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5
votes
1answer
85 views

In the definition of the Heegard Floer surgery exact triangle, what exactly is the correspondence between Whitney triangles and periodic domains?

I'm reading Osváth-Szabó's notes on Heegard Floer homology, in particular about the surgery exact triangle. On page 14 (numbered 42 on the document), they describe an isomorphism between the space of ...
2
votes
3answers
226 views

Heegaard Floer Homology of double branched cover

The question is the following: Let $K\subset S^{3}$ be a knot, consider the double branched cover $Y$ of $S^{3}$ over $K$. We know $Y$ has a unique spin structure $\mathfrak{s}$, now the question is: ...
8
votes
0answers
104 views

When were bordered Heegaard Floer homology's DA bimodules invented?

This is less of a strictly mathematical question and more of a reference request. In their paper "Bordered Heegaard Floer Homology (http://arxiv.org/pdf/0810.0687.pdf)," Lipshitz-Ozsvath-Thurston ...
11
votes
1answer
176 views

Wanted: a nontrivial weakly inadmissible Heegaard diagram

This is a question asked by a student in my lecture. After drawing pictures for awhile, I thought it was a good one. I seek a nontrivial example of a pointed Heegaard diagram ...
9
votes
1answer
220 views

Is there a reasonable definition of TQFTs for n-cobordisms with connected inputs/outputs?

A question that's been on my mind for a while is whether any precise statement to the effect of "Heegaard Floer homology is a TQFT," for some reasonable definition of TQFT, can be made. Of course, a ...
6
votes
1answer
260 views

Untwisting Heegaard diagrams

Most Heegaard diagrams contain many rectangles, for instance from loops that circle around one of the handle disks. You can always `twist' a Heegaard diagram to get more and more rectangles (as in ...
0
votes
1answer
212 views

On the proof of Robert Lipshitz's formula on Maslov index.

Hello. I am a beginning graduate student who wants to study Heegaard Floer Homologies. I am now reading the paper http://front.math.ucdavis.edu/1301.4919 Errata to 'A cylindrical reformulation of ...
2
votes
0answers
150 views

maslov index of a holomorphic disk

I am studying some introductory papers on heegard floer homology and I do not understand the meaning of the Maslov index of a holomorphic disk. I could not find any definition in any of the papers. I ...
3
votes
1answer
218 views

Decorations in Szabo's combinatorial spectral sequence

Szabo in http://arxiv.org/abs/1010.4252 gives a combinatorial candidate for what an explicit calculation of the spectral sequence of branched double covers should yield. In other words he gives a ...
6
votes
2answers
443 views

heegard diagram

It seems like there is an algorithm to find the Heegard diagram of a 3 manifold obtained by surgery on a link. Also someone told me I can find it in the Gompf and Stipciz's book. But I could not find ...
2
votes
1answer
269 views

Heegard Floer homology [closed]

I am new to Heegard Floer. So far I understand that different HF groups are invariants of a three manifold. But I do not understand what these groups actually measure. I mean it seems to me that they ...
3
votes
1answer
270 views

Sarkar's Maslov index formula

I have difficulty understanding Sarkar's maslov index formula in symmetric products from http://arxiv.org/abs/math/0609673. If $D$ is an $n$-sided region with corner points $p_1,\ldots, p_n$ then it ...
7
votes
1answer
426 views

Why is Heegaard Floer Homology defined in terms of Sym$^g\Sigma_g$ instead of Pic$^g\Sigma_g$?

Recall the definition of Heegaard Floer homology: $\Sigma_g$ is a closed surface, and $\{\alpha_1,\ldots,\alpha_g\}$ and $\{\beta_1,\ldots,\beta_g\}$ are sets of attaching circles. Then Heegaard ...
3
votes
1answer
391 views

Wanted: differential coming from higher genus surface in Heegaard Floer Homology

I am interested in studying moduli of complex surfaces which arise in computing the differential on the Heegaard Floer Homology chain complex. In particular, I am interested in the generic case, when ...
4
votes
1answer
297 views

path of almost complex structure in the definition of heegaard floer homology

In order to define Heegaard Floer Homology for a connected, closed, oriented 3 manifold, we fix a generic path of nearly symmetric almost complex strucutre $J_s$ over $Sym^g(\Sigma)$. By ...
34
votes
7answers
3k views

Why should I care about Heegaard-Floer theory?

I would like to collect a list of applications of Heegaard-Floer theory. By applications, I don't mean things like "it can detect the unknot" or "it can detect knot genus". Algorithms for these ...
7
votes
1answer
779 views

Is $Sym^g$ of a Riemann Surface of genus $g$ Calabi-Yau?

The $g$-fold symmetric product of a Riemann surface of genus $g$ naturally has both a symplectic structure as well as a complex structure. Is it in fact Calabi-Yau? If so, is anything known about a ...
4
votes
3answers
729 views

Is there a version of Seiberg-Witten-Floer or Heegard-Floer homology for 3-manifolds with boundary?

Recently, the Seiberg-Witten-Floer homology created by Kronheimer and Mrowka has important applications in Taubes' proofs of Weinstein conjecture and Arnold Chord Conjecture. Also, Cagatay Kutluhan, ...