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Floer homology. (English) Zbl 0837.58011
Hofer, Helmut (ed.) et al., The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 105-108 (1995).
The most important contributions of A. Floer to the elucidation of links between classical geometry and quantum field theory are reviewed. The main emphasis is given on what is known now as Floer homology. The existence of Morse inequalities for the fixed points of global symplectic diffeomorphisms is considered.
This conjecture was finally established by Floer in its great generality. Floer’s main idea was to define appropriate homology groups related to the critical points of an action functional as in usual Morse theory. The relation between Donaldson invariants and Floer homology is also discussed. The possible progress in developing methods for calculating Floer homology groups is analyzed. This could be realized by considering the moduli spaces of flat connections on compact orientable surfaces.
For the entire collection see [Zbl 0824.00019].
Reviewer: G.Zet (Iaşi)

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
81T70 Quantization in field theory; cohomological methods
55N35 Other homology theories in algebraic topology
58D27 Moduli problems for differential geometric structures
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