Maslov Index Computation

Cylindrical Formulation for Heegaard Floer Homology

Existence of Holomorphic Representative
Suppose $A\in {\pi }_2(\vec{x},\vec{y})$ is positive. Then there exists a Riemann surface with boundary and corners $S$ and smooth map
$u:S\to \Sigma \times [0,1]\times \mathbb{R}$ satisfying conditions (1)–(6) as in the original theorem.

Maslov Index Formula

For a domain $D\in {\pi }_2(\vec{x},\vec{y})$, $ind(D)=e(D)+n_x(D)+n_y(D)$ where $e(D)$ is the Euler measure and $n_x(D)$ is the point measure.

Euler Measure

Let $D$ have Euler characteristic $\chi (D)$, with $k$ acute angles and $l$ obtuse angles.
$e(D)=\chi (D)-\frac{k}{4}+\frac{l}{4}.$

Remark

Euler measure is additive: $e(D_1+D_2)=e(D_1)+e(D_2)$.

Point Measure

$n_x(D)={\sum }_i{n}_{x_i}(D)$
where $4n_{x_i}$ is the number of $j\in \{0,1,2,3\}$ with $D\cap ϵe^{i(\pi /4+\pi /2j)}\ne \varnothing$, $ϵ<1$.

Pairing Evaluation

$⟨c_1(\mathfrak{s}),D⟩=e(D)+2n_x(D)$
for $D\in {\hat{\pi }}_2(x,x)$.
Also, for $P\in {\pi }_2(x,x)$:
$ind(P)=⟨c_1(\mathfrak{s}),P⟩+2n_{\mathfrak{z}}(P).$