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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?

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Yes, this is correct. It is spelled out explicitly on page 20 of András Juhász' A survey of Heegaard Floer homology.


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.

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  • $\begingroup$ Thanks, the edit really clears things up -- and my question was admittedly unclear. I've accepted the answer since it fully handles the torsion case I asked about, but I'm also curious about the non-torsion case (where it's pointed out that strong admissibility still implies weak admissibility). My follow-up question won't fit here, so I've edited it into the original post. Would you mind taking a look? $\endgroup$
    – hfquestion
    Feb 9 '16 at 14:03
  • $\begingroup$ 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? $\endgroup$
    – hfquestion
    Feb 9 '16 at 14:17
  • $\begingroup$ This does sound correct, indeed. $\endgroup$ Feb 9 '16 at 22:12

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