arXiv research

Positive braids linked to knot invariants and geometric monodromy groups.

The geometric monodromy of a plane curve singularity is a quasi-finite diffeomorphism. In this paper we locate the reduction curves of the geometric monodromy and the quadratic vanishing cycles of the singularity. An application to the geometric monodromy group is given.

We consider spaces of plane curves in the setting of algebraic geometry and of singularity theory. On one hand there are the complete linear systems, on the other we consider unfolding spaces of bivariate polynomials of Brieskorn-Pham type. For suitable open subspaces we can define the bifurcation braid monodromy takin…

In this paper, Hamiltonian monodromy is studied from the point of view of geometric quantization abd theta functions, and various differential geometric aspects thereof are dealt with, all related to holonomies of suitable flat connections.

We show that a variety of monodromy phenomena arising in geometric topology and algebraic geometry are most conveniently described in terms of quandle homomorphisms from a knot quandle associated to the base to a quandle associated to a fiber. We consider the cases of the monodromy of a branched covering, braid monodro…

This paper generalizes monodromy maps for projective structures with poles.

We describe all the situations in which the Kontsevich-Zorich cocycle has zero Lyapunov exponents. Confirming a conjecture of Forni, Matheus, and Zorich, this only occurs when the cocycle satisfies additional geometric constraints. We also describe the real Lie groups which can appear in the monodromy of the Kontsevich…

A question of Griffiths-Schmid asks when the monodromy group of an algebraic family of complex varieties is arithmetic. We resolve this in the affirmative for the class of algebraic surfaces known as Atiyah-Kodaira manifolds, which have base and fibers equal to complete algebraic curves. Our methods are topological in …

This work classifies monodromy in vineyards using singularity theory.

For isolated complex hypersurface singularities with real defining equation we show the existence of a monodromy vector field such that complex conjugation intertwines the local monodromy diffeomorphism with its inverse. In particular, it follows that the geometric monodromy is the composition of the involution induced…

New isolated geometric triangulations found in once-punctured torus bundles.

Characterizes monodromies of projective structures on finite-type surfaces.

Study of translation covers of platonic solids reveals monodromy group structures.

We introduce several new families of relations in the mapping class groups of planar surfaces, each equating two products of right-handed Dehn twists. The interest of these relations lies in their geometric interpretation in terms of rational blowdowns of 4-manifolds, specifically via monodromy substitution in Lefschet…

This study examines arithmetic properties of GIB manifolds and their monodromy representations.

This work computes integral variation and monodromy maps for plane curve singularities.

For a punctured surface $S$, we characterize the representations of its fundamental group into $\mathrm{PSL}_2 (\mathbb{C})$ that arise as the monodromy of a meromorphic projective structure on $S$ with poles of order at most two and no apparent singularities. This proves the analogue of a theorem of Gallo-Kapovich-Mar…

Study plane curve singularities to determine vanishing cycles and monodromy groups.

We consider the space of smooth complex projective plane curves of degree d. Defined over this is the tautological family of plane curves, and hence there is a monodromy representation into the mapping class group of the fiber. We show two results concerning this monodromy group. First, we show that the presence of an …

The paper describes handle decompositions and Kirby diagrams for plane algebraic curves.

The study provides obstructions and unusual subgroup properties in mapping class groups.

We consider components of Hurwitz moduli space of G-Galois covers and set up a powerful algebraic framework to study the set of corresponding equivalence classes of monodromy maps. Within that we study geometric stabilisation by various G-covers branched over the disc. Our results addresses the problem to decide equiva…

We study several geometric and group theoretical problems related to Kodaira fibrations, to more general families of Riemann surfaces, and to surface-by-surface groups. First we provide constraints on Kodaira fibrations that fiber in more than two distinct ways, addressing a question by Catanese and Salter about their …

Study complex surfaces fibered over Teichmüller curves with Veech fibers.

A surface automorphism is strongly irreducible if every essential simple closed curve in the surface has nontrivial geometric intersection with its image. We show that a three-manifold admits only finitely many inequivalent surface bundle structures with strongly irreducible monodromy.

This paper is devoted to a systematic study of the geometry of nondegenerate $\bbR^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamil…

We give a topological and geometrical description of focus-focus singularities of integrable Hamiltonian systems. In particular, we explain why the monodromy around these singularities is non-trivial, a result obtained before by J.J. Duistermaat and others for some concrete systems.

Any endomorphism of a finitely generated free group naturally descends to an injective endomorphism of its stable quotient. In this paper, we prove a geometric incarnation of this phenomenon: namely, that every expanding irreducible train track map inducing an endomorphism of the fundamental group gives rise to an expa…

Geometric methods show cosets of mapping class groups for 3-manifolds.

In this paper we continue our study of equivariant minimal Lagrangian surfaces in $\mathbb{C} P^2$, characterizing the rotationally equivariant cases and providing explicit formulae for relevant geometric quantities of translationally equivariant minimal Lagrangian surfaces in terms of Weierstrass elliptic functions.

Paper studies Lagrangian submanifolds and their homological monodromy.

New conditions found for hyperbolic bicycle tracks.

We compute the Floer homology of mapping classes which do not have any pseudo-Anosov components in the sense of Thurston's theory of surface diffeomorphisms. The formula for the Floer homology is obtained from a topological separation of fixed points and a separation mechanism for Floer connecting orbits. As examples, …

Noninjective monodromy found in polynomial critical point tracking.

The study finds arithmetic groups often in square-tiled surface monodromies.

Characterizes conical angles for metrics with dihedral symmetry.

For any topological groupoid G and any homomorphism from a locally compact Hausdorff topological group K to G, we construct an associated monodromy group. We prove that Morita equivalent topological groupoids have the same monodromy groups. We show how the monodromy groups can be used to test if a Lie groupoid lacks fa…

This is a survey on contact open books and contact Dehn surgery. The relation between these two concepts is discussed, and various applications are sketched, e.g. the monodromy of Stein fillable contact 3-manifolds, the Giroux-Goodman proof of Harer's conjecture on fibred links, construction of symplectic caps to filli…

Study of monodromy and vanishing cycles for complete intersection curves.

In this work we describe a method to reconstruct the braid monodromy of the preimage of a curve by a Kummer cover. This method is interesting, since it combines two techniques, namely, the reconstruction of a highly non-generic braid monodromy with a systematic method to go from a non-generic to a generic braid monodro…

The paper studies the index of a specific monodromy for origamis in a particular stratum.

Polynomials with distinct critical values have braid monodromy groups equal to braid groups.

The monodromy conjecture states that every pole of the topological (or related) zeta function induces an eigenvalue of monodromy. This conjecture has already been studied a lot; however, in full generality it is proven only for zeta functions associated to a polynomial in two variables. In this article we consider zeta…

For every genus $g\geq 2$, we construct an infinite family of strongly quasipositive fibred knots having the same Seifert form as the torus knot $T(2,2g+1)$. In particular, their signatures and four-genera are maximal and their homological monodromies (hence their Alexander module structures) agree. On the other hand, …

This is a glossary of notions and methods related with the topological theory of collections of affine planes, including braid groups, configuration spaces, order complexes, stratified Morse theory, simplicial resolutions, complexes of graphs, Orlik--Solomon rings, Salvetti complex, matroids, Spanier--Whitehead duality…

Arithmetic Kontsevich-Zorich monodromy found in a specific origami surface.

Holomorphic connections found on Riemann surfaces with Fuchsian monodromy.

This paper classifies commensurability of Deligne-Mostow lattices.