Realizing isomorphisms between first homology groups of closed 3-manifolds by borromean surgeries

We refine Matveev's result asserting that any two closed oriented 3-manifolds can be related by a sequence of borromean surgeries if and only if they have isomorphic first homology groups and linking pairings. Indeed, a borromean surgery induces a canonical isomorphism between the first homology groups of the involved 3-manifolds, which preserves the linking pairing. We prove that any such isomorphism is induced by a sequence of borromean surgeries. As an intermediate result, we prove that a given algebraic square finite presentation of the first homology group of a 3-manifold, which encodes the linking pairing, can always be obtained from a surgery presentation of the manifold

Finite type invariants of rational homology 3-spheres

We consider the rational vector space generated by all rational homology spheres up to orientation-preserving homeomorphism, and the filtration defined on this space by Lagrangian-preserving rational homology handlebody replacements. We identify the graded space associated with this filtration with a graded space of augmented Jacobi diagrams

Slice genus, TT-genus and 44-dimensional clasp number

The $T$-genus of a knot is the minimal number of borromean-type triple points on a normal singular disk with no clasp bounded by the knot; it is an upper bound for the slice genus. Kawauchi, Shibuya and Suzuki characterized the slice knots by the vanishing of their $T$-genus. We generalize this to provide a $3$-dimensional characterization of the slice genus. Further, we prove that the $T$-genus majors the $4$-dimensional positive clasp number and we deduce that the difference between the $T$-genus and the slice genus can be arbitrarily large. We introduce the ribbon counterpart of the $T$-genus and prove that it is an upper bound for the ribbon genus. Interpreting the $T$-genera in terms of $\Delta$-distance, we show that the $T$-genus and the ribbon $T$-genus coincide for all knots if and only if all slice knots are ribbon. We work in the more general setting of algebraically split links and we also discuss the case of colored links. Finally, we express Milnor's triple linking number of an algebraically split $3$-component link as the algebraic intersection number of three immersed disks bounded by the three components.Comment: 19 pages, 21 figures, comments welcom

Torsions and intersection forms of 4-manifolds from trisection diagrams

Gay and Kirby introduced trisections which describe any closed oriented smooth 4-manifold $X$ as a union of three four-dimensional handlebodies. A trisection is encoded in a diagram, namely three collections of curves in a closed oriented surface $\Sigma$, guiding the gluing of the handlebodies. Any morphism $\varphi$ from $\pi_1(X)$ to a finitely generated free abelian group induces a morphism on $\pi_1(\Sigma)$. We express the twisted homology and Reidemeister torsion of $(X;\varphi)$ in terms of the first homology of $(\Sigma;\varphi)$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;\varphi)$ in terms of the intersection form of $(\Sigma;\varphi)$.Comment: Comments are welcom

Equivariant triple intersections

International audienceGiven a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\widetilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\widetilde{X}$ by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $\phi$ on $\mathfrak{A}^{\otimes 3}$, where $\mathfrak{A}$ is the Alexander module of $(M,K)$, and that the isomorphism class of $\phi$ is an invariant of the pair $(M,K)$. For a fixed Blanchfield module $(\mathfrak{A},\mathfrak{b})$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(\mathfrak{A},\mathfrak{b})$ equipped with a marking, i.e. a fixed isomorphism from $(\mathfrak{A},\mathfrak{b})$ to the Blanchfield module of $(M,K)$. In this setting, we compute the variation of $\phi$ under null Borromean surgeries and we describe the set of all maps $\phi$. Finally, we prove that the map $\phi$ is a finite type invariant of degree 1 of marked pairs $(M,K)$ with respect to null Lagrangian-preserving surgeries, and we determine the space of all degree 1 invariants with rational values of marked pairs $(M,K)$

Equivariance and finite type invariants in dimension 3

Cette thèse a pour objet l'étude des invariants de type fini des sphères d'homologie rationnelle de dimension 3, et des nœuds homologiquement triviaux dans ces sphères. Les principaux résultats sont présentés dans le chapitre 2. Ils sont démontrés dans les chapitres 3 à 6. Le chapitre 3 est un article intitulé ``Finite type invariants of rational homology 3-spheres'', à paraître dans Algebraic & Geometric Topology. Il décrit le gradué associé à la filtration de l'espace vectoriel rationnel engendré par les sphères d'homologie rationnelle, définie par les chirurgies rationnelles préservant le lagrangien. Le chapitre 4 est un article intitulé ``On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres'', publié dans Journal of Knot Theory and its Ramifications. Il contient la classification des modules d'Alexander des nœuds homologiquement triviaux dans les sphères d'homologie rationnelle, et une étude des formes de Blanchfield définies sur ces modules. Dans la suite, on considère les paires (M,K) formées d'une sphère d'homologie rationnelle M et d'un nœud K homologiquement trivial dans M. Dans le chapitre 5, on montre que deux telles paires ont des modules d'Alexander rationnels munis de leurs formes de Blanchfield isomorphes si et seulement si elles s'obtiennent l'une de l'autre par une suite finie de chirurgies rationnelles nulles préservant le lagrangien, c'est-à-dire effectuées sur des corps en anses d'homologie rationnelle homologiquement triviaux dans le complémentaire du nœud. Dans le chapitre 6, on étudie le gradué associé à la filtration de l'espace vectoriel rationnel engendré par les paires (M,K) définie par les chirurgies rationnelles nulles préservant le lagrangien. Ces deux derniers chapitres comportent des travaux en progrès.This thesis contains a study of finite type invariants of rational homology 3-spheres, and of null-homologous knots in these spheres. The main results are described in Chapter 2, and proved in Chapters 3 to 6. Chapter 3 is an article entitled ``Finite type invariants of rational homology 3-spheres'', to appear in Algebraic & Geometric Topology. In this article, we describe the graded space associated with the filtration of the rational vector space generated by rational homology spheres, defined by rational Lagrangian-preserving surgeries. Chapter 4 is an article entitled ``On Alexander modules and Blanchfield forms of null-homologous knots in rational homology spheres'', published in Journal of Knot Theory and its Ramifications. It contains the classification of the Alexander modules of null-homologous knots in rational homology spheres, and a study of the Blanchfield forms defined on these modules. In the sequel, we consider pairs (M,K) made of a rational homology sphere M and a null-homologous knot K in M. In Chapter 5, we prove that two such pairs have isomorphic rational Alexander modules endowed with their Blanchfield forms if and only if they can be obtained from one another by a finite sequence of null rational Lagrangian-preserving surgeries, i.e. Lagrangian-preserving surgeries performed on rational homology handlebodies homologically trivial in the complement of the knot. In Chapter 6, we study the graded space associated with the filtration of the rational vector space generated by pairs (M,K) defined by null rational Lagrangian-preserving surgeries. These last two chapters contain work in progress