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Morse homotopy, \(A^ \infty\)-category, and Floer homologies. (English) Zbl 0853.57030

Kim, Hong-Jong (ed.), Proceedings of the GARC workshop on geometry and topology ’93 held at the Seoul National University, Seoul, Korea, July 1993. Seoul: Seoul National University, Lect. Notes Ser., Seoul. 18, 1-102 (1993).
From the introduction: This is an extended version of the lecture by the author at the Seoul National University in July 1993. This paper is a mixture of a survey article and a research announcement. Chapters 2 and 3 are surveys of results by Gromov, Floer, Ruan, and others. Many of the material in the other chapters are new. However these chapters also include several parts which survey earlier works or restate them in a bit different way. Also there may be a possible overlap of results with those of other authors. In the following is given a rough summary of each chapter.
Chapter 1 is devoted to a construction which detects some information of homotopy types of manifolds using Morse functions. The result of this chapter is a “toy model” of the construction we will perform in later chapters. Chapter 2 is a rough summary of Floer’s idea on the Arnold conjecture. In this chapter we assume a rather restrictive hypothesis and try to discuss the basic points without studying various difficulties. Those troubles we meet are discussed in Chapter 3. But in this chapter we rather discuss the case of pseudo-holomorphic sphere. (While one needs to study the pseudo-holomorphic disk for the Arnold conjecture.) We apply a result on pseudo-holomorphic sphere to define Gromov-Ruan’s [Y. Ruan, Topological sigma model and Donaldson type invariant in Gromov theory (preprint), Duke Math. J. 83, No. 2, 461–500 (1996; Zbl 0864.53032)] invariant which justify several constructions in the topological \(\sigma\)-model.
In Chapter 4, first we join the ideas in Chapters 2 and 3 to Maslov index and Novikov ring and define Floer homology for the Lagrangian intersection in a pseudo-Einstein symplectic manifold of nonnegative curvature. Then we combine the construction of Floer homology to one in chapter 1 and define an \(A^\infty\)-category.
In Chapter 5 we first recall the definition of Floer homology of 3-manifold and the results by S. Dostoglou and D. Salamon [Ann. Math. (2) 139, 581–640 (1994; Zbl 0812.58031)] and T. Yoshida [Floer homology and holomorphic curves – Atiyah conjecture (preprint)] which relate it to symplectic Floer theory. Then, using the result of the last section, we define the Floer homology for 3-manifolds with boundary and discuss its properties.
For the entire collection see [Zbl 0812.00028].

MSC:

57R45 Singularities of differentiable mappings in differential topology
53D40 Symplectic aspects of Floer homology and cohomology
53D12 Lagrangian submanifolds; Maslov index
57R58 Floer homology
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