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Noncommutative Hodge-to-de Rham spectral sequence and the Heegaard Floer homology of double covers

Robert Lipshitz; David Treumann — 2016

Journal of the European Mathematical Society

Let $A$ be a dg algebra over $𝔽_{2}$ and let $M$ be a dg $A$ -bimodule. We show that under certain technical hypotheses on $A$ , a noncommutative analog of the Hodge-to-de Rham spectral sequence starts at the Hochschild homology of the derived tensor product $M \otimes_{A}^{L} M$ and converges to the Hochschild homology of $M$ . We apply this result to bordered Heegaard Floer theory, giving spectral sequences associated to Heegaard Floer homology groups of certain branched and unbranched double covers.

A faithful linear-categorical action of the mapping class group of a surface with boundary

Robert Lipshitz; Peter Ozsváth; Dylan P. Thurston — 2013

Journal of the European Mathematical Society

We show that the action of the mapping class group on bordered Floer homology in the second to extremal spin $^{c}$ -structure is faithful. This paper is designed partly as an introduction to the subject, and much of it should be readable without a background in Floer homology.

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