Abstract. Using Taubes' periodic ends theorem, Auckly gave examples of toroidal and hyperbolic irreducible integer homology spheres which are not surgery on a knot in the three-sphere. We give an obstruction to a homology sphere being surgery on a knot coming from Heegaard Floer homology. This is used to construct infinitely many small Seifert fibered examples.

We introduce a notion of complexity for Seifert homology spheres by establishing a correspondence between lattice point counting in tetrahedra and the Heegaard-Floer homology. This complexity turns out to be equivalent to a version of Casson invariant and it is monotone under a natural partial order on the set of Seifert homology spheres. Using this interpretation we prove that there are finitely many Seifert homology spheres with a prescribed Heegaard-Floer homology. As an application, we characterize L-spaces and weakly elliptic manifolds among Seifert homology spheres. Also, we list all the Seifert homology spheres up to complexity two.

An isolated complex surface singularity induces a canonical contact structure on its link. In this paper, we initiate the study of the existence problem of Stein cobordisms between these contact structures depending on the properties of singularities. As a first step, we construct an explicit Stein cobordism from any contact 3-manifold to the canonical contact structure of a proper almost rational singularity introduced by Némethi. We also show that the construction cannot always work in the reverse direction: in fact, the U-filtration depth of contact Ozsváth–Szabó invariant obstructs the existence of a Stein cobordism from a proper almost rational singularity to a rational one. Along the way, we detect the contact Ozsváth–Szabó invariants of those contact structures fillable by an AR plumbing graph, generalizing an earlier work of the first author.

Abstract. In this paper, we show that the Ozsváth-Szabó contact invariant c + (ξ) ∈ HF + (−Y ) of a contact 3-manifold (Y, ξ) can be calculated combinatorially if Y is the boundary of a certain type of plumbing X, and ξ is induced by a Stein structure on X. Our technique uses an algorithm of Ozsváth and Szabó to determine the Heegaard-Floer homology of such 3-manifolds. We discuss two important applications of this technique in contact topology. First, we show that it simplifies the calculation of the Ozsváth-StipsiczSzabó obstruction to admitting a planar open book. Then we define a numerical invariant of contact manifolds that respects a partial ordering induced by Stein cobordisms. We do a sample calculation showing that the invariant can get infinitely many distinct values.

We establish three rank inequalities for the reduced flavor of Heegaard Floer homology of Seifert fibered integer homology spheres. Combining these inequalities with the known classifications of non-zero degree maps between Seifert fibered spaces, we prove that a map f : Y ′ → Y f:Y’ \to Y between Seifert homology spheres yields the inequality | deg ⁡ f | r a n k H F r e d ( Y ) ≤ r a n k H F r e d ( Y ′ ) |\deg f|\mathrm {rank} HF_{\mathrm {red}}(Y) \leq \mathrm {rank} HF_{\mathrm {red}}(Y’) . These inequalities are also applied in conjunction with an algorithm of Némethi to give a method to solve the botany problem for the Heegaard Floer homology of these manifolds.

Abstract. We show that an infinite family of contractible 4-manifolds have the same boundary as a special type of plumbing. Consequently their Ozsváth-Szabó invariants can be calculated algorithmically. We run this algorithm for the first few members of the family and list the resulting Heegaard-Floer homologies. We also show that the rank of the Heegaard-Floer homology can get arbitrary large values in this family by using its relation with the Casson invariant. For comparison, we list the ranks of Floer homologies of all the examples of Briekorn spheres that are known to bound contractible manifolds.

Abstract. Thanks to a result of Lisca and Matić and a refinement by Plamenevskaya, it is known that on a 4-manifold with boundary Stein structures with non-isomorphic Spin c structures induce contact structures with distinct Ozsváth-Szabó invariants. Here we give an infinite family of examples showing that converse of Lisca-Matić-Plamenevskaya theorem does not hold in general. Our examples arise from Mazur type corks. IntroductionFor any contact structure ξ on a 3-manifold Y , let c + (ξ) ∈ HF + (−Y ) denote its Ozsváth-Szabó invariant. Recall Lisca-Matić-Plamenevskaya theorem: In the light of the above theorem a natural question to ask is whether the Spin c structure of a Stein filling completely determines the Ozsváth-Szabó invariant of the induced contact structure. An evidence towards a positive answer was provided in a work of Karakurt [Kar14, Proposition 1.2] where it was shown that the Ozsváth-Szabó invariant depends only on the first Chern class of the Stein filling on W when the total space of the filling is a special type of plumbing. Our main result suggests that the answer is in general negative. To state it let π : HF + (−∂Y ) → HF red (−∂Y ) be the natural projection map from the plus flavor to reduced Heegaard Floer homology. Theorem 1.2. There exists an infinite family {W n : n ∈ N} of compact contractible 4-manifolds with boundary and Stein structures J n 1 and J n 2 on W n satisfying the following properties: (1) The Spin c structures s n 1 and s n 2 associated to J n 1 and J n 2 , respectively, are the same for every n ∈ N.(2) The induced contact structures ξ n 1 and ξ n 2 on ∂W n have distinct Ozsváth-Szabó invariants, in fact π(c + (ξ n 1 )) = 0 and π(c + (ξ n 2 )) = 0, for every n ∈ N. (3) the Casson invariant of ∂W n is given by λ(∂W n ) = 2n for every n ∈ N. (4) ∂W n is irreducible for every n ∈ N.Our examples W n are Mazur type manifolds obtained from the symmetric link L n in Figure 1 by putting a dot on one of the components and attaching a 0-framed 2-handle to the other component as in Figure 2. Note that the manifold W 1 is the Akbulut cork. A Stein structure J n 1 on W n can immediately be obtained by drawing a Legendrian representative of the attaching circle of the 2-handle and stabilizing as necessary to make the framing one less than the Thurston-Bennequin Date: February 26, 2018.

We define a new 4-dimensional symplectic cut and paste operation which is analogous to Fintushel and Stern's rational blow-down. We use this operation to produce multiple constructions of symplectic smoothly exotic complex projective spaces blown up eight, seven, and six times. We also show how this operation can be used in conjunction with knot surgery to construct an infinite family of minimal exotic smooth structures on the complex projective space blown-up seven times. 53Dxx; 57R57