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Floer homotopy theory, revisited. (English) Zbl 1476.57057

Miller, Haynes (ed.), Handbook of homotopy theory. Boca Raton, FL: CRC Press. CRC Press/Chapman Hall Handb. Math. Ser., 369-404 (2020).
The Handbook of homotopy theory has been reviewed in [Zbl 1468.55001].
Summary: In three seminal papers in 1988 and 1989 A. Floer introduced Morse theoretic homological invariants that transformed the study of low dimensional topology and symplectic geometry. A dramatic application of the ideas of Floer homotopy theory appeared in the work of Lipshitz and Sarkar on the homotopy theoretic foundations of M. Khovanov’s homological invariants of knots and links [R. Lipshitz and S. Sarkar, J. Am. Math. Soc. 27, No. 4, 983–1042 (2014; Zbl 1345.57014) and J. Topol. 7, No. 3, 817–848 (2014; Zbl 1345.57013)]. The Floer homotopy theory used was a type of Hamiltonian Floer theory for the cotangent bundle. The Khovanov chain complex is generated by all possible configurations of resolutions of the crossings of a link diagram. Monopole Floer homology is similar in nature to Floer’s Instanton homology theory, but it is based on the Seiberg-Witten equations rather than the Yang-Mills equations. The chapter describes the Floer homotopy theoretic methods of T. Kragh [Geom. Topol. 17, No. 2, 639–731 (2013; Zbl 1267.53081)] and of M. Abouzaid and T. Kragh [J. Topol. 9, No. 1, 232–244 (2016; Zbl 1405.53120)] in the study of the symplectic topology of the cotangent bundle of a closed manifold, and how they were useful in studying Lagrangian immersions and embeddings inside the cotangent bundle.
For the entire collection see [Zbl 1468.55001].

MSC:

57R58 Floer homology
57K18 Homology theories in knot theory (Khovanov, Heegaard-Floer, etc.)
53D12 Lagrangian submanifolds; Maslov index
53D40 Symplectic aspects of Floer homology and cohomology
55P35 Loop spaces
55P42 Stable homotopy theory, spectra
57Q15 Triangulating manifolds
57-02 Research exposition (monographs, survey articles) pertaining to manifolds and cell complexes
53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
55-02 Research exposition (monographs, survey articles) pertaining to algebraic topology
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