arXiv research

Holomorphic handle attaching proves complex surface properties.

Study of 3-handle attachments in 4D manifolds using Kirby calculus.

Defines extended TQFTs using handle attachments.

J. Boyle classified 1-handles attached to surface-knots, that are closed and connected surfaces embedded in the Euclidean 4-space, in the case that the surfaces are oriented and 1-handles are orientable with respect to the orientations of the surfaces. In that case, the equivalence classes of 1-handles correspond to th…

Formulae for 1-3 handle attachments in 4-manifolds.

We review Giroux's contact handles and contact handle attachments in dimension three and show that a bypass attachment consists of a pair of contact 1 and 2-handles. As an application we describe explicit contact handle decompositions of infinitely many pairwise non-isotopic overtwisted 3-spheres. We also give an alter…

A standard convexity condition on the boundary of a symplectic manifold involves an induced positive contact form (and contact structure) on the boundary; the corresponding concavity condition involves an induced negative contact form. We present two methods of symplectically attaching 2-handles to convex boundaries of…

We prove that pseudo-holomorphic discs attached to a maximal totally real submanifold inherit their regularity from the regularity of the submanifold and of the almost complex structure. The proof is based on the computation of an explicit lower bound for the Kobayashi metric in almost complex manifolds, which also yie…

We prove a localization formula for a "holomorphic equivariant cohomology" attached to the Atiyah algebroid of an equivariant holomorphic vector bundle. This generalizes Feng-Ma, Carrell-Liebermann, Baum-Bott and K. Liu's localization formulas.

For spin manifolds with boundary we consider Riemannian metrics which are product near the boundary and are such that the corresponding Dirac operator is invertible when half-infinite cylinders are attached at the boundary. The main result of this paper is that these properties of a metric can be preserved when the met…

We prove that a crossing change along a double point circle on a 2-knot is realized by ribbon-moves for a knotted torus obtained from the 2-knot by attaching a 1-handle. It follows that any 2-knots for which the crossing change is an unknotting operation, such as ribbon 2-knots and twist-spun knots, have trivial Khovan…

Paper introduces tensor product of quandles for knot classification.

We prove generic fibre of Painlevé moduli spaces are Weinstein handlebodies.

We prove that knots obtained by attaching a band to a split link satisfy the cabling conjecture. We also give new proofs that unknotting number one knots are prime and that genus is superadditive under band sum. Additionally, we prove a collection of results comparing two 2-handle additions to a genus two boundary comp…

We show that the smooth $4$-manifold $M$ obtained by attaching a $2$-handle to $B^4$ along a certain knot $K\subset \partial B^4$ admits infinitely many absolutely exotic copies $M_n$, $n=0,1,2..$, such that each copy $M_n$ is obtained by attaching $2$-handle to a fixed compact smooth contractible manifold $W$ along th…

We extend the spectral theory of generalized Laplacians to integrable metrics on compact Riemann surfaces. As a consequence, we attach in a direct way, a holomorphic analytic torsion to any integrable metrics. We also provide a different approach to define the holomorphic analytic torsion. We prove that both approaches…

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

In this paper we give explicit, handle-by-handle constructions of concave symplectic fillings of all closed, oriented contact 3-manifolds. These constructions combine recent results of Giroux relating contact structures and open book decompositions of 3-manifolds, earlier results of the author on attaching 4-dimensiona…

We compactify the classical moduli variety $A_g$ of principally polarized abelian varieties of complex dimension $g$ by attaching the moduli of flat tori of real dimensions at most $g$ in an explicit manner. Equivalently, we explicitly determine the Gromov-Hausdorff limits of principally polarized abelian varieties. Th…

We show that the first eigenvalue of a closed Riemannian surface normalized by the area can be strictly increased by attaching a cylinder or a cross cap. As a consequence we obtain the existence of maximizing metrics for the normalized first eigenvalue on any closed surface of fixed topological type. Since these metric…

Proves finiteness for 3-manifolds from handlebodies with 2-handles.

We give a definition of symplectic homology for pairs of filled Liouville cobordisms, and show that it satisfies analogues of the Eilenberg-Steenrod axioms except for the dimension axiom. The resulting long exact sequence of a pair generalizes various earlier long exact sequences such as the handle attaching sequence, …

On a compact Kähler manifold, we introduce a notion of almost nonpositivity for the holomorphic sectional curvature, which by definition is weaker than the existence of a Kähler metric with semi-negative holomorphic sectional curvature. We prove that a compact Kähler manifold of almost nonpositive holomorphic sectional…

Let M be a closed, irreducible, genus two 3-manifold, and F a maximal collection of pairwise disjoint, closed, orientable, incompressible surfaces embedded in M. Then each component manifold M_i of M-F has handle number at most one, i.e. admits a Heegaard splitting obtained by attaching a single 1-handle to one or two …

We emphasize some properties of coherent state groups, i.e. groups whose quotient with the stationary groups, are manifolds which admit a holomorphic embedding in a projective Hilbert space. We determine the differential action of the generators of the representation of coherent state groups on the symmetric Fock space…

Attaching a 2-handle to a genus two or greater boundary component of a 3-manifold is a natural generalization of Dehn filling a torus boundary component. We prove that there is an interesting relationship between an essential surface in a sutured 3-manifold, the number of intersections between the boundary of the surfa…

Let $M$ be a simple manifold, and $F$ be a component of $\partial M$ of genus two. For a slope $γ$ on $F$, we denote by $M(γ)$ the manifold obtained by attaching a 2-handle to $M$ along a regular neighborhood of $γ$ on $F$. In this paper, we shall prove that there is at most one separating slope $γ$ on $F$ so that $M(γ…

The paper simplifies embedding spaces in manifolds by attaching handles.

We show that the pre-order defined on the category of contact manifolds by arbitrary symplectic cobordisms is considerably less rigid than its counterparts for exact or Stein cobordisms: in particular, we exhibit large new classes of contact 3-manifolds which are symplectically cobordant to something overtwisted, or to…

A new method converts knot Floer homology to immersed curves.

We construct a small, hyperbolic 3-manifold $M$ such that, for any integer $g\geq 2$, there are infinitely many separating slopes $r$ in $\partial M$ so that $M(r)$, the 3-manifold obtained by attaching a 2-handle to $M$ along $r$, is hyperbolic and contains an essential separating closed surface of genus $g$. The resu…

The paper calculates a new invariant for 4-manifolds using handle decompositions and skein relations.

For the oriented 3-dimensional handlebody constructed from a 3-ball by attaching g 1-handles, it is shown that the natural surjection from the group of orientation preserving diffeomorphisms of it to the mapping class group of it has no section when g is at least 6.

We propose a global invariant $σ_c$ for contact manifolds which admit a strictly pseudoconvex CR structure, analogous to the Yamabe invariant $σ$. We prove that this invariant is non-decreasing under handle attaching and under connected sum. We then give a lower bound on $σ_c$ in a particular case.

Two codimension-one submanifolds are cobordant if they have the same homology class.

For a smoothing Y of a 2-dimensional cyclic quotient singularity X, we construct a simple handle decomposition of Y by using a particular birational map from Y to the projective plane. The manifold Y is built up from the product of an annulus with a disk by attaching 2-handles in a manner which can be described by mean…

We give series of explicit examples of Levi-nondegenerate real-analytic hypersurfaces in complex spaces that are not transversally holomorphically embeddable into hyperquadrics of any dimension. For this, we construct invariants attached to a given hypersurface that serve as obstructions to embeddability. We further st…

The paper studies invariants of surfaces in the 3-sphere using handlebody-links.

We construct a hyperbolic 3-manifold $M$ (with $\partial M$ totally geodesic) which contains no essential closed surfaces, but for any even integer $g> 0$ there are infinitely many separating slopes $r$ on $\partial M$ so that $M[r]$, the 3-manifold obtained by attaching 2-handle to $M$ along $r$, contains an essential…

We prove the existence of complete, embedded, constant mean curvature 1 surfaces in 3 dimensional hyperbolic space when g, the genus of the surface, and n, the number of ends of the surface, satisfy either g=0 and $n\geq 1$ or $g \geq 1$ and $2n\geq g+5$. The surfaces are all regular points of their corresponding modul…

New method constructs non-arithmetic hyperbolic orbifolds from complex arithmetic ball quotients.

Let $H$ be a strongly irreducible Heegaard surface in a closed oriented Riemannian $3$-manifold. We prove that $H$ is either isotopic to a minimal surface of index at most one or isotopic to the boundary of a tubular neighborhood about a non-orientable minimal surface with a vertical handle attached. This confirms a lo…

For all $m \in \mathbb N - \{0\}$, we prove the existence of a one dimensional family of genus $m$, constant mean curvature (equal to 1) surfaces which are complete, immersed in $\mathbb R^3$ and have two Delaunay ends asymptotic to nodoïdal ends. Moreover, these surfaces are invariant under the group of isometries of …

The study distinguishes exotic R^4's using knot Floer homology and Casson handles.

We prove a homological stability theorem for moduli spaces of manifolds of dimension $2n$, for attaching handles of index at least $n$, after these manifolds have been stabilised by countably many copies of $S^n \times S^n$. Combined with previous work of the authors, we obtain an analogue of the Madsen--Weiss theorem …

The generalized coherent states attached to the Jacobi group realize the squeezed states. Imposing hermitian conjugacy to the generators of the Jacobi algebra, we find out the form of the weight function appearing in the scalar product. We show effectively the orthonormality of the base functions with respect to the sc…

Study Weinstein structures on toric divisors' complements.

New findings about twists in 4-sphere diffeomorphisms.