Questions tagged [floer-homology]
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121 questions
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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 ?
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Introduction to Floer Theory?
Can anyone suggest a good overview/introduction of the Floer machine for a beginner? (Someone pointed out some intriguing connections to surface mapping class groups, which might be enough incentive ...
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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 ...
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$\pi_0${plane fields}$\to\mathbb{Z}_2$
On a 3-manifold $Y$, oriented 2-plane fields $\xi$ are oriented rank-2 subbundles of $TY$. Denote the set of such (up to homotopy) by $\Theta=\pi_0\lbrace\xi\rbrace$. What is an explicit canonical map ...
- 17.5k
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Maslov index and Heegard Floer homology
I am an undergraduate who wants to learn Knot Floer homology. I was told to start with this expository paper which was working quite well until I reached the actual definition of the differential. ...
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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 ...
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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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Path of almost complex structure in the definition of Heegaard Floer homology
$\DeclareMathOperator\Sym{Sym}$In order to define Heegaard Floer Homology for a connected, closed, oriented 3 manifold, we fix a generic path of nearly symmetric almost complex structure $J_s$ over $\...
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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 ...
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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}(...
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Maslov Index in heegaard floer homology
Can anyone explain what is definition of maslov index in Heegaard Floer homology? I am puzzled> Thank you.,
- 841
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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 ...
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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 ...
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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 ...
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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 \...
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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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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.
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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 ...
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The "miracle" of Heegard Floer.
Taking tori in symmetric products and "miraculously" proving that the Floer homology is independent of choices always seemed, well, miraculous. Some time ago Max Lipyanski explained to me the origins ...
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What is Floer homology of a knot?
I've heard that there are different theories providing knot invariants in form of homologies. My understanding is that if you embed knot in a special way into a space, there is a special homology ...
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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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