arXiv research

Study shows no smooth embeddings of rational homology balls into complex projective plane.

The paper constructs new rational homology 3-spheres bounding rational homology 4-balls.

Study shows surgeries on certain knots bound rational homology 4-balls.

New 3-manifolds bound rational 4-balls through specific operations.

We present complete classifications of links in the 3-sphere modulo framed and twisted Whitney towers in a rational homology 4-ball. This provides a geometric characterization of the vanishing of the Milnor invariants of links in terms of Whitney towers. Our result also says that the higher order Arf invariants, which …

The study classifies $χ-$slice pretzel links and Seifert fiber spaces.

We give a complete classification of the spherical 3-manifolds that bound smooth rational homology 4-balls. Furthermore, we determine the order of spherical 3-manifolds in the rational homology cobordism group of rational homology 3-spheres. To this end, we use constraints for 3-manifolds to bound rational homology bal…

Classifies torus bundles bounding 4-manifolds with rational homology.

New knots not slice in rational 4-balls found.

New balls smoothly fit in CP² but not symplectically.

We generalise theorems of Khodorovskiy and Park-Park-Shin, and give new topological proofs of those theorems, using embedded surfaces in the 4-ball and branched double covers. These theorems exhibit smooth codimension-zero embeddings of certain rational homology balls bounded by lens spaces.

This paper describes a method to construct standard 4-balls from homotopy 4-balls in $\mathbb{C}^2$.

Paper proves triple linking form vanishes under specific conditions.

We show that for rational surface singularities with odd determinant the mu-bar invariant defined by W. Neumann is an obstruction for the link of the singularity to bound a rational homology 4-ball. We identify the mu-bar invariant with the corresponding correction term in Heegaard Floer theory.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

New Stein fillings found for non-weighted homogeneous singularities.

The paper examines rational homology spheres that admit special generic maps into Euclidean spaces.

The smooth rational homology cobordism group of rational homology three spheres, T, contains subgroups T_p generated by 3-manifolds with first homology p-torsion, where p is a prime. Rochlin's theorem and gauge theoretic methods show that the inclusion of the direct sum of the T_p into T has infinitely generated kernel…

Given a spin$^c$ rational homology sphere $(Y,\mathfrak{s})$ with $\mathfrak{s}$ self-conjugate and for which the reduced monopole Floer homology $\mathit{HM}_{\bullet}(Y,\mathfrak{s})$ has rank one, we provide obstructions to the intersection forms of its Stein fillings which are not negative definite. The proof of th…

In this article, using combinatorial techniques of mapping class groups, we show that a Stein fillable integral homology $3$-sphere supported by an open book decomposition with page a $4$-holed sphere admits a unique Stein filling up to diffeomorphism. Furthermore, according to a property of deforming symplectic fillin…

The article classifies cubiquitous sublattices and applies them to branched covers.

Let L be a nonunimodular definite lattice. Using a theorem of Elkies we show that whether L embeds in the standard definite lattice of the same rank is completely determined by a collection of lattice correction terms, one for each metabolizing subgroup of the discriminant group. As a topological application this gives…

We consider a homology sphere $M_n(K_1,K_2)$ presented by two knots $K_1,K_2$ with linking number 1 and framing $(0,n)$. We call the manifold {\it Matsumoto's manifold}. We show that there exists no contractible bound of $M_n(T_{2,3},K_2)$ if $n<2τ(K_2)$ holds. We also give a formula of Ozsváth-Szabó's $τ$-invariant as…

We show that there are contact 3-manifolds of support genus one which admit infinitely many Stein fillings, but do not admit arbitrarily large ones. These Stein fillings arise from genus-1 allowable Lefschetz fibrations with distinct homology groups, all filling a fixed minimal genus open book supporting the boundary c…

New techniques in Khovanov homology help distinguish exotic surfaces in 4-ball.

Paper tackles which 3-spheres bound contractible 4-manifolds or homology 4-balls.

We study the topology of exact and Stein fillings of the canonical contact structure on the unit cotangent bundle of a closed surface $Σ_g$, where $g$ is at least 2. In particular, we prove a uniqueness theorem asserting that any Stein filling must be s-cobordant rel boundary to the disk cotangent bundle of $Σ_g$. For …

Let $LHT$ be a left handed trefoil knot and $K$ be any knot. We define $M_n(K)$ to be the homology $3$-sphere which is represented by a simple link of $LHT$ and $LHT \sharp K$ with framings $0$ and $n$ respectively. Starting with this link, we construct homotopy $K3$ and spin rational homology $K3$ surfaces containing …

Study on contact type hypersurfaces in 4-space, proving no Brieskorn spheres can embed.

Classifies knots that bound equivariant surfaces with free symmetries.

The paper calculates homology and intersection pairing of branched covers using disoriented homology.

Study embeds ruled surfaces into symplectic manifolds, finds Stein fillability results.

We exhibit a knot $P$ in the solid torus, representing a generator of first homology, such that for any knot $K$ in the 3-sphere, the satellite knot with pattern $P$ and companion $K$ is not smoothly slice in any homology 4-ball. As a consequence, we obtain a knot in a homology 3-sphere that does not bound a piecewise-…

New spanning tree model connects knot homology, s-invariant, and exotic discs.

This paper investigates the symplectic and contact topology associated to circular spherical divisors. We classify, up to toric equivalence, all concave circular spherical divisors $D$ that can be embedded symplectically into a closed symplectic 4-manifold and show they are all realized as symplectic log Calabi-Yau p…

We prove symplectic hypersurfaces in Weinstein domains and give obstructions for manifold boundaries.

For an integer $n$, write $X_n(K)$ for the 4-manifold obtained by attaching a 2-handle to the 4-ball along the knot $K\subset S^3$ with framing $n$. It is known that if $n< \overline{\text{tb}}(K)$, then $X_n(K)$ admits the structure of a Stein domain, and moreover the adjunction inequality implies there is an upper bo…

New Stein fillings found for rational surface singularities.

Whitehead link surgeries are not L-spaces if they support taut foliations.

We study lens space surgeries along two different families of 2-component links, denoted by $A_{m,n}$ and $B_{p,q}$, related with the rational homology 4-ball used in J.\ Park's (generalized) rational blow down. We determine which coefficient $r$ of the knotted component of the link yields a lens space by Dehn surgery.…

Using 1-twist rim surgery, we construct infinitely many smoothly embedded, orientable surfaces in the 4-ball bounding a knot in the 3-sphere that are pairwise topologically isotopic, but not ambient diffeomorphic. We distinguish the surfaces using the maps they induce on perturbed sutured Floer homology. Along the way,…

We compute the Heegaard Floer homology of any rational homology 3-sphere with an open book decomposition of the form (T,φ), where T is a genus one surface with one boundary component. In addition, we compute the Heegaard Floer homology of any T^2-bundle over S^1 with first Betti number equal to one, and we compare our …

The paper computes a tau-invariant for holomorphic curves in Stein domains and links.

New Seifert surfaces in 4-ball differ even when pushed in.

We use contact handle decompositions and a stabilization process to compute the cylindrical contact homology of a subcritical Stein-fillable contact manifold with vanishing first Chern class, and show that it is completely determined by the homology of a subcritical Stein-filling of the contact manifold.

The study of symplectic fillings for rational cuspidal curves.

Characterizes Stein surfaces with finite homotopy rank-sum.

Legendrian surgery describes canonical contact structures and calculates Gompf's θ-invariant.