On the infinity flavor of Heegaard Floer homology and the integral cohomology ring

Abstract

For a three-manifold $Y$ and torsion ${\mathrm{Spin}}^c$ structure $\mathfrak{s}$, Ozsváth and Szabóconstruct a spectral sequence with $E^2$ term an exterior algebra over $H^1(Y;\mathbb{Z})$ converging to $H{F}^{\infty }(Y,\mathfrak{s})$. They conjecture that the differentials are completely determined by the integral triple cup product form. In this paper, we prove that $H{F}^{\infty }(Y,\mathfrak{s})$ is in fact determined by the cohomology ring when $\mathfrak{s}$ is torsion. Furthermore, we give a complete calculation of such $H{F}^{\infty }(Y,\mathfrak{s})$, with mod 2 coefficients, in the case where $b_1(Y)$ is 3 or 4.