# Admissibility in Heegaard Floer, especially with torsion Spin^c structures

I'm confused about the relationship between strong admissibility and weak admissibility for pointed diagrams in Heegaard Floer theory. For reference, here are Ozsváth-Szàbo's original definitions:

A pointed Heegaard diagram is called strongly admissible for the ${Spin}^c$ structure $\mathfrak{s}$ if for every nontrivial periodic domain $\mathcal{D}$ with $$\langle c_1(\mathfrak{s}),H(\mathcal{D})\rangle=2n \geq 0,$$ $\mathcal{D}$ has some coefficient $>n$. A pointed Heegaard diagram is called weakly admissible for $\mathfrak{s}$ if for each nontrivial periodic domain $\mathcal{D}$ with $$\langle c_1(\mathfrak{s}),H(\mathcal{D})\rangle =0,$$ $\mathcal{D}$ has both positive and negative coefficients.

Here's a concrete case that puzzles me: If $c_1(\mathfrak{s})$ is torsion, then it evaluates to zero on every homology class. In that case, it seems that a diagram is

• strongly admissible for $\mathfrak{s}$ if every nontrivial periodic domain $\mathcal{D}$ has some positive coefficient, and
• weakly admissible for $\mathfrak{s}$ if every nontrivial periodic domain $\mathcal{D}$ has both positive and negative coefficients.

If these conditions are to coincide for $c_1(\mathfrak{s})$ torsion (or at least for "strong" to be stronger), then it seems like any nontrivial periodic domain with positive coefficients must also have negative coefficients. Is this true?

Edit: it was not clear to me that you were also asking why, for torsion spin$^c$ structures, strong admissibility implies weak admissibility. Here's a possible proof: if $\mathcal D$ is periodic, then so is $-\mathcal D$. Hence, both $\mathcal D$ and $-\mathcal D$ have some positive coefficient, i.e. $\mathcal D$ has both positive and negative coefficients.
• Ah, I think I see it now. Weak admissibility only requires a nontrivial periodic domain to have positive and negative coefficients if $c_1(\mathfrak{s})$ vanishes on it for the reasons you've explained, and makes no demands of periodic domains seen by $c_1(\mathfrak{s})$. Strong admissibility makes the same requirement of periodic domains on which $c_1(\mathfrak{s})$ vanishes, but now also requires periodic classes on which $c_1(\mathfrak{s})$ is positive to have some sufficiently positive coefficient (and doesn't care when $c_1(\mathfrak{s})$ is negative on the domain). Correct? Feb 9 '16 at 14:17