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 first construct a genus zero positive allowable Lefschetz fibration over the disk (a genus zero PALF for short) on the Akbulut cork and describe the monodromy as a positive factorization in the mapping class group of a surface of genus zero with five boundary components. We then construct genus zero PALFs on infinitely many exotic pairs of compact Stein surfaces such that one is a cork twist of the other along an Akbulut cork. The difference of smooth structures on each of exotic pairs of compact Stein surface is interpreted as the difference of the corresponding positive factorizations in the mapping class group of a common surface of genus zero.
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.
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