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Three approaches to Morse-Bott homology. (English) Zbl 1311.57043
The paper under review presents a comprehensive survey of three alternative approaches to Morse-Bott homology. Recall that a Morse-Bott function on a closed manifold is a function \(f\) whose critical points are a union of connected submanifolds such that the Hessian of \(f\) on the normal bundle of each critical component is non-degenerate. Starting from this data one may build a chain complex which computes the homology of \(M\) using the following three methods:
1) perturb \(f\) in the neighborhood of its critical submanifolds to get a Morse function and then consider the Morse-Smale-Witten complex of the latter.
2) define a complex generated by the critical points of chosen Morse functions \(f_{j}\) on the critical submanifolds of \(f\), the differential of which is defined using “cascades”: these are broken orbits consisting of concatenation of flow lines of some (chosen) gradients of \(f_{j}\) and of \(f\).
3) define a multicomplex using singular cubical chains on the critical submanifolds and the flow lines of some gradient of \(f\). Such a complex yields a genuine complex, called “assembled complex”.
The author describes :
- in the case 1) the proof that the Morse-Bott inequalities which relate the critical submanifolds to the Betti numbers of \(M\) are valid, following the lines of his former proof with A. Banyaga [A. Banyaga and D. E. Hurtubise, Expo. Math. 22, No. 4, 365–373 (2004; Zbl 1078.57031)].
- in the case 2) his proof with A. Banyaga of the existence of a one-to-one correspondence between the moduli spaces defined by the cascades and the corresponding ones defined as in 1) for a sufficiently small perturbation [A. Banyaga and D. E. Hurtubise, Algebr. Geom. Topol. 13, No. 1, 237–275 (2013; Zbl 1261.57029)].
- in the case 3) his proof with A. Banyaga of the invariance of the homology of the assembled complex with respect to the choice of the Morse-bott function and the gradient. This is done using a continuation method in the spirit of Floer [A. Banyaga and D. E. Hurtubise, Trans. Am. Math. Soc. 362, No. 8, 3997–4043 (2010; Zbl 1226.57038)]. Note that this gives an immediate alternative proof of the Morse homology theorem.

MSC:
57R70 Critical points and critical submanifolds in differential topology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
37D15 Morse-Smale systems
58K05 Critical points of functions and mappings on manifolds
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Full Text: Euclid arXiv
References:
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