I'm reading Osváth-Szabó's notes on Heegard Floer homology, in particular about the surgery exact triangle.
On page 14 (numbered 42 on the document), they describe an isomorphism between the space of homology classes of Whitney triangles $\pi_2(\mathbf{x},\mathbf{y},\mathbf{z})$ in $\mbox{Sym}^g(\Sigma)$ and $\mathbb{Z} \times \mathcal{P} $, where $\mathcal{P}$ denotes the group of periodic domains in $\Sigma$.
I'm not sure if I understand this isomorphism correctly, or if there are some typos, or both. Here is how I understand the isomorphism works:
Given two elements $\psi, \psi_0 \in \pi_2(\mathbf{x},\mathbf{y},\mathbf{z})$, we can associate domains $\mathcal{D}(\psi), \mathcal{D}(\psi_0)$, by taking $$\mathcal{D}(\psi) = \sum n_{z_i}(\psi) D_i,$$
where $D_i$ are the components of $\Sigma - \{\mathbf{\alpha} \cup \mathbf{\beta}\cup\mathbf{\gamma}\}$, and $z_i \in D_i$; and $n_{z_i}(\psi)$ is the algebraic intersection of $\psi$ with $z_i \times \mbox{sym}^{g-1}(\Sigma)$.
It follows that if $n_z(\psi) = n_z(\psi_0)$, the domain $E = \mathcal{D}(\psi) - \mathcal{D}(\psi_0)$ is periodic: i.e. it satisfies $n_z(E) = 0$ (here, $n_z(E)$ denotes the coefficient of the component of $\Sigma - \{\mathbf{\alpha} \cup \mathbf{\beta}\cup\mathbf{\gamma}\}$ containing $z$ in $E$).
So all we have to do to define the isomorphism is pick a $\psi_0$ such that $n_z(\psi_0) = 1$: then we can subtract off $n \mathcal{D}(\psi_0)$ from $\mathcal{D}(\psi)$ to get a periodic domain. Given a fixed $\psi_0$, the isomorphism is then given by
$$\psi \mapsto (n, \mathcal{D}(\psi) - n\mathcal{D}(\psi_0)).$$
The inverse of the isomorphism should then be given by sending $(n, E)$ to something like $n \psi_0 + \phi$, where $\mathcal{D}(\phi) = E$; but this seems dependent on choice and I'm not entirely sure how to make sense of $n$ times a holomorphic triangle.
Is this right? How do I fix the last part?
Thanks very much.
