I'm reading [Osváth-Szabó's notes on Heegard Floer homology](http://math.mit.edu/~petero/Lectures.pdf), 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.



In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains?
If I had to guess, I'd go for the typo *and* a small misunderstanding.


I think that the isomorphism should indeed depend on the choice of a triangle $\psi\_0$ with $n\_z(\psi\_0) = 0$ (careful here: this has to be 0 and not 1, as you wrote above), but that it should be defined as


$$\psi\mapsto (n\_z(\psi), \mathcal{D}(\psi)-\mathcal{D}(\psi\_0)-n\_z(\psi)\cdot\Sigma)$$


whose inverse is given by (notice that now you only sum a triangle and some periodic domain, plus multiples of the Heegaard surface)


$$(n,P)\mapsto \psi\_0 + P + n\Sigma$$


The isomorphism I gave above agrees with the isomorphism in the notes only for domains with $n\_z = 0$ (which are the only ones you consider in the hat version).




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I don't think that you need a canonical choice for $\psi\_0$ (hence for the isomorphism), nor that it's possible to have one.


One final remark: here you're not really summing *holomorphic triangles*, but rather *domains*, so you shouldn't worry about the "holomorphic meaning" of the sum of domains.


