This thesis is focused on Floer homology, a theory elaborated in the 1980s by Andreas Floer in order to prove the famous Arnold's conjecture about the number of fixed points of an Hamiltonian diffeomorphism. Floer homology is still an important tool in symplectic geometry and we present here two of its main applications: Arnold's conjecture and Conley's conjecture. It is striking that the estimate on the number of fixed points in Arnold's conjecture recalls the classical one in Morse theory about the number of non-degenerate critical points of a function on a compact manifold. This resemblance is very deep, in facts Floer's theory can be thought as a version of Morse theory adapted to an infinite dimensional space. The link between Morse theory and Floer's idea is the fact that fixed points of an Hamiltonian diffeomorphism are in correspondence with the 1-periodic orbits of a Hamiltonian system which in turn are the critical points of the Hamiltonian action, which is a functional defined on the space of loops of the manifold. Because of this connection we present in the first part of the thesis the basics of singular homology, Morse theory and Morse homology, and in the second part we develop Floer homology.

### A close look into Floer homology and applications

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*NARDI, ALESSANDRA*

##### 2020/2021

#### Abstract

This thesis is focused on Floer homology, a theory elaborated in the 1980s by Andreas Floer in order to prove the famous Arnold's conjecture about the number of fixed points of an Hamiltonian diffeomorphism. Floer homology is still an important tool in symplectic geometry and we present here two of its main applications: Arnold's conjecture and Conley's conjecture. It is striking that the estimate on the number of fixed points in Arnold's conjecture recalls the classical one in Morse theory about the number of non-degenerate critical points of a function on a compact manifold. This resemblance is very deep, in facts Floer's theory can be thought as a version of Morse theory adapted to an infinite dimensional space. The link between Morse theory and Floer's idea is the fact that fixed points of an Hamiltonian diffeomorphism are in correspondence with the 1-periodic orbits of a Hamiltonian system which in turn are the critical points of the Hamiltonian action, which is a functional defined on the space of loops of the manifold. Because of this connection we present in the first part of the thesis the basics of singular homology, Morse theory and Morse homology, and in the second part we develop Floer homology.The text of this website © Università degli studi di Padova. Full Text are published under a non-exclusive license. Metadata are under a CC0 License

`https://hdl.handle.net/20.500.12608/29764`