Hurtubise, David E. Three approaches to Morse-Bott homology. (English) Zbl 1311.57043 Afr. Diaspora J. Math. 14, No. 2, 145-177 (2012). 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. Reviewer: Mihai Damian (Strasbourg) Cited in 4 Documents 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 Keywords:Morse-Bott theory; Morse-Smale-Witten chain complex; cascade chain complex; Morse-Bott multi complex; Morse-Bott inequalities Citations:Zbl 1078.57031; Zbl 1261.57029; Zbl 1226.57038 × Cite Format Result Cite Review PDF Full Text: arXiv Euclid References: [1] Alberto Abbondandolo and Pietro Majer, Lectures on the Morse complex for infinite-dimensional manifolds , Morse theoretic methods in nonlinear analysis and in symplectic topology, NATO Sci. Ser. II Math. Phys. Chem., vol. 217, Springer, Dordrecht, 2006, pp. 1-74. · Zbl 1089.37007 · doi:10.1007/1-4020-4266-3_01 [2] David M. Austin and Peter J. 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