Chain Complex of Heegaard Floer Homology

Chain Complex

We are now ready to define the Heegaard Floer chain complex $\widehat{CF}(Y)$.

The Chain Complex

For a fixed ${\text{Spin}}^{\mathbb{C}}$ structure $\mathfrak{s}$,
the chain complex $\widehat{CF}(Y,\mathfrak{s})$ is generated by intersection points corresponding to the same $\mathfrak{s}$.

The boundary operator is defined by:

$$\hat{\partial }\vec{x}=\sum _{\vec{y}\in \mathfrak{s}}\sum _{\begin{array}{c}A\in {\hat{\pi }}_2(\vec{x},\vec{y})\\ \text{ind}(A)=1\end{array}}\#{\mathcal{M}}^A\vec{y}.$$

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