On the Infinity Flavor of Heegaard Floer Homology and the Integral Cohomology Ring

TL;DR

This paper proves that the infinity flavor of Heegaard Floer homology for torsion Spin^c structures is determined by the integral cohomology ring, and provides explicit calculations for certain cases with mod 2 coefficients.

Contribution

It establishes that HF^ extinfty(Y,s) is fully determined by the integral cohomology ring for torsion structures, confirming a conjecture and computing specific instances.

Findings

HF^ extinfty(Y,s) determined by integral cohomology ring

Complete calculation of HF^ extinfty with mod 2 coefficients for b_1=3 or 4

Confirmation of the conjecture relating differentials to the triple cup product

Abstract

Ozsvath and Szabo construct a spectral sequence with E_2 term \Lambda^*(H^1(Y;Z))\otimes Z[U,U^{-1}] converging to HF^\infty(Y,s) for a torsion Spin^c structure s. They conjecture that the differentials are completely determined by the integral triple cup product form via a proposed formula. In this paper, we prove that HF^\infty(Y,s) is in fact determined by the integral cohomology ring when s is torsion. Furthermore, for torsion Spin^c structures, we give a complete calculation of HF^\infty with mod 2 coefficients when b_1 is 3 or 4.