arXiv research

We study connected sum at infinity on smooth, open manifolds. This operation requires a choice of proper ray in each manifold summand. In favorable circumstances, the connected sum at infinity operation is independent of ray choices. For each m at least 3, we construct an infinite family of pairs of m-manifolds on whic…

Proofs knot homology connected sums using grid complexes.

We show that for a suitable class of ``Dirac-like'' operators there holds a Gluing Theorem for connected sums. More precisely, if $M_1$ and $M_2$ are closed Riemannian manifolds of dimension $n\ge 3$ together with such operators, then the connected sum $M_1 # M_2$ can be given a Riemannian metric such that the spectrum…

We give inequalities for the Manolescu invariants $α,β,γ$ under the connected sum operation. We compute the Manolescu invariants of connected sums of some Seifert fiber spaces. Using these same invariants, we provide a proof of Furuta's Theorem, the existence of a $\mathbb{Z}^\infty$ subgroup of the homology cobordism …

The goal of the paper is to calculate the limit sectrum of the Hodge-Laplace operator under the perturbation of collapse of one part of a connected sum. This gives some new results concerning the 'conformal spectrum' on differential forms.

We study the behavior of Legendrian and transverse knots under the operation of connected sums. As a consequence we show that there exist Legendrian knots that are not distinguished by any known invariant. Moreover, we classify Legendrian knots in some non-Legendrian simple knot types.

Developed Gompf connected sum for orbifolds, constructing symplectic and K-contact manifolds.

The article provides formulas for the number of terms in connected sums of sphere products associated with dual-neighborly polytopes.

We record an answer to the question "In which dimensions is the connected sum of two closed almost complex manifolds necessarily an almost complex manifold?". In the process of doing so, we are naturally led to ask "For which values of l is the connected sum of l closed almost complex manifolds necessarily an almost co…

New constructions in Legendrian embeddings space yield novel invariants.

The paper studies relationships between twisted torsion and connected sums of 3-manifolds.

Let $N$ be a closed enlargeable manifold in the sense of Gromov-Lawson and $M$ a closed spin manifold of equal dimension, a famous theorem of Gromov-Lawson states that the connected sum $M\# N$ admits no metric of positive scalar curvature. We present a potential generalization of this result to the case where $M$ is n…

In this note we take some initial steps in the investigation of a fourth order analogue of the Yamabe problem in conformal geometry. The Paneitz constants and the Paneitz invariants considered are believed to be very helpful to understand the topology of the underlined manifolds. We calculate how those quantities chang…

Study bounds Urysohn width of manifolds under surgeries.

The Lusternik-Schnirelmann category and topological complexity are important invariants of manifolds (and more generally, topological spaces). We study the behavior of these invariants under the operation of taking the connected sum of manifolds. We give a complete answer for the LS-categoryof orientable manifolds, $\c…

The paper studies how knots and links behave under connected sum operations.

Whyte used the index theory of Dirac operators and Block-Weiberger uniformly finite homology to show that certain infinite connected sums do not carry a metric with nonnegative scalar curvature in their bounded geometry class. His proof uses a coarse version of the $\hat{A}$-class to obstruct such metrics. In this note…

New findings on knot operations challenge a long-standing conjecture.

The main theorem describes the behaviour of the stable cohomotopy invariant defined in the first article (joint with M. Furuta) in this series of two under the operation of taking connected sums of four-manifolds: The invariant of a connected sum is the smash product (in the sense of equivariant spectra) of the invaria…

Characterizes homology d-manifolds with g2=3 for d≥3.

Normal 4-pseudomanifolds with one or two singular vertices are derived from specific operations.

Given a link L in the 3-sphere, we ask whether the components of L bound disjoint, nullhomologous disks properly embedded in a simply-connected positive-definite smooth 4-manifold; the knot case has been studied extensively in work of Cochran-Harvey-Horn. Such a 4-manifold is necessarily homeomorphic to a (punctured) c…

In this article, we consider the maximal value of the Thurston--Bennequin invariant of Legendrian knots which topologically represent a fixed knot type in the standard contact 3-space and we prove a formula of the value under the connected sum operation of knots.

Connected sum affects crossing numbers of flat virtual knots.

We establish a general `gluing theorem', which states roughly that if two nondegenerate constant mean curvature surfaces are juxtaposed, so that their tangent planes are parallel and very close to one another, but oppositely oriented, then there is a new constant mean curvature surface quite near to this configuration …

In the symplectic category there is a `connect sum' operation that glues symplectic manifolds by identifying neighborhoods of embedded codimension two submanifolds. This paper establishes a formula for the Gromov-Witten invariants of a symplectic sum Z=X#Y in terms of the relative GW invariants of X and Y. Several appl…

New flag-no-square 4-manifolds discovered with unique triangulations.

Classifies exceptional Legendrian realizations of Hopf link connected sums.

Extends Jones' construction to map Thompson group to pointed links.

We show that a band-connected sum of knots $K_0$ and $K_1$ along a band $b$ is equal to the connected sum $K_0\# K_1$ if and only if $b$ is a trivial band.

Knots' Morse-Novikov number behaves additively under connected sum and unchanged by cabling.

New method proves Jones Polynomial's connect sum property.

Minimal crystallizations of simply connected PL 4-manifolds are very natural objects. Many of their topological features are reflected in their combinatorial structure which, in addition, is preserved under the connected sum operation. We present a minimal crystallization of the standard PL K3 surface. In combination w…

Contact connected sums do not increase support genus.

We derive constraints on Lagrangian embeddings in completions of certain stable symplectic fillings with semisimple symplectic cohomologies. Manifolds with these properties can be constructed by generalizing the boundary connected sum operation to our setting, and are related to certain birational surgeries like blow-d…

The paper connects diffeomorphism groups and sphere embeddings, proving a group structure.

Proves a general connected sum formula for families Seiberg-Witten invariants.

Study satellite operations on knot invariant θ, proving additivity and distinguishing knots.

The study characterizes homology 4-manifolds with $g_2\leq 5$ combinatorially.

The purpose of this paper is to introduce Liouville hypersurfaces in contact manifolds, which generalize ribbons of Legendrian graphs and pages of supporting open books. Liouville hypersurfaces are used to define a gluing operation for contact manifolds called the Liouville connect sum. Performing this operation on a c…

A small cover was introduced by Davis and Januszkiewicz as an $n$-dimensional closed manifold with a locally standard $Z_2)^n$-action such that its orbit space is a simple convex polytope. There exist a one-to-one correspondence between small covers and $(Z_2)^n$-colored polytopes. In this paper we study a construction…

Genus 3 Heegaard groups of lens space connected sums are finitely generated.

Researchers solved a conjecture about a mathematical structure of connected sums of solid tori.

This paper is a survey of some of the most elementary consequences of the JSJ-decomposition and geometrization for knot and link complements in the 3-sphere. Formulated in the language of graphs, the result is the construction of a bijective correspondence between the isotopy classes of links in $S^3$ and a class of ve…

We show that essential punctured spheres in the complement of links with distance three bridge spheres have bounded complexity. We define the operation of tangle product, a generalization of both connected sum and Conway product. Finally, we use the bounded complexity of essential punctured spheres to show that the bri…

New hyperbolic knots with convex Upsilon invariants constructed.

We prove a formula for the conjugation action on the knot Floer complex of the connected sum of two knots. Using the formula we construct a homomorphism from the smooth concordance group to an abelian group consisting of chain complexes with homotopy automorphisms, modulo an equivalence relation. Using our connected su…

Classifies multigraphs for torus actions on 6D manifolds with isolated fixed points.