The floer-homology tag has no wiki summary.

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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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**3**answers

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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 does Yang-Mills and mass gap problem has to do with mathematics?

I'm not very experienced in this topic, but I read a short description of the Yang-Mills existence and mass gap problem, and as long as I understood it has mainly physical consequences and ...

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**1**answer

584 views

### 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.
...

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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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774 views

### 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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420 views

### 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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435 views

### 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 ...

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**1**answer

421 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 ...

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366 views

### 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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**2**answers

714 views

### 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 ...

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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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**1**answer

488 views

### 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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937 views

### 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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**1**answer

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### Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?

Consider the following question:
If $K\subset S^{3}$ is a nontrivial knot. Let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $Y$ ...

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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 ...

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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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**1**answer

288 views

### 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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294 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 ...

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### What can be said about compact embedded exact Lagrangians in the generalized pair of pants?

What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation:
$$ 1+\Sigma_i z_i = ...

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214 views

### 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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284 views

### $\pi_0${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 ...

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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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**1**answer

111 views

### use Floer homology to prove the fixed points

I read paper, in page 21, there is a proposition:
Let $(M,\omega)$ be a closed symplectic manifold with $\pi_2(M)=0$. Let {$f_t$}, $f_0$ = id, $f_1= f$ be a Hamiltonian path on M generated by a ...

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**1**answer

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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 ...

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185 views

### Regular points for solutions of Floer's equation

Let $(W,\omega)$ be a symplectic manifold and $H\in C^{\infty}(W\times S^{1};\mathbb{R})$.
Let $(J_{t})_{t\in S^{1}}$ denote a family of $\omega$-compatible
almost complex structures.
Fix a map $u\in ...

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**1**answer

408 views

### 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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362 views

### 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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### Computation of symplectic quasi-state

A subset of a symplectic manifold is called strongly non-displaceable if it cannot be displaced by symplectomorphisms. A meridian in a $2$-torus is displaceable by a symplectomorphism, but not by a ...

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### $C^0$ estimates in wrapped Lagrangian Floer cohomology

Let $(M, d\theta, \theta, Z)$, be an exact Liouville domain, where $Z$ is the Liouville vector field and $\theta$ is the primitive of the symplectic form. Let $\bar{M}$, be the symplectic completion ...

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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 ...

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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.,

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**1**answer

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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 ...

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### Gromov-Floer compactness for C^0 convergence of complex structure/ C^1 convergence of Hamiltonian

Let $M$ be a compact symplectic manifold, $J$ a possibly surface dependent complex structure, and $H$ a Hamiltonian on $M$. I am interested in a variant of Gromov-Floer convergence for solutions of ...

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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} : ...