The decategorification of bordered Heegaard Floer homology

Bordered Heegaard Floer homology is an invariant for $3$-manifolds, which associates to a surface $F$ an algebra $\mathcal{A}(\mathcal{Z})$, and to a $3$-manifold $Y$ with boundary, together with an orientation-preserving diffeomorphism $\phi : F \to \partial Y$, a module over $\mathcal{A}(\mathcal{Z})$. We study the Grothendieck group of modules over $\mathcal{A}(\mathcal{Z})$, and define an invariant lying in this group for every bordered $3$-manifold $(Y, \partial Y, \phi)$. We prove that this invariant recovers the kernel of the inclusion $i_{*} : H_1 (\partial Y ; \mathcal{\mathbb{Z}}) \to H_1 (Y ; \mathcal{\mathbb{Z}})$ if $H_1(Y, \partial Y ; \mathcal{\mathbb{Z}})$ is finite, and is $0$ otherwise. We also study the properties of this invariant corresponding to gluing. As one application, we show that the pairing theorem for bordered Floer homology categorifies the classical Alexander polynomial formula for satellites.

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Received 19 October 2014

Accepted 31 October 2017

Published 20 April 2018