= Heegaard Floer chain complex
{c}
{title2=$CF^-(Y,\mathfrak s)$}
For an admissible pointed <Heegaard diagram>, $CF^-$ is a free $\mathbb Z[U]$ <chain complex> with one generator for each point of $\mathbb T_\alpha\cap\mathbb T_\beta$. Each generator is an unordered selection containing exactly one intersection on each alpha circle and exactly one on each beta circle. After choosing coherent orientations,
$$
\partial^-\mathbf x=\sum_{\mathbf y}\sum_{\substack{\psi\in\pi_2(\mathbf x,\mathbf y)\\\mu(\psi)=1}}\#\widehat{\mathcal M}(\psi)\,U^{n_z(\psi)}\mathbf y.
$$
Here $\widehat{\mathcal M}$ is the moduli space modulo strip translation, $\mu$ is the <Maslov index>, and $n_z$ is multiplicity at the basepoint. Holomorphic representatives have nonnegative domain multiplicities. The differential lowers grading by one and $U$ lowers it by two. Boundary strata of one-dimensional moduli spaces give the cancellation proving $\partial^2=0$. Its homology is $HF^-$.
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