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Morse homology. (English) Zbl 0806.57020

Progress in Mathematics (Boston, Mass.). 111. Basel: Birkhäuser Verlag. ix, 235 p. (1993).
The classical Morse theory relates the critical points of a generic function to the global topology of the underlying manifold. The stable and unstable manifolds of the negative gradient flow associated to a Morse function are cells and have a transversal intersection \(W^ u(x) \pitchfork W^ s(y)\) for all \(x,y\in \text{Crit}(f)\). Thus one is able to define a chain complex generated by the critical points and the boundary operator is given by \(\partial_ k(x)= \sum_{y\in \text{Crit}_{k-1}(f)} \#[ W^ u(x)\cap W^ s(y)) ]y\) and the homology of this complex – the Morse homology – is isomorphic to the standard homology of the compact manifold. Andreas Floer recognized that in the situation of an infinite-dimensional manifold the Morse homology can be still defined – today known as Floer homology – if one can handle the space of connecting orbits \(M_{x,y}\).
The book under review contains a complete construction of the Morse homology in case of a finite-dimensional manifold. Introducing a suitable topology the author studies the trajectory space \(M_{x,y}\) and proves in particular the compactness as well as the gluing properties of this space. Chapter 3 of the book discusses the orientation of the trajectory spaces in order to obtain a \(Z\)-homology theory. The second part of the book is devoted to the construction of the Morse homology as a homology theory in the sense of Eilenberg-Steenrod. The book ends with the discussion of the products as well as the duality in this homology theory.

MSC:

57R70 Critical points and critical submanifolds in differential topology
57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes
55N10 Singular homology and cohomology theory
57R19 Algebraic topology on manifolds and differential topology
58A05 Differentiable manifolds, foundations
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