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Floer’s infinite dimensional Morse theory and homotopy theory. (English) Zbl 0843.58019
Hofer, Helmut (ed.) et al., The Floer memorial volume. Basel: Birkhäuser. Prog. Math. 133, 297-325 (1995).
The authors describe a set of ideas which could lead to a “Floer homotopy theory”. In his work on the Arnold conjecture A. Floer [J. Differ. Geom. 28, No. 3, 513-547 (1988; Zbl 0674.57027)] developed an infinite dimensional Morse theory for the action functional defined on the loop space \({\mathcal L} M\) of a compact symplectic manifold \(M\). In this paper the authors suggest that the Floer homology of the action functional should be the classical homology of a pro-spectrum somehow associated to the functional. They also discuss the question what structure an infinite-dimensional manifold \(X\) like \({\mathcal L} M\) should have in order to define this pro-spectrum as an invariant of \(X\). The concepts and ideas are well motivated either by going back to classical Morse theory on compact manifolds, or by working out certain infinite dimensional examples where \(X\) is \(\mathbb{R} P^\infty\), \(\mathbb{C} P^\infty\) or \({\mathcal L} \mathbb{C} P^\infty\).
For the entire collection see [Zbl 0824.00019].

58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
55P99 Homotopy theory
53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
55N35 Other homology theories in algebraic topology