arXiv research

New families of Brieskorn spheres bound rational homology balls.

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

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

New invariant computed for Brieskorn homology spheres.

We classify SL(2;C)-representations of a Brieskorn homology 3-sphere. We show any irreducible representation into SL(2;C) is conjugate to that into either SU(2) or SL(2;R). We also give a construction of SL(2;R)-representations for a Brieskorn homology 3-sphere from PSL(2;R)-representations of the base orbifold fundame…

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

It is known by the author that there exist 20 families of Dehn surgeries in the Poincaré homology sphere yielding lens spaces. In this paper, we give the concrete knot diagrams of the families and extend them to families of lens space surgeries in Brieskorn homology spheres. We illustrate families of lens space surgeri…

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…

Fintushel and Stern showed that the Brieskorn sphere $Σ(2,3,7)$ bounds a rational homology ball, while its non-trivial Rokhlin invariant obstructs it from bounding an integral homology ball. It is known that their argument can be modified to show that the figure-eight knot is rationally slice, and we use this fact to p…

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…

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

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…

Study obstructs symplectic structures on Mazur 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…

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

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 …

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…

A conjecture due to Gompf asserts that no nontrivial Brieskorn homology sphere admits a pseudoconvex embedding in ${\mathbb C}^2$, with either orientation. A related question asks whether every compact contractible 4-manifold admits the structure of a Stein domain. We verify Gompf's conjecture, with one orientation, fo…

We perform a resurgence analysis of the $SU(2)$ Chern-Simons partition function on a Brieksorn homology sphere $Σ(2,5,7)$. Starting from an exact Chern-Simons partition function, we study the Borel resummation of its perturbative expansion.

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

Using $S^1$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn-Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of $S^2\times S^3$ a…

For the link $M$ of a normal complex surface singularity $(X,0)$ we ask when a knot $K\subset M$ exists for which the answer to whether $K$ is the link of the zero set of some analytic germ $(X,0)\to (\mathbb C,0)$ affects the analytic structure on $(X,0)$. We show that if $M$ is an integral homology sphere then such a…

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

Study finds all Brieskorn spheres with at most two fillable contact structures.

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…

We show that Brieskorn manifolds with their standard contact structures are contact branched coverings of spheres. This covering maps a contact open book decomposition of the Brieskorn manifold onto a Milnor open book of the sphere.

We corrected a few errors in the previous submission. These do not affect any of the topological conclusions of the earlier version. We have also included a few observations about the Casson invariant of the Brieskorn homology spheres.

Method calculates $d$-invariants for a family of Brieskorn spheres.

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

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

Survey of homology cobordism group with open problems.

In this paper we investigate the question of when different surgeries on a knot can produce identical manifolds. We show that given a knot in a homology sphere, unless the knot is quite special, there is a bound on the number of slopes that can produce a fixed manifold that depends only on this fixed manifold and the h…

Study volume growth in Milnor fibers using real Lagrangians.

We express the number of lattice points inside certain simplices via Dedekind-Rademacher sums. As an application, we prove a conjecture of Kronheimer and Mrowka in the special case of Brieskorn spheres (with at most 4 singular fibers). This conjecture relates the Euler characteristic of the Seiberg-Witten-Floer homolog…

We recognize the Gromoll-Meyer sphere Sigma^7 as the geodesic join of a simple closed geodesic and a minimal subsphere Sigma^5, which can be equivariantly identified with the Brieskorn sphere W^5_3. As applications we in particular determine the full isometry group of Sigma^7, classify all closed subgroups that act fre…

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\…

This paper analyzes $Δ_a$ invariants in non-perturbative complex Chern-Simons theory.

An invariant of orientable 3-manifolds is defined by taking the minimum $n$ such that a given 3-manifold embeds in the connected sum of $n$ copies of $S^2 \times S^2$, and we call this $n$ the embedding number of the 3-manifold. We give some general properties of this invariant, and make calculations for families of le…

The equivariant rho-invariants studied in this paper are a version of the classical rho-invariants of Atiyah, Patodi, and Singer in the presence of an isometric involution. We compute these rho-invariants for all involutions on the 3-dimensional lens spaces with 1-dimensional fixed point sets, as well as for some invol…

Study compares two methods to extend $\widehat{Z}$ invariants, finding incompatibility for Brieskorn spheres.

New contact structures found on Brieskorn spheres.

Characterizes symplectic rational homology ball fillings of Seifert fibered spaces.

Constructs families of monotone Lagrangians in Brieskorn-Pham hypersurfaces.

While the topological types of {normal} surface singularities with homology sphere link have been classified, forming a rich class, until recently little was known about the possible analytic structures. We proved in [Geom. Topol. 9(2005) 699-755] that many of them can be realized as complete intersection singularities…

Quantum invariants of 3-manifolds linked to splice diagrams.

New examples show Sasaki manifolds without extremal metrics.