Questions tagged [floer-homology]
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121 questions
3
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A question about solutions to Floer's equation which are asymptotic to a stationary point
Let $M$ be a compact symplectic manifold and $H$ a time independent Hamiltonian on $M$. Let $\alpha$ be a solution to Floer's equation
$$ u(t,s): S^1 \times \mathbb{R} \to M$$
$$(du+X_H\otimes dt)^...
- 1,242
4
votes
1
answer
565
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Question about transversality for PSS map in Hamiltonian Floer cohomology
Let X be a compact symplectic manifold and $H_t,J_t$ a Floer regular pair of $\mathbb{S}^1$ dependent Hamiltonians and complex structures. The PSS maps are defined by considering $\mathbb{C}$ with a ...
- 1,628
6
votes
0
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270
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Unobstructed Lagrangian tori
Let $X$ be a compact symplectic manifold, $U$ a Darboux chart, and $L$ a standard Lagrangian torus in $U$. Is $H^1(L)$ weakly unobstructed in the sense of Fukaya-Oh-Ohta-Ono (that is, does $b \in H^1(...
- 1,628
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2
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Orientations for pseudoholomorphic curves with totally real boundary condition
I am trying to understand what the obstructions are to orienting moduli spaces of pseudoholomorphic curves with totally real boundary condition.
I believe that Fukaya-Oh-Ohta-Ono have shown that if ...
- 275
10
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2
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2k
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Floer homology and Invariants for Einstein Field Equations?
Motivation: There have been the instanton (anti-self dual connection) solutions to the Yang-Mills equation $d_A^\ast F_A=0$ which extremize the YM energy $\int_M|F_A|^2$, leading to the Donaldson ...
- 17.5k
12
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1
answer
986
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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 ...
- 13.2k
5
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0
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381
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Lie-infinity structure in Lagrangian Floer theory ?
Is there (besides the A-infinity structure) also a L-infinity structure in Lagrangian Floer theory (forming together a G-infinity structure) - like in Hochschild cohomology ?
8
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1
answer
533
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How to compute the Monopole Floer Homology for Surface $\times S^1$ ?
We know that Monopole Floer homology of a 3-manifold $M$ depends on a spin-c structure. My question is that if $M$ is $F\times S^1$ ($F$ is a surface of genus larger than 1) then how can we compute ...
- 215
14
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4
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2k
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Other Homology Theories still Count Holes?
This may be a naive question, but since first learning homology I considered it as a tool which counts appropriate holes in your space (on top of orientation and torsion phenomena). Then I was ...
CommunityBot
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205
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Spin-c structures for a closed,oriented 3-manifold equipped with a null homologous knot
Let $Y$ be a closed, oriented 3-manifold equipped with an oriented null homologous knot $K$. I want to understand the relative $spin^c$ structures for $(Y,K)$. There is canonical zero surgery $Y_0(K)$...
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3
votes
2
answers
798
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Dimension of moduli space in Lagrangian Floer homology
Let $(M,\omega)$ be symplectic manifold with $\omega=c_{1}=0$ on
$\pi_{2}M$. Let $\Lambda\subseteq M$ be Lagrangian submanifold.
Let $H:M\times S^{1}\rightarrow\mathbf{R}$ be Hamiltonian and $J$
be ...
- 1,242
0
votes
0
answers
182
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why $H_{1}(\Sigma)\cong H_{1}(Sym^{g}(\Sigma))$ ?
In paper holomorphic disks and 3-manifold invariants, Ozsvath and Szabo connstruct two
homeomoephisms
$\mathcal {f} : H_{1}(\Sigma)\rightarrow H_{1}(Sym^{g}(\Sigma))$ and
$\mathcal {g} : H_{1}(...
- 841
1
vote
2
answers
938
views
Maslov Index in heegaard floer homology
Can anyone explain what is definition of maslov index in Heegaard Floer homology? I am puzzled> Thank you.,
- 431
15
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1
answer
1k
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Where are $+$, $-$ and $\infty$ in bordered Heegaard-Floer theory?
Here goes my first MO-question. I've just read Lipshitz, Ozsváth and Thurston's recently updated "A tour of bordered Floer theory". To set the stage let me give two quotes from this paper.
Heegaard ...
- 13.2k
8
votes
0
answers
694
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SFT gluing on chain level in Floer homology?
I came across a new type of gluing in the FOOO paper http://arxiv.org/abs/1002.1660 (maybe not so new to experts). The situation is as follows: one considers a relative (to lagrangian) class and ...
- 211
5
votes
0
answers
514
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Degenerate moduli spaces in Floer homology
Let $(W,\omega)$ be a closed symplectially aspherical symplectic
manifold, and fix a Hamiltonian $H\in C^{\infty}(W\times S^{1};\mathbb{R})$
and a compatible almost complex structure $J$ on $W$. Given ...
- 51
10
votes
3
answers
1k
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Index theorem interpretation of the spectral flow for a pseudo holomorphic curve
Let $(M , \omega)$ be a symplectic manifold, $J$ a compatible almost complex structure. We call pseudo holomorphic strip a solution $u : \mathbb R \times I \to M$ of the equation $\partial_s u + J \...
- 3,439
3
votes
1
answer
621
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Length of Floer flow lines
Suppose $(X,\omega)$ is a closed symplectic manifold. Let $H$ denote a time-dependent Hamiltonian, all of whose critical points are non-degenerate, and fix an $\omega$-compatible time dependent family ...
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2
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1
answer
599
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Instanton homology - reference request
What is the best reference for someone (i.e. me) trying to learn Instanton Floer homology? Assume I already know symplectic Floer homology.
- 4,414
6
votes
1
answer
791
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Almost complex structures in Floer theory
When defining the Floer cohomology $HF(L_0,L_1)$ of 2 Lagrangians in a symplectic manifold $(M,\omega)$, one first has to choose some extra data such a 1-parameter family of almost complex structures ...
6
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2
answers
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Computations in Knot Homology Theories
1) Relative to one another, how computable are the various knot homology theories? For example, how many crossings can we allow a knot and still hope to compute its Khovanov homology, versus its knot ...
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