Heegaard Floer correction terms, with a twist

TL;DR

This paper introduces a new class of numerical invariants for 3-manifolds using twisted Heegaard Floer homology, extending the classical correction terms to more general settings.

Contribution

It generalizes the correction terms to arbitrary closed 3-manifolds with torsion spin^c structures using twisted coefficients in Heegaard Floer homology.

Findings

Defines new twisted correction terms for 3-manifolds.

These invariants share properties with classical correction terms.

Provides topological restrictions on 4-manifolds bounding the 3-manifolds.

Abstract

We use Heegaard Floer homology with twisted coefficients to define numerical invariants for arbitrary closed 3-manifolds equipped torsion spin$^c$ structures, generalising the correction terms (or $d$--invariants) defined by Ozsv\'ath and Szab\'o for integer homology 3-spheres and, more generally, for 3-manifolds with standard ${\rm HF}^\infty$. Our twisted correction terms share many properties with their untwisted analogues. In particular, they provide restrictions on the topology of 4-manifolds bounding a given 3-manifold.