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

TL;DR

This paper develops a noncommutative spectral sequence framework relating Hochschild homology and applies it to Heegaard Floer homology of double covers, revealing new algebraic structures in low-dimensional topology.

Contribution

It introduces a noncommutative Hodge-to-de Rham spectral sequence for dg algebras and applies it to bordered Heegaard Floer theory, connecting algebraic and topological invariants.

Findings

Spectral sequences relate Hochschild homology of bimodules to their derived tensor products.

Application to Heegaard Floer homology yields new spectral sequences for double covers.

Provides algebraic tools for studying topological invariants of 3-manifolds.

Abstract

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 of M with itself 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.