M. (2014). Morse theory and Floer homology. Universitext. Springer, London; EDP Sciences, Les Ulis.

[4] Abbondandolo, A. and Schwarz, M. (2006). On the Floer homology of cotangent bundles. Comm. Pure Appl. Math., 59(2):254-316.

[5] Abbondandolo, A. and Schwarz, M. (2014). Corrigendum: On the Floer homology of cotangent bundles. Comm. Pure Appl. Math., 67(4):670-691.

[6] Abouzaid, M. (2015). Symplectic cohomology and Viterbo’s theorem. In Free loop spaces in geometry and topology, volume 24 of IRMA Lect. Math. Theor. Phys., pages 271-485. Eur. Math. Soc., Zurich.

[7] Bourgeois, F. and Oancea, A. (2017). $S^1$ -equivariant symplectic homology and linearized contact homology. Int. Math. Res. Not. IMRN, (13):3849-3937.

See also

miguelmsantos_29042020.pdf