Questions tagged [floer-homology]
The floer-homology tag has no usage guidance.
121 questions
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Linearization of $\bar \partial_J$ in the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2
Reading the paper Floer cohomology of Lagrangian intersecitons and pseudo-holomorphic discks 2, in page $1004$ the authors want to prove that the linearization of $\bar \partial_J$ is surjective for ...
user174565
1
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0
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189
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Relating the Morse index with the Maslov index
In the following paper https://arxiv.org/pdf/math/0408280.pdf there is created an isomorphism between the Floer Homology of an hamiltonian functional $H$ in the cotangent bundle and the the Morse ...
- 791
10
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0
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898
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Roadmap to Floer homotopy theory?
I am a young postdoc working in symplectic topology.
Recently I became intrigued by Floer homotopy, especially after seeing it had been applied to classical questions in symplectic topology. (e.g. ...
6
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1
answer
718
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Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends
I need some clarification about the reason why we have a sphere bubbling off in the situation described by Seidel in The Symplectic Floer Homology of a Dehn Twist.
I’ll try to summarize to the best ...
- 2,018
2
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0
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71
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Using the removal of singularities theorem in $\mathbb{C}\mathbb{P}^1-\{0,\infty\}$ with lagrangian boundary conditions
Reading the paper "Floer Cohomology of Lagrangian intersections" the authors construct a map $f: \mathbb{R}^n \times [0,2^N]\rightarrow \mathbb{C}\mathbb{P}^n$ such that $f(\tau,0)=f(\tau,2^...
user174565
3
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0
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115
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Properties of $I_{\mu}$ for Lagrangian Floer Homology in the Cotangent bundle
Following the notation of the book "Lagrangian intersection Floer theory anomaly and obstruction" suppose we have that our symplectic manifold is a cotangent bundle $T^*M$ with the canonical ...
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21
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1
answer
1k
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How is Chern-Simons theory related to Floer homology?
Chern-Simons theory (say, with gauge group $G$) is the quantum theory of the Chern-Simons functional
$$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$
...
- 1,430
13
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2
answers
1k
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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. ...
- 133
2
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0
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115
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Higher genus (Hamiltonian perturbed) holomorphic curves in cotangent bundle of S^1
Consider $T^*S^1$ as symplectic manifold, with hamiltonian function $H(x,y) = y^2$ (y is the fiber direction, I know this is morse bott but it can be perturbed). consider the set of maps $u: \Sigma \...
- 113
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93
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Gluing maps in Floer Homology and boundary conditions
Recently, I have been trying to a construction of a gluing map regarding the Lagrangian Floer Homology of two fibers in the cotangent bundle $T^*M$ of a manifold , in order to prove that the map $\...
- 791
5
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0
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98
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Lagragian floer homology vs homology of $\Omega(L_0,L_1)$
I'm very new to this subject, so apologies for a very naive question and probably many mistakes. Let $M$ be some compact sympletic manifold with $L_0,L_1$ Lagrangian submanifolds which intersects ...
- 2,356
13
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3
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2k
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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 ...
- 804
6
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332
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Spectral flow of Dirac operator twisted by instanton
Suppose $E$ is a $SU(2)$-bundle over a closed three manifold $M$ and $S$ is the spinor bundle over $M$. Also assume $D_{A(t)}:\Gamma(S\otimes_{\mathbb c} E)\to \Gamma(S\otimes_{\mathbb c} E)$ is a ...
- 171
4
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1
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712
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The singular cohomology embeds into the symplectic cohomology
Viterbo's theorem on cotangent bundles $M=T^*N$ tells you in particular that singular cohomology $H^*(M)$ gets embedded in $SH^*(M)$ via the $c^*$ map. Having a Weinstein manifold (or more generally ...
- 1,687
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276
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Is there an symplectic field theory compactness theorem applicable in the context of Floer cohomology of a symplectomorphism?
Is there any reference in the literature about results regarding symplectic field theory (SFT) compactness for a neck-stretch in the context of Floer homology of a symplectomorphism $\phi \colon (M,\...
- 2,018
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0
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79
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Gluing of hybrid trajectories in Floer homology
In the paper by A. Abbondadolo and M.Schwarz, "On the Floer homology of cotangent bundles" arXiv link, to prove the desired isomorphism between the Floer homology and the Morse homology of ...
- 791
3
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200
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Conley Zehnder index for Floer homology of a symplectomorphism
I'm trying to get some intuition for the Conley-Zehnder index in the setting of Floer homology of a symplectomorphism $\phi : (M,\omega) \to (M,\omega)$. Let's assume that $\phi$ only has non-...
- 2,018
6
votes
1
answer
741
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Intuition about bubbling off a ghost bubble
I'm trying to improve my intuition about the bubbling phenomenon for $J$-holomorphic curves $\Sigma \to (M,\omega)$, where $\Sigma$ is a compact Riemann surface with possibly boundary. I assume that ...
- 2,018
3
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0
answers
102
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Continuation map interpolating two quadratic Hamiltonians with respect to different contact boundaries
Let $(M,\lambda)$ be a Liouville manifold. Consider two different contact boundaries $\partial_{\infty}^1M$ and $\partial_{\infty}^2M$ with respect to the same Liouville flow $Z$. Each of them ...
- 461
9
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1
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356
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Is the instanton homology for webs and foams a categorified Chern-Simons?
In their paper, Kronheimer and Mrowka constructed an instanton homology $J^{\#}$ for webs and foams and conjectured that for planar webs, $\dim J^{\#}=\#\text{ of Tait colorings}$. According to my ...
- 1,430
2
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Definition of signs of isomorphisms $c_u : o(x_1) \to o(x_0)$ in the definition of Floer cohomology via Seidel's book
I'm reading Paul Seidel's book "Fukaya Categories and Picard-Lefschetz Theory", chapter 12, and I'm currently trying to understand the differential on Floer cohomology in terms of ...
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4
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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 ...
- 17.5k
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1
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395
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Floer equation and Cauchy Riemann equation
Consider a symplectic manifold $(M,\omega)$ with the property that $\pi_2(M) = 0$. Given a time dependent hamiltonian $H_t$ on $M$, and a $\omega$-compatible almost complex structure J on M, we may ...
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11
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962
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Monopole Floer Homology vs. Heegaard-Floer theory
I have a (possibly very naive) question: what is the relation between Monopole Floer Homology and Heegaard-Floer theory? (both known and conjectured)
Is there some version of Atiyah-Floer conjecture ...
- 1,981
5
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2
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371
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Manifold of mappings between $M$ and $N$, with non-compact source $M$
EDIT: Let $M$ and $N$ are two smooth manifold and suppose $N$ is compact but $M$ is not necessarily compact. For my purpose, I just need to consider the case $M=\mathbb R \times S^1$ or $\mathbb R \...
- 2,789
3
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0
answers
233
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Locality in Floer theory
There appears to be a dearth of resources and references for the question of 'locality' in Floer theory. In particular, I cannot seem to find any complete statement of what people refer to as '...
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408
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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
5
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2
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370
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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
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$A_{\infty}$ multiplications on Morse cochain complex
Can the higher order $A_{\infty}$ multiplications defined by Fukaya be made trivial(by perturbing gradient trees) when Morse cochain complex is isomorphic to Morse cohomology, in which case the cup ...
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9
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1
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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 ...
- 15.1k
6
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1
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602
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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 $\...
- 83
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527
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Floer cohomology from mapping spaces of $\infty$ categories
There's a meta-observation (of Urs Schreiber, who attributes it to Ken Brown and Lurie) that 'cohomology theories come from mapping spaces of $(\infty,1)$ categories'. This is described in detail at ...
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Differential of the Rabinowitz Action Functional
On an exact Hamiltonian system $(M,d\alpha,H)$ define the Rabinowitz action functional
$$\mathcal{A}^H \colon C^\infty(\mathbb{S}^1,M) \times (0,+\infty) \to \mathbb{R}$$
by
$$\mathcal{A}^H(\gamma,\...
- 548
11
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2
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445
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Fredholm property about $L^p$-extension $(p\neq 2)$ of differential operators
The following is a well-known result for elliptic operators.
Theorem. Let $P: \Gamma(E)\to \Gamma(F)$ be an elliptic operator of order $m$ between vector bundles $E$ and $F$ over a compact ...
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2
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Reference Request: "Neck Stretching Procedure" (In Symplectic Field Theory)
I've been reading some papers in Symplectic Geometry which refer to something called "Stretching the neck", and give reference to Eliashberg, Givental and Hofer's SFT paper (http://arxiv.org/abs/math/...
- 1,893
4
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0
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105
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How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration
My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. ...
- 1,155
5
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0
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Does there exist a candidate for 'holomorphic' instanton Floer homology?
The Euler characteristic of instanton Floer homology agrees with the Casson invariant. Thomas introduced the notion of holomorphic Casson invariant, defined using the holomorphic Chern-Simons ...
- 31
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2
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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
7
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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
3
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Reference Request: Central Curvature "Fix"
Context: In Lagrangian-Floer theory, the (an) $\mathbf{A}_\infty$-algebra of a Lagrangian is curved. However, the curvature is central. One consequence of this is that you can get an uncurved $\mathbf{...
- 101
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Morse theory for pairs of submanifolds of complementary dimension
If you have a closed monotone symplectic manifold $M$, then to any pair of closed monotone Lagrangian submanifolds $L_1$, $L_2$ you can associate (modulo some bubbling assumptions) a $\mathbb{Z}_N$-...
user129710
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2
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1k
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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 ...
- 1,242
10
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3
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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 \...
- 363
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Symplectic displacement energy for several intersection points?
Let $(X, \omega)$ be a symplectic manifold. For any non-empty subset $Y \subset X$ we may define the displacement energy as
$$
e(Y)=\mathrm{inf}\{||\phi||_H \: | \phi \in Ham(X, \omega), \phi(Y) \cap ...
user74900
8
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0
answers
251
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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
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1
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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 ...
- 724
4
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1
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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 ...
- 1,331
12
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1
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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 ...
- 18.7k
8
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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
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 ...
- 31