CONTRIBUTIONS À LA THÉORIE DE MORSE DISCRÈTE ET À L'HOMOLOGIE DE HEEGAARD-FLOER COMBINATOIRE

Abstract : This thesis deals with two aspects of Morse theory: Forman discrete Morse theory (finite dimension case) and link Floer homology (infinite dimension case).
In the first part, we focus on the sign refinement problem for combinatorial link Floer homology. We give another construction which is more conceptual than the original one done by Manolescu, Ozsváth, Szabó and D. Thurston. Then we give the link between both constructions and describe and algorithm to compute signs.
The second part deals with Forman's discrete Morse theory. After explaining the relation between it and algebra over chain complexes we prove that any combinatorial Thom-Smale complex given by a smooth Morse function on a smooth closed manifold can be combinatorially realized by a triangulation and a discrete Morse function on it. We use this to obtain a representation by a complete matching of any Euler structure on a closed oriented 3-manifold.
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Étienne Gallais. CONTRIBUTIONS À LA THÉORIE DE MORSE DISCRÈTE ET À L'HOMOLOGIE DE HEEGAARD-FLOER COMBINATOIRE. Mathématiques [math]. Université de Bretagne Sud, 2007. Français. ⟨tel-00265283⟩

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