arXiv research

Classifies SL(2;C) representations of a Brieskorn homology 3-sphere.

Smooth actions on certain 3-spheres can't extend to acyclic 4-manifolds.

In this paper we answer the question posed by M.~Atiyah and give an explicit formula for Floer homology of Brieskorn homology spheres in terms of their branching sets over the 3--sphere. We further show how Floer homology is related to other invariants of knots and 3--manifolds, among which are the $\barμ$--invariant o…

There are known infinite families of Brieskorn homology 3-spheres which can be realized as boundaries of smooth contractible 4-manifolds. In this paper we show that free periodic actions on these Brieskorn spheres do not extend smoothly over a contractible 4-manifold. We give a new infinite family of examples in which …

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

The paper proves new applications of knot invariants and smooth group actions on 3-spheres.

Survey of homology cobordism group with open problems.

New method shows some 3D shapes can't be filled in certain ways.

Study Brieskorn spheres using Floer homology, generating infinite rank summands in homology cobordism.

New families of Brieskorn spheres bound rational homology balls.

New Brieskorn spheres found to bound rational homology balls but not integral ones.

Using Seiberg-Witten Floer spectrum and Pin(2)-equivariant KO-theory, we prove new Furuta-type inequalities on the intersection forms of spin cobordisms between homology $3$-spheres. As an application, we give explicit constrains on the intersection forms of spin $4$-manifolds bounded by Brieskorn spheres $\pmΣ(2,3,6k\…

New invariant computed for Brieskorn homology spheres.

We define invariants $\frak{ds}$ and $\overline{\frak{ds}}$, which are the maximal and minimal second Betti number divided by $8$ among definite spin boundings of a homology sphere. The similar invariants $g_8$ and $\overline{g_8}$ are defined by the maximal (or minimal) product sum of $E_8$-form of bounding 4-manifold…

The study confirms the non-existence of rational homology ball symplectic fillings for certain Brieskorn spheres.

We compute the Heegaard-Floer homology for the family Sigma(2,3,6n+1) of Brieskorn spheres using the algorithm by P. Ozsvath and Z. Szabo

Refines Ozsváth-Szabó d-invariants for knot concurrence.

The paper identifies and illustrates families of knot diagrams yielding lens spaces from various homology spheres.

Cylindrical contact homology studied for a specific class of 3-manifolds.

Study volume growth in Milnor fibers using real Lagrangians.

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

In this paper, we give a parameterization of the SU(2,1) representation space of the Brieskorn homology spheres using the trace coordinates. As applications, we give an example which shows that the orbifold Toledo invariant in \cite{krebs} does not distinguish the connected components of the PU(2,1) representation spac…

Ozsváth and Szabó gave a combinatorial description for the Heegaard Floer homology of boundaries of certain negative-definite plumbings. Némethi constructed a remarkable algorithm for executing these computations for almost-rational plumbings, and his work gives a formula computing the invariants for the Brieskorn homo…

Constructs families of monotone Lagrangians in Brieskorn-Pham hypersurfaces.

Study obstructs symplectic structures on Mazur manifolds.

Let S^3_r(K) be the oriented 3--manifold obtained by rational r-surgery on a knot K in S^3. Using the contact Ozsvath-Szabo invariants we prove, for a class of knots K containing all the algebraic knots, that S^3_r(K) carries positive, tight contact structures for every r not= 2g_s(K)-1, where g_s(K) is the slice genus…

In math.DG/9903083 (henceforth referred to as EA) we defined an integer invariant $h(Y)$ for oriented integral homology 3-spheres $Y$ which only depends on the rational homology cobordism class of $Y$ and is additive under connected sums. In this paper we establish lower bounds for $h(Y)$ when $Y$ is the boundary of a …

Research confirms Gompf's conjecture for some Brieskorn spheres and finds a new 4-manifold without Stein structure.

Study contact structures and Sasakian geometry on manifolds.

We develop techniques for computing the integer valued SU(3) Casson invariant. Our method involves resolving the singularities in the flat moduli space using a twisting perturbation and analyzing its effect on the topology of the perturbed flat moduli space. These techniques, together with Bott-Morse theory and the spl…

New rational homology 3-spheres found that can't bound definite 4-manifolds.

New examples of Kirby-Ramanujam spheres found, leading to contractible 4-manifolds.

It has long been known that every quasi-homogeneous normal complex surface singularity with Q-homology sphere link has universal abelian cover a Brieskorn complete intersection singularity. We describe a broad generalization: First, one has a class of complete intersection normal complex surface singularities called "s…

In this survey, we give an overview of Brieskorn manifolds and varieties, and their role in contact topology. We discuss open books, fillings and invariants such as contact and symplectic homology. We also present some new results involving exotic contact structures, invariants and orderability. The main tool for the r…

This paper concerns the problem of existence of taut foliations among 3-manifolds. Since the contribution of David Gabai, we know that closed 3-manifolds with non-trivial second homology group admit a taut foliations. The essential part of this paper focuses on Seifert fibered homology 3-spheres. The result is quite di…

Study on 3-sphere Goeritz group's twisted first homology group.

New coverings of lens spaces identified from Brieskorn manifolds and Sieradski groups.

We show that a graph manifold which is a Z-homology 3-sphere not homeomorphic to either the 3-sphere or the Poincaré homology 3-sphere admits a horizontal foliation. This combines with known results to show that the conditions of not being an L-space, of having a left-orderable fundamental group, and of admitting a co-…

The article provides formulas for homological blocks of Seifert fibered homology 3-spheres.

Study finite type invariants for knots in rational homology 3-spheres.

The paper examines conditions for contact surgeries on rational homology 3-spheres.

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

Resurgence analysis of $SU(2)$ Chern-Simons on a specific homology sphere.

Paper finds infinitely many 3-spheres without nice orderings.

Study invariants of $\mathbb{Z}/p$-homology 3-spheres from abelianization of mapping class groups.

We show that the perturbative ${\frak g}$ invariant of rational homology 3-spheres can be recovered from the LMO invariant for any simple Lie algebra ${\frak g}$, i.e, the LMO invariant is universal among the perturbative invariants. This universality was conjectured in [25]. Since the perturbative invariants dominate …

Identifies a mod-$p$ triple cup product for rational homology 3-spheres with specific first homology.

New invariant fully describes finite type invariants of knots in homology 3-spheres.