Heegaard Floer homology of certain mapping tori

Abstract

We calculate the Heegaard Floer homologies $H{F}^{+}\left(M,\mathfrak{s}\right)$ for mapping tori $M$ associated to certain surface diffeomorphisms, where $\mathfrak{s}$ is any ${spin}^c$ structure on $M$ whose first Chern class is non-torsion. Let $\gamma$ and $\delta$ be a pair of geometrically dual nonseparating curves on a genus $g$ Riemann surface ${\Sigma }_g$, and let $\sigma$ be a curve separating ${\Sigma }_g$ into components of genus $1$ and $g-1$. Write $t_{\gamma }$, $t_{\delta }$, and $t_{\sigma }$ for the right-handed Dehn twists about each of these curves. The examples we consider are the mapping tori of the diffeomorphisms $t_{\gamma }^m\circ t_{\delta }^n$ for $m,n\in \mathbb{Z}$ and that of $t_{\sigma }^{\pm 1}$.