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Comultiplicativity of the Ozsvath-Szabo contact invariant

John A. Baldwin
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Abstract

Suppose that S is a surface with boundary and that g and h are diffeomorphisms of S which restrict to the identity on the boundary. Let Y_g, Y_h, and Y_{hg} be the three-manifolds with open book decompositions given by (S,g), (S,h), and (S,hg), respectively. We show that the Ozsvath-Szabo contact invariant is natural under a comultiplication map on Heegaard Floer homology. It follows that if the contact invariants associated to the open books (S, g) and (S, h) are non-zero then the contact invariant associated to the open book (S, hg) is also non-zero. We extend this comultiplication to a map on HF^+, and as a result we obtain obstructions to the three-manifold Y_{hg} being an L-space. We also use this to find restrictions on contact structures which are compatible with planar open books.

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John A. Baldwin. 2007-04-10. Comultiplicativity of the Ozsvath-Szabo contact invariant. https://arxiv.org/abs/math/0702603

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