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Groupe fondamental de Morse stable

Abstract : The object of this thesis is to define a presentation of the fundamental group of a manifold exclusively out of the dynamics of the gradient of a stable Morse function. In general, relating topological invariants to dynamical properties is a fruitful tool to explore the interplay of the two worlds; it is in particular the main idea behind the celebrated Floer theory, for which the stable Morse theory can be thought of as a finite dimensional model. It is well known that the homology of the manifold can be recovered from the dynamics of the gradient even in the stable Morse case, but the case of the fundamental group is much more challenging. In particular, as shown by M. Damian, stable Morse functions may have strictly less critical points than the minimal number of generators of the fundamental group. The question of the generators was solved by J.-F. Barraud in "J.-F. Barraud. A Floer fundamental group. Ann. Sci. Éc. Norm. Supér. (4), 51(3) :773–809, 2018. », but an intrinsic description of the relations was still missing and of interest to deal with injectivity questions arising naturally, in particular when working with techniques based on the s-cobordism theorem. The main idea to get an appropriate definition of the relations is to reformulate the action of the flow on topological loops in terms of moduli spaces of trajectories. The main result is a definition of a presentation of the stable Morse fundamental group dollar pi_1(M,F,g,star) dollar and the proof that the resulting group is invariant and isomorphic to the usual fundamental group. The first part of the thesis describes the known presentation of the fundamental group in Morse theory, taking a point of view which can naturally be extended towards the stable Morse setting. The spaces of trajectories involved in this description are detailed, as are their geometry and the combinatorics of their boundary components. The second part of the thesis defines a presentation of the "stable Morse fundamental group". On top of the generators, new spaces of trajectories are introduced and used to define generators of the subgroup of relations. It is then shown that the resulting quotient group is invariant and isomorphic to the usual fundamental group.
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Submitted on : Monday, July 18, 2022 - 5:39:11 PM
Last modification on : Wednesday, September 7, 2022 - 9:57:11 AM


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Florian Bertuol. Groupe fondamental de Morse stable. Variables complexes [math.CV]. Université Paul Sabatier - Toulouse III, 2022. Français. ⟨NNT : 2022TOU30073⟩. ⟨tel-03726679⟩



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