arXiv subjects

Delphine Moussard

Publications and source records attributed to Delphine Moussard.

At least 19 recordsLinked to original sources

arXiv2026

Multisections and bridge positions in arbitrary dimensions

Multisections were defined by Ben Aribi--Courte--Golla--Moussard as a way to decompose closed manifolds into $1$-handlebodies, which is a generalization of Heegaard splittings and trisections. Previously, their existence was only known in dimensions up to $5$. We show that multisections exist for closed manifolds in arbitrary dimensions. We also show that submanifolds of codimension at least $2$ in multisected manifolds can always be put into an appropriate bridge position, generalizing the existence result on bridge trisections in dimension $4$.

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arXiv2025

A universal finite type invariant of knots in homology 3-spheres

An essential goal in the study of finite type invariants of some objects (knots, manifolds) is the construction of a universal finite type invariant, universal in the sense that it contains all finite type invariants of the given objects. Such a universal finite type invariant is known for knots in the 3-sphere -- the Kontsevich integral -- and for homology 3-spheres -- the Le-Murakami-Ohtsuki invariant. For knots in homology 3-spheres, an invariant constructed by Garoufalidis and Kricker as a lift of the Kontsevich integral has been considered for the last two decades as the best candidate to be a universal finite type invariant. Although this invariant is eventually universal in restriction to knots whose Alexander polynomial is trivial, we prove here that it is not powerful enough in general. For that we provide a refinement of its construction which produces a strictly stronger invariant, and we prove that this new invariant is a universal finite type invariant of knots in homology 3-spheres. This provides a full diagrammatic description of the graded space of finite type invariants of knots in homology 3-spheres.

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arXiv2025

On diffeomorphisms of 4-dimensional 1-handlebodies

We give a new proof of Laudenbach and Poénaru's theorem, which states that any diffeomorphism of the boundary of a 4-dimensional 1-handlebody extends to the whole handlebody. Our proof is based on the cassification of Heegaard splittings of double handlebodies and a result of Cerf on diffeomorphisms of the 3-ball. Further, we extend this theorem to 4-dimensional compression bodies, namely cobordisms between 3-manifolds constructed using only 1-handles: when the negative boundary is a product of a compact surface by interval, we show that every diffeomorphism of the positive boundary extends to the whole compression body. This invlolves a strong Haken theorem for sutured Heegaard splittings and a classification of sutured Heegaard splittings of double compression bodies. Finally, we show how this applies to the study of relative trisection diagrams for compact 4-manifolds.

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arXiv2024

Slice genus, $T$-genus and $4$-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 $Δ$-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.

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arXiv2023

Multisections of surface bundles and bundles over the circle

A multisection is a decomposition of a manifold into 1-handlebodies, where each subcollection of the pieces intersects along a 1-handlebody except the global intersection which is a closed surface. These generalizations of Heegaard splittings and Gay-Kirby trisections were introduced by Ben Aribi, Courte, Golla and the author, who proved in particular that any 5-manifold admits such a multisection. In arbitrary dimension, we show that two classes of manifolds admit multisections: surface bundles and fiber bundles over the circle whose fiber itself is multisected. We provide explicit constructions, with examples.

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arXiv2023

Multisections of higher-dimensional manifolds

Generalizing Heegaard splittings of 3-manifolds and trisections of 4-manifolds, we consider multisections of higher-dimensional smooth (or PL) closed orientable manifolds, namely decompositions into 1-handlebodies whose subcollections intersect along 1-handlebodies, with global intersection a closed surface. With such a multisection one can associate a diagram. We prove that a multisection diagram determines a unique PL-manifold in all dimensions and a unique smooth manifold up to dimension 6. Further, we show that any closed orientable smooth 5-manifold admits a multisection.

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arXiv2021

The algebraic topology of 4-manifolds multisections

A multisection of a 4-manifold is a decomposition into 1-handlebodies intersecting pairwise along 3-dimensional handlebodies or along a central closed surface; this generalizes the Gay-Kirby trisections. We show how to compute the twisted absolute and relative homology, the torsion and the twisted intersection form of a 4-manifold from a multisection diagram. The homology and torsion are given by a complex of free modules defined by the diagram and the intersection form is expressed in terms of the intersection form on the central surface. We give efficient proofs, with very few computations, thanks to a retraction of the (possibly punctured) 4-manifold onto a CW-complex determined by the multisection diagram. Further, a multisection induces an open book decomposition on the boundary of the 4-manifold; we describe the action of the monodromy on the homology of the page from the multisection diagram.

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arXiv2021

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 $Σ$, guiding the gluing of the handlebodies. Any morphism $φ$ from $π_1(X)$ to a finitely generated free abelian group induces a morphism on $π_1(Σ)$. We express the twisted homology and Reidemeister torsion of $(X;φ)$ in terms of the first homology of $(Σ;φ)$ and the three subspaces generated by the collections of curves. We also express the intersection form of $(X;φ)$ in terms of the intersection form of $(Σ;φ)$.

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arXiv2020

A splicing formula for the LMO invariant

We prove a "splicing formula" for the LMO invariant, which is the universal finite-type invariant of rational homology $3$-spheres. Specifically, if a rational homology $3$-sphere $M$ is obtained by gluing the exteriors of two framed knots $K_1 \subset M_1$ and $K_2\subset M_2$ in rational homology $3$-spheres, our formula expresses the LMO invariant of $M$ in terms of the Kontsevich-LMO invariants of $(M_1,K_1)$ and $(M_2,K_2)$. The proof uses the techniques that Bar-Natan and Lawrence developed to obtain a rational surgery formula for the LMO invariant. In low degrees, we recover Fujita's formula for the Casson-Walker invariant and we observe that the second term of the Ohtsuki series is not additive under "standard" splicing. The splicing formula also works when each $M_i$ comes with a link $L_i$ in addition to the knot $K_i$, hence we get a "satellite formula" for the Kontsevich-LMO invariant.

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arXiv2019

Splitting formulas for the rational lift of the Kontsevich integral

Kricker defined an invariant of knots in homology 3-spheres which is a rational lift of the Kontsevich integral, and proved with Garoufalidis that this invariant satisfies splitting formulas with respect to a surgery move called null-move. We define a functorial extension of the Kricker invariant and prove splitting formulas for this functorial invariant with respect to null Lagrangian-preserving surgery, a generalization of the null-move. We apply these splitting formulas to the Kricker invariant.

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arXiv2019

Toward universality in degree 2 of the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant

In the setting of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries, there are two candidates to be universal invariants, defined respectively by Kricker and Lescop. In a previous paper, the second author defined maps between spaces of Jacobi diagrams. Injectivity for these maps would imply that Kricker and Lescop invariants are indeed universal invariants; this would prove in particular that these two invariants are equivalent. In the present paper, we investigate the injectivity status of these maps for degree 2 invariants, in the case of knots whose Blanchfield modules are direct sums of isomorphic Blanchfield modules of Q-dimension two. We prove that they are always injective except in one case, for which we determine explicitly the kernel.

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arXiv2019

Finite type invariants of knots in homology 3-spheres with respect to null LP-surgeries

We study a theory of finite type invariants for null-homologous knots in rational homology 3-spheres with respect to null Lagrangian-preserving surgeries. It is an analogue in the setting of the rational homology of the Goussarov-Rozansky theory for knots in integral homology 3-spheres. We give a partial combinatorial description of the graded space associated with our theory and determine some cases when this description is complete. For null-homologous knots in rational homology 3-spheres with a trivial Alexander polynomial, we show that the Kricker lift of the Kontsevich integral and the Lescop equivariant invariant built from integrals in configuration spaces are universal finite type invariants for this theory; in particular it implies that they are equivalent for such knots.

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arXiv2019

A Fox-Milnor theorem for the Alexander polynomial of knotted $2$-spheres in $S^4$

For knots in $S^3$, it is well-known that the Alexander polynomial of a ribbon knot factorizes as $f(t)f(t^{-1})$ for some polynomial $f(t)$. By contrast, the Alexander polynomial of a ribbon $2$-knot is not even symmetric in general. Via an alternative notion of ribbon $2$-knots, we give a topological condition on a $2$-knot that implies the factorization of the Alexander polynomial.

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arXiv2016

Finite braid group orbits in Aff(C)-character varieties of the punctured sphere

We give a complete description of finite braid group orbits in Aff(C)-character varieties of the punctured Riemann sphere. This is performed thanks to a coalescence procedure and to the theory of finite complex reflection groups. We then derive consequences in the theory of differential equations. These concern algebraicity of isomonodromic deformations for reducible rank two logarithmic connections on the sphere, the Riemann-Hilbert problem and F_D-type Lauricella hypergeometric functions.

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arXiv2015

Equivariant triple intersections

Given a null-homologous knot $K$ in a rational homology 3-sphere $M$, and the standard infinite cyclic covering $\tilde{X}$ of $(M,K)$, we define an invariant of triples of curves in $\tilde{X}$, by means of equivariant triple intersections of surfaces. We prove that this invariant provides a map $ϕ$ on $\Al^{\otimes 3}$, where $\Al$ is the Alexander module of $(M,K)$, and that the isomorphism class of $ϕ$ is an invariant of the pair $(M,K)$. For a fixed Blanchfield module $(\Al,\bl)$, we consider pairs $(M,K)$ whose Blanchfield modules are isomorphic to $(\Al,\bl)$, equipped with a marking, {\em i.e.} a fixed isomorphism from $(\Al,\bl)$ to the Blanchfield module of $(M,K)$. In this setting, we compute the variation of $ϕ$ under null borromean surgeries, and we describe the set of all maps $ϕ$. Finally, we prove that the map $ϕ$ 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 of marked pairs $(M,K)$ with rational values.

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arXiv2014

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.

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arXiv2013

Rational Blanchfield forms, S-equivalence, and null LP-surgeries

Null Lagrangian-preserving surgeries are a generalization of the Garoufalidis and Rozansky null-moves, that these authors introduced to study the Kricker lift of the Kontsevich integral, in the setting of pairs (M,K) composed of a rational homology sphere M and a null-homologous knot K in M. They are defined as replacements of null-homologous rational homology handlebodies of M\K by other such handlebodies with identical Lagrangian. A null Lagrangian-preserving surgery induces a canonical isomorphism between the rational Alexander modules of the involved pairs, which preserves the Blanchfield form. Conversely, we prove that a fixed isomorphism between rational Alexander modules which preserves the Blanchfield form can be realized, up to multiplication by a power of t, by a finite sequence of null Lagrangian-preserving surgeries. We also prove that such classes of isomorphisms can be realized by rational S-equivalences. In the case of integral homology spheres, we prove similar realization results for a fixed isomorphism between integral Alexander modules.

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arXiv2012

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.

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