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Lectures on Floer homology. (English) Zbl 1031.53118

Eliashberg, Yakov (ed.) et al., Symplectic geometry and topology. Lecture notes from the graduate summer school program, Park City, UT, USA, June 29-July 19, 1997. Providence, RI: American Mathematical Society. IAS/ Park City Math. Ser. 7, 145-229 (1999).
This series of lectures provides an introduction to symplectic Floer theory and the proof of the Arnold conjecture on the lower bound for the number of 1–periodic solutions of a 1–periodic Hamiltonian system in terms of Betti numbers.
The first lecture surveys different statements of the Arnold conjecture, the monotonicity condition for symplectic manifolds, the Morse–Smale–Witten complex computing homology of a compact smooth Riemannian manifolds, the symplectic action functional, and the properties of its gradient flow lines and their moduli spaces.
The second lecture is dedicated to the Fredholm theory of the linearized operator. The main issues are the \(L^p\)–estimates, the Conley–Zehnder index, the spectral flow, transversality of the moduli spaces of flow lines and the proof of the fact that all finite energy solutions decay exponentially to periodic orbits.
The third lecture introduces Floer homology. The main issue in this case is the compactification of the moduli spaces of flow lines, namely the convergence modulo bubbling, and the gluing theory for flow lines, which ensure that the fundamental relation \(\partial^2=0\) is satisfied. In the same lecture, the author discusses two approaches to the identification of Floer homology groups and ordinary homology: one by proving independence of the Hamiltonian, and then reducing the computation of Floer homology to the Morse complex of a smooth time independent Morse function; the other using \(J\)–holomorphic spiked disks which provide a chain homotopy between the chain complexes of Floer homology and ordinary homology. Furthermore, the same lecture contains a discussion of the Calabi–Yau case, and of Novikov rings.
The fourth lecture is dedicated to Gromov compactness and stable maps. The lecture covers bubbling, soft rescaling, and Kontsevich’s notion of stable maps. To illustrate this notion when the target space is a point, the moduli space of Riemann surfaces of genus zero with marked points is presented with its Deligne–Mumford compactification.
The fifth lecture returns to discuss the Arnold conjecture, under the assumption of a negative monotonicity condition, so that all \(J\)–holomorphic spheres have negative Chern number. The technical issue in the compactification created by the possible presence of multiply covered \(J\)–holomorphic spheres is addressed, using a setup with multi–valued perturbations. The resulting moduli spaces are no longer smooth manifolds, but branched manifolds with rational weights. The remaining of the lecture analyzes the main properties of such moduli spaces, which lead to a definition of Floer homology over the rationals.
For the entire collection see [Zbl 0921.00023].

MSC:

53D40 Symplectic aspects of Floer homology and cohomology
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
57R17 Symplectic and contact topology in high or arbitrary dimension
37J45 Periodic, homoclinic and heteroclinic orbits; variational methods, degree-theoretic methods (MSC2010)
53D45 Gromov-Witten invariants, quantum cohomology, Frobenius manifolds
57R58 Floer homology
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