arXiv research

Developed algorithms to compute three polynomial invariants of veering triangulations.

Study of fibred faces of Thurston polyhedra for 2-component 2-bridge links

The taut polynomial equals a twisted Alexander polynomial.

We show that there is a fully faithful embedding of the category of manifolds with corners into the Cahiers topos, one of the premier models for Synthetic Differential Geometry. This embedding is shown to have a number of nice properties, such as preservation of open covers and transverse fibre products. We develop a t…

Given a triangulation of a closed, oriented, irreducible, atoroidal 3-manifold every oriented, incompressible surface may be isotoped into normal position relative to the triangulation. Such a normal oriented surface is then encoded by non-negative integer weights, 14 for each 3-simplex, that describe how many copies o…

Describes spectral data for singular fibres of a specific Hitchin system.

We classify semi-Riemannian submersions with connected totally geodesic fibres from a real pseudo-hyperbolic space onto a semi-Riemannian manifold under the assumption that the dimension of the fibres is less than or equal to three and the metrics induced on fibres are negative definite. Also, we obtain the classificat…

Ozsváth and Szabó conjectured that knot Floer homology detects fibred links. We will verify this conjecture for closed 3-braids, by classifying fibred closed 3-braids. In particular, given a nontrivial closed 3-braid, either it is fibred, or it differs from a fibred link by a half twist. The proof uses Gabai's method o…

Given a (smooth) complex analytic family of compact complex manifolds, we prove that the central fibre must be Moishezon if the other fibres are Moishezon. Using a "strongly Gauduchon metric" on the central fibre whose existence was proved in our previous work on limits of projective manifolds, we show that the irreduc…

Following on from work of Dunfield, we determine the fibred status of all the unknown hyperbolic 3-manifolds in the cusped census. We then find all the fibred hyperbolic 3-manifolds in the closed census and use this to find over 100 examples each of closed and cusped virtually fibred non-fibred census 3-manifolds, incl…

In this paper, we extend the theory of sutured Floer homology developed by the author. We first prove an adjunction inequality, and then define a polytope P(M,g) in H^2(M,\partial M; R) that is spanned by the Spin^c-structures which support non-zero Floer homology groups. If (M,g) --> (M',g') is a taut surface decompos…

We consider vector fields on knot/link complements in $S^3$ which are transverse to the fibres of a fibration of the complement over a circle. We prove that a large class of fibred knots/links, including all non-torus fibred 2-bridge knots, has the following property: any vector field transverse to the fibres of the fi…

Profinite rigidity studied for algebraic fibring of groups.

The stabilisation height of a fibre surface in the 3-sphere is the minimal number of Hopf plumbing operations needed to attain a stable fibre surface from the initial surface. We show that families of fibre surfaces related by iterated Stallings twists have unbounded stabilisation height.

Study shows volume and genus unrelated for hyperbolic fibred knots.

Any leafwise connection on a fibre bundle over a foliated manifold is proved to come from a connection on this fibre bundle.

Survey on fibring in manifolds and groups, focusing on recent developments and conjectures.

Defines and parametrizes $\mathfrak{sl}(2)$-type singular fibres in symplectic and odd orthogonal Hitchin systems.

We explore algebraic characterizations of 2-knots whose associated knot manifolds fibre over lower-dimensional orbifolds, and consider also some issues related to the groups of higher-dimensional fibred knots.

We consider Schrödinger operators $H=-Δ_{g_\varepsilon} + V$ on a fibre bundle $M\stackrelπ{\to}B$ with compact fibres and a metric $g_\varepsilon$ that blows up directions perpendicular to the fibres by a factor ${\varepsilon^{-1}\gg 1}$. We show that for an eigenvalue $λ$ of the fibre-wise part of $H$, satisfying a l…

In this paper we give a generalisation of the Radius Rigidity theorem of F.Wilhelm. This is done by showing that if a Riemannian submersion of $S^{15}$ with 7-dimensional fibres has at least one fibre which is a great sphere then all the fibres are so. Some weaker than known conditions which force the existence of such…

Study low energy resolvent behavior on fibred boundary metrics.

Knot Floer homology matches fixed point Floer for fibred knots.

We define the pull-back of a smooth principal fibre bundle, and show that it has a natural principal fibre bundle structure. Next, we analyse the relationship between pull-backs by homotopy equivalent maps. The main result of this article is to show that for a principal fibre bundle over a paracompact manifold, there i…

Formula calculates homology groups of Milnor fibres for real hypersurface singularities.

We develop some techniques to study the adiabatic limiting behaviour of Calabi-Yau metrics on the total space of a fibration, and obtain strong control near the singular fibres by imposing restrictions on the singularity types. We prove a uniform lower bound on the metric up to the singular fibre, under fairly general …

Study finds lower bounds for energy on fibred manifolds using fiberwise symmetrization.

Real Milnor fibres become contractible after attaching handles, matching classical results.

Given a closed 3--manifold $Y$, we show that the Heegaard Floer homology determines whether $Y$ fibres over the circle with a fibre of negative Euler characteristic. This is an analogue of an earlier result about knots proved by Ghiggini and the author.

We introduce the fibred toric varieties as equivariant $\mathbb{C}P^r$ bundles over lower dimensional toric varieties. An equivalent characterization is that the natural morphisms on them degenerate to bundle projections in the context of variation of toric varieties as GIT quotients. Our main observation is that these…

In a fibre bundle, natural derivatives of a section are defined as tangent vector fields on the image of a section of the fibre bundle. A local extension to vector fields in the tangent bundle leads to a direct proof of the formula expressing the curvature of a connection in terms of covariant derivatives. The result i…

A review of the parallel transport (translation) in fibre bundles is presented. The connections between transports along paths and parallel transports in fibre bundles are examined. It is proved that the latter ones are special cases of the former.

Let $X$ be a compact manifold with boundary. Suppose that the boundary is fibred, $φ:\pa X\longrightarrow Y,$ and let $x\in\CI(X)$ be a boundary defining function. This data fixes the space of `fibred cusp' vector fields, consisting of those vector fields $V$ on $X$ satisfying $Vx=O(x^2)$ and which are tangent to the f…

We get obstructions to existence of semi-Riemannian submersions with totally umbilic fibres.

We get obstructions to existence of semi-Riemannian submersions with totally umbilic fibres.

Paper studies curvature in Finsler geometry, proving curvature constancy under isotropy.

An interesting question is whether two 3-manifolds can be distinguished by computing and comparing their collections of finite covers; more precisely, by the profinite completions of their fundamental groups. In this paper, we solve this question completely for closed orientable Seifert fibre spaces. In particular, all…

Classifies 3D spaces using specific invariants.

This work has the purpose of applying the concept of Geometric Calculus (Clifford Algebras) to the Fibre Bundle description of Quantum Mechanics. Thus, it is intended to generalize that formulation to curved spacetimes [the base space of the fibre bundle in question] in a more natural way.

It was proved by Chern, Hirzebruch and Serre that the signature of a fibre bundle is multiplicative if the fundamental group of the base acts trivially on the cohomology ring of the fibre with real coefficients, in which case the signature of the total space equals the product of the signatures of base and fibre. Hambl…

We put together some of the efforts by several people of making aspects of fibre bundle theory into algebra. The initiator of these efforts was Charles Ehresmann, who put the notion of groupoid and groupoid action in the focus of fibre bundle theory in general, and in connection theory in particular.

Ozsváth and Szabó conjectured that knot Floer homology detects fibred knots in $S^3$. We will prove this conjecture for null-homologous knots in arbitrary closed 3--manifolds. Namely, if $K$ is a knot in a closed 3--manifold $Y$, $Y-K$ is irreducible, and $\hat{HFK}(Y,K)$ is monic, then $K$ is fibred. The proof relies …

In classical field theory, the composite fibred manifolds Y -> Z -> X provides the adequate mathematical formulation of gauge models with broken symmetries, e.g., the gauge gravitation theory. This work is devoted to connections on composite fibred manifolds. In particular, we get the horizontal splitting of the vertic…

Classifies contact structures on negative-definite Seifert fibred spaces.

The triangulation complexity of a closed orientable 3-manifold is the minimal number of tetrahedra in any triangulation of the manifold. The main theorem of the paper gives upper and lower bounds on the triangulation complexity of any closed orientable hyperbolic 3-manifold that fibres over the circle. We show that the…

Study shows constraints on slopes for knot manifolds with specific tori.

The Milnor fibre of any isolated hypersurface singularity contains many exact Lagrangian spheres: the vanishing cycles associated to a Morsification of the singularity. Moreover, for simple singularities, it is known that the only possible exact Lagrangians are spheres. We construct exact Lagrangian tori in the Milnor …

We study the behaviour of Laplace-type operators H on a complex vector bundle E $\rightarrow$ M in the adiabatic limit of the base space. This space is a fibre bundle M $\rightarrow$ B with compact fibres and the limit corresponds to blowing up directions perpendicular to the fibres by a factor 1/$ε$. Under a gap condi…