Admissible Criterion in Heegaard Floer Homology

Admissible Diagram

We need to ensure that the sum in the differential $\hat{\partial }$ runs over finitely many positive domains
$A\in {\hat{\pi }}_2(x,y)$.
If the Heegaard diagram satisfies the weakly admissible criterion, then this finiteness holds, and $\hat{\partial }$ is well defined.

For $H{F}^{-}$ version, we need Strongly Admissible.

Weak Admissibility Criterion

The pointed Heegaard diagram $(\Sigma ,\alpha ,\beta ,\mathfrak{z})$ is weakly admissible for the ${\text{Spin}}^{\mathbb{C}}$–structure $\mathfrak{s}$
if every nontrivial periodic domain $P$ with

$$⟨c_1(\mathfrak{s}),P⟩=0$$

(as defined in Theorem [Pairing Evaluation]) has both positive and negative coefficients.

Equivalent Definition

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$b_1(Y)=0$

If $b_1(Y)=0$, for example for ${\mathbb{S}}^3$ or any rational homology sphere, the admissibility criterion is satisfied automatically.

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$b_1(Y)>0$

$S^1\times S^2$

If the diagram is non-generic, e.g. for $S^1\times S^2$,
one interesting weakly admissible diagram appears as shown below.

Figure: Weakly admissible diagram for $S^1\times S^2$.

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Existence of Admissible Diagrams and Heegaard Moves

(Lipshitz, 2006 — Cylindrical Reformulation of Heegaard Floer Theory)
There exists an admissible diagram for each ${\text{Spin}}^{\mathbb{C}}$ structure $\mathfrak{s}$.

Moreover, if two pointed Heegaard diagrams $H_1$ and $H_2$ are admissible,
then there is a sequence of pointed Heegaard moves connecting $H_1$ to $H_2$ such that
every intermediate diagram is also weakly (respectively strongly) admissible for $\mathfrak{s}$.