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Title:
Obstruction bundles and counting holomorphic disks in Heegaard Floer homology
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Abstract:
We discuss an approach to count the holomorphic disks which give the boundary operator in Heegaard Floer homology. Fix some Heegaard diagram and a domain $D$ from a generator $\mathbf x$ to a generator $\mathbf y$. We consider the moduli space $B$ of ramified maps $S\to\Sigma_g$ which represent $D$. We will discuss how counting the number of points in the moduli space of holomorphic disks from $\mathbf x$ to $\mathbf y$ is equivalent to calculating the Euler class of a certain ``obstruction bundle'' over $B$. More specifically, we observe that given a Riemann surface $S$ with boundary colored with $\alpha$ and $\beta$, there is a natural obstruction $\lambda\in H^1(S,\mathbb R)$ which vanishes if and only if $S$ is a ramified cover of the standard disk $\mathbb D$ with colored boundary.
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