arXiv research

Formula connects surgeries to Seiberg-Witten invariants.

Two approaches study the homotopy of blow ups in algebraic and symplectic geometry.

We review the geometric theory of \emp{smooth systems of smooth maps}, of \emp{smooth systems of smooth sections} of a smooth double fibred manifold and of \emp{smooth systems of smooth connections} of a smooth fibred manifold. Moreover, after reviewing the concept of \emp{F-smooth space} due to A. Frölicher, we discus…

We show that deformations of a coisotropic submanifold inside a fibrewise entire Poisson manifold are controlled by the $L_\infty$-algebra introduced by Oh-Park (for symplectic manifolds) and Cattaneo-Felder. In the symplectic case, we recover results previously obtained by Oh-Park. Moreover we consider the extended de…

Unique optimal symplectic connections found for submersions.

Constructs small bundle gerbes and proves index theorems for manifolds.

Study of torsion forms for positive line bundles.

Given a holomorphic vector bundle $E$ on the twistor space $\mathrm{Tw}(M)$ of a simple hyperkähler manifold $M$, we view it as a family of bundles $\left\{E_I\right\}$ on the fibres $π^{-1}(I)$ of the twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$, and study the relationship between stability of $E$ and its …

We calculate the eta-invariant for the odd signature operator relative to a specific submersion metric on the Milnor fibration of a quasihomogeneous hypersurface singularity using certain global boundary conditions in terms of the data of its fibre intersection form, monodromy and variation mapping resp. the monomial d…

We describe a construction of fibrewise inner products on the cotangent bundle of the smooth free loop space of a Riemannian manifold. Using this inner product, we construct an operator over the loop space of a string manifold which is directly analogous to the Dirac operator of a spin manifold.

This paper studies the formal adiabatic limit of coassociative K3 fibred torsion free $G_2$ manifolds fibred over a contractible base, shows how to put this structure on a different fibration obtained by fibrewise performing Mukai duality of K3 surfaces, and furthermore relates the gauge theories on both fibrations by …

The paper studies distributions on surfaces and their connection to twistor spaces.

We study $n$-dimensional Kähler manifolds whose geodesic flows possess $n$ first integrals in involution that are fibrewise hermitian forms and simultaneously normalizable. Under some mild assumption, one can associate with such a manifold an $n$-dimensional commutative Lie algebra of infinitesimal automorphisms. This,…

Let xi be a smooth oriented vector bundle, with n-dimensional fibre, over a smooth manifold M. Denote by xi-hat the fibrewise one-point compactification of xi. The main purpose of this paper is to define geometrically a canonical element Upsilon(xi) in H^n(xi-hat,Q) (H^n(xi-hat,Z) tensor 1/2, to be more precise). The e…

We construct an explicit example of a stable bundle on the twistor space $\mathrm{Tw}(M)$ of a hyperkähler manifold $M$ whose restrictions to all the fibres of the natural twistor projection $π: \mathrm{Tw}(M) \to \mathbb{CP}^1$ are nonstable. We also describe the relationship between bundles on $\mathrm{Tw}(M)$ that d…

Generalised spin structures, or r-spin structures, on a 2-dimensional orbifold Σare r-fold fibrewise connected coverings (also called r-th roots) of its unit tangent bundle STΣ. We investigate such structures on hyperbolic orbifolds. The conditions on r for such structures to exist are given. The action of the diffeomo…

Develops moment map theory for twisted scalar curvature in Kähler geometry.

In this paper, we use the parametrised strict deformation quantization of C*-bundles obtained in a previous paper, and give more examples and applications of this theory. In particular, it is used here to classify H_3-twisted noncommutative torus bundles over a locally compact space. This is extended to the case of gen…

We consider structures analogous to symplectic Lefschetz pencils in the context of a closed 4-manifold equipped with a `near-symplectic' structure (ie, a closed 2-form which is symplectic outside a union of circles where it vanishes transversely). Our main result asserts that, up to blowups, every near-symplectic 4-man…

Let K be a knot in the 3--sphere. An r-surgery on K is left-orderable if the resulting 3--manifold K(r) of the surgery has left-orderable fundamental group, and an r-surgery on K is called an L-space surgery if K(r) is an L-space. A conjecture of Boyer, Gordon and Watson says that non-reducing surgeries on K can be cla…

Study confirms contact cosmetic surgery for most knots, with exceptions.

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

Cosmetic surgeries on pretzel knots are unique.

Study pochette surgery on 4-manifolds, focusing on 4-spheres.

New method proves cosmetic surgery conjecture for certain knots.

Diffeology explores $k$-forms and bundles with more information than traditional differential forms.

New Heegaard Floer homology findings block chirally cosmetic surgeries.

Round surgery diagrams represent 3-manifolds in $S^3$.

The study calculates and analyzes alternating surgeries for various knots.

The paper constructs homotopy 4-spheres using pochette surgery.

Contact round surgeries on $(\mathbb{S}^3,ξ_{st})$ help in constructing and understanding contact 3-manifolds.

This paper concerns the truly or purely cosmetic surgery conjecture. We give a survey on exceptional surgeries and cosmetic surgeries. We prove that the slope of an exceptional truly cosmetic surgery on a hyperbolic knot in $S^3$ must be $\pm 1$ and the surgery must be toroidal but not Seifert fibred. As consequence we…

Defines contact surgery distance and shows it's bounded by topological surgery distance by 5.

We show that all exceptional surgeries on hyperbolic alternating knots in the 3-sphere are integral surgeries.

A Seifert surgery is an integral surgery on a knot in S^3 producing a Seifert fiber space which may contain an exceptional fiber of index 0. The Seifert Surgery Network is a 1-dimensional complex whose vertices correspond to Seifert surgeries; its edges correspond to single twistings along "seiferters" or "annular pair…

New proof for a knot type not admitting certain surgeries.

Study finds chirally cosmetic surgeries on knots and manifolds, contradicting previous conjectures.

New surgery operation preserves monotonicity of Lagrangians.

New insights into cosmetic surgeries using Heegaard Floer homology.

New surgeries found in 3D shapes without 2-spheres.

A Seifert surgery is a pair (K, m) of a knot K in the 3-sphere and an integer m such that m-Dehn surgery on K results in a Seifert fiber space allowed to contain fibers of index zero. Twisting K along a trivial knot called a seiferter for (K, m) yields Seifert surgeries. We study Seifert surgeries obtained from those o…

The paper defines and studies contact surgery numbers for contact 3-manifolds.

We study chirally cosmetic surgeries, that is, a pair of Dehn surgeries on a knot producing homeomorphic 3-manifolds with opposite orientations. Several constraints on knots and surgery slopes to admit such surgeries are given. Our main ingredients are the original and the $SL(2,\mathbb{C})$ version of Casson invariant…

Two knot families meet cosmetic surgery conjecture.

Extends knot surgery to exotic four-manifolds.

Study exact surgery formula in involutive Heegaard Floer homology.

The paper develops surgery theories for foliations and solves a problem posed by Weinberger.

The study examines quasi-alternating surgeries on knots and their properties.