All Questions
Tagged with floer-homology gt.geometric-topology
17 questions
23
votes
7
answers
9k
views
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 ...
15
votes
1
answer
1k
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.
Heegaard ...
- 724
13
votes
2
answers
1k
views
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. ...
- 133
12
votes
1
answer
986
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 ...
- 18.7k
9
votes
3
answers
668
views
Do we get a instanton $S^{3}$ if we do $1/n$ surgery on a knot in $S^{3}$?
Consider the following question:
Let $K\subset S^{3}$ be a nontrivial knot, and let $Y$ be the manifold obtained by doing $1/n$-surgery ($n\geq1$). Is it possible that the instanton Floer homology of $...
- 1,069
8
votes
1
answer
533
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 ...
- 81
8
votes
0
answers
251
views
Exact triangle for monopole Floer homology with $\mathbb{Z}$-coefficient
Let $Y$ be oriented 3 manifold with torus boundary and let $\gamma_{j}$ (j=0,1,2) be three curves on its boundary with $\#(\gamma_{j}\cap \gamma_{j+1})=-1$. We denote by $Y_{j}$ the manifold obtained ...
- 1,069
7
votes
0
answers
404
views
Finding basis of cohomology of a symplectic manifold by using Symplectic Minimal Model Program
My question is about Floer theory via symplectic surgery of Minimal Model program for finding basis of cohomology.
Motivation: Perelman for solving Thurston's Geometrization Conjecture used some sort ...
user21574
6
votes
1
answer
602
views
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 $\...
- 83
6
votes
0
answers
225
views
Is Heegaard-Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$?
Is Heegaard Floer homology the Lagrangian Floer homology of $M=\text{Sym}^g(\Sigma), L_0=\mathbb{T}_{\alpha},L_1=\mathbb{T}_{\beta}$? I am interested in the relationship between the theories
...
- 131
5
votes
2
answers
1k
views
Heegard diagrams for three-manifolds
I have a basic question about the Heegaard diagrams involved in providing a framework
for calculation of Floer-Homology of three-manifolds.
Typically such diagrams look like Figure 1 and Figure 2 here ...
- 5,966
5
votes
2
answers
370
views
Ozsváth-Szabó's contact invariant on the Brieskorn sphere $\Sigma(2,3,6m+1)$
According to Theorem 1.7 of Mark-Tosun's paper, the Brieskorn sphere $\Sigma(2,3,6m+1)$ admits two tight contact structure $\xi_{i}\ (i=0,1)$. They are both Stein fillable and they are contactomorphic ...
- 1,069
5
votes
0
answers
408
views
Is there any known relationship between sutured contact homology and Legendrian contact homology?
On one hand, Colin-Ghiggini-Honda-Hutchings' construction (https://arxiv.org/abs/1004.2942) provides an invariant of a Legendrian $L$ in a closed contact manifold $(M,\xi)$ via the sutured contact ...
- 299
3
votes
1
answer
223
views
Algebraic variations of the full knot Floer complex
In Hom's paper (arXiv link), p.20, Section 3.3 ends with
"There are other algebraic modifications one may consider, such as setting $U^n =
0$ or $UV = 0$",
referring to the knot Floer ...
- 171
2
votes
0
answers
205
views
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 $Y_0(K)$...
- 83
1
vote
0
answers
95
views
Laplace eigenvalue and floer theory
I'm trying to study the spectrum operator via Floer Theory. The idea is to consider the Rayleigh quotient as an operator on the Hilbert space $W^{1,2}$ and to study it's negative gradient flow.
Any ...
- 11
1
vote
0
answers
133
views
Regularity of the taut foliation
In Eliashberg-Thurston's famous paper "Confoliations" Corollary 3.2.11, they proved that Irreducible three manifold with $b_{1}>0$ admits semi-fillable contact structure using Gabai's theorem in ...
- 1,069