Homotopy invariants of vector fields in 3-manifolds

Abstract : In 1998, R. Gompf defined a homotopy invariant of oriented 2-plane fields in 3-manifolds. This invariant is defined for oriented 2-plane fields xi in a closed oriented 3-manifold M when the first Chern class c_1(xi) is a torsion element of H_2(M;Z). In Chapter I, we define an extension of the Gompf invariant for all compact oriented 3-manifolds with boundary and we study its iterated variations under Lagrangian-preserving surgeries. It follows that the extended Gompf invariant has degree two for a suitable finite type invariant theory.The Theta-invariant is an invariant of parallelized 3-manifolds constructed from the degree one part of the perturbative expansion of Chern–Simons theory. G. Kuperberg and D. Thurston identified the invariant Theta(M,tau) of a rational homology 3-sphere M equipped with a parallelization tau with 3·lambda_cw(M) + 1/4·p_1(tau) where lambda_cw denotes Walker’s generalization of the Casson invariant and where p_1 is an invariant of parallelizations defined using a first Pontrjagin class. C. Lescop extended the Theta-invariant to rational homology 3-spheres equipped with a homotopy class of combings and she showed that for all rational homology 3-sphere M equipped with a combing X, the relation Theta(M,[X]) = 3·lambda_cw(M) + 1/4·p_1([X]) still holds using an ad hoc extension of the Pontrjagin numbers for combings. She also gave a combinatorial formula for the Theta-invariant of a rational homology 3-sphere represented by a Heeagaard diagram and equipped with a combing associated to the diagram, and she proved that this formula defines a homotopy invariant of the pair (M,[X]), in a combinatorial way. Following this work, Chapter II presents a combinatorial proof of the decomposition of this combinatorial invariant as 3·lambda_cw(M) + 1/4·p_1([X]). This proof relies on the finite type invariant theory for rational homology 3-spheres with respect to Lagrangian-preserving surgeries established by D. Moussard in 2012
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Jean-Mathieu Magot. Homotopy invariants of vector fields in 3-manifolds. K-Theory and Homology [math.KT]. Université Grenoble Alpes, 2016. English. ⟨NNT : 2016GREAM070⟩. ⟨tel-01681440v2⟩

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