Questions tagged [floer-homology]

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1answer
94 views

Dismissing pseudoholomorphic curves in embedded contact homology

In the papers https://arxiv.org/abs/math/0410059, https://arxiv.org/abs/math/0410061, https://arxiv.org/abs/1608.07988; Hutchings, Sullivan and Choi use a interesting trick to dismiss the existence of ...
2
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0answers
33 views

Associativity of orientations on Moduli spaces in Floer Homology

I have been reading the paper "Coherent Orientations for periodic orbits problems in symplectic geometry" by Floer and Hofer , trying to understand how we can orient the moduli spaces that ...
2
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0answers
68 views

Compactness of Moduli spaces in Lagrangian Floer Cohomology

I have been reading Denis Aurox lecture notes on Fukaya Categories https://arxiv.org/pdf/1301.7056.pdf , and in page $9$ he starts to discuss the compactness properties of the moduli spaces and how we ...
6
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0answers
194 views

Computation of the Fredhom index in Floer theory

I have been reading Salamon's lecture notes on Floer homology, and to compute the Fredholm index of operators they use the fact that the spectral flow of $A(s)$ is the Fredholm index. Now in the proof ...
3
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0answers
116 views

Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology

I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...
2
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1answer
130 views

Computing the Fredholm index in Floer theory

In Salamon's notes on Floer homology, it's claimed that under some non-degenerancy assumptions the operator $$D:= \partial_s+J_0\partial_t+S(s,t): W^{1,p}(\mathbb{R}\times S^1,\mathbb{R}^{2n})\...
1
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0answers
86 views

Compactness properties in floer homology of cotangent bundles in the non-periodic case

Following the paper https://arxiv.org/pdf/math/0408280.pdf I have been interested in studying the case of solutions $x:[0,1] \rightarrow T^*M$ such that $x(0)\in T_{q_0}^*M$ and $x(1)\in T_{q_1}^*M$ ...
1
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0answers
59 views

Constant holomorphic strips in Lagrangian floer cohomology

Let $(P,\omega)$ be a symplectic manifold , when defining the lagrangian floer cohomology of a pair $(L_0,L_1)$ of lagrangian submanifolds with $\Sigma_{L}\geq 3$ we will want to look at the space $\...
2
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0answers
99 views

Examples and classification of holomorphic strips in $(\mathbb{C}\mathbb{P}^n,\mathbb{R}\mathbb{P}^n)$

Consider an exact isotopy $\phi_t$ of $\mathbb{C}\mathbb{P}^n$ such that $\phi_1(\mathbb{R}\mathbb{P}^n)\pitchfork \mathbb{R}\mathbb{P}^n$. When trying to compute the Lagrangian Floer cohomology of $(\...
2
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0answers
96 views

Bubbling of disks when proving compactness properties in Lagrangian floer cohomology

When defining lagrangian floer cohomology , as it's done in the paper "Floer cohomology of lagrangian Intersections and pseudo holomorphic disks I,II" we will need to look into the ...
3
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0answers
57 views

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^...
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50 views

Definition of multiply covered $J$-holomorphic disk

In the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks in page $1009$ it's mentioned that a $J-$holomorphic disk $w:D^2\rightarrow S^2$ with $\mu(w)=2$ cannot be ...
1
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1answer
125 views

Maslov index equal to $2$ implies that the disk is not multiply covered

In the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks in page $1009$ the authors claim that a map $w:D^2\rightarrow S^2$ with $w|_{\partial D^2}\subset L$, where $L$ ...
2
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0answers
89 views

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 ...
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0answers
33 views

Question about index formula for $k$-cusp trajectories

In the paper Floer cohomology of Lagrangian intersections and pseudo-holomorphic disks $1$, the following is claimed : Let $(J_{\alpha}, L_0^{\alpha}, L_1^{\alpha})\rightarrow (J,L_0,L_1) $ be a ...
2
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0answers
382 views

$\phi_1^{2^{N-1}}u$ cannot be a translation of $u$ in $\mathcal{M}_{J,\phi}(x,y)$

I have been reading and trying to understand the paper Floer cohomology of lagrangian intersections and pseudo-holomorphic disks II , but there is a part where I am not understanding a justification ...
2
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1answer
144 views

$\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$, Lagrangian floer cohomology

Following the paper "Floer cohomology of lagrangian intersections and Pseudo-Holomoprhic discks 2" by OH, it is mentioned that $\mathbb{R}\mathbb{P}^n$ is monotone in $\mathbb{C}\mathbb{P}^n$...
6
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130 views

Composition of coproduct and product in Lagrangian Floer (co)homology

Let's take a Riemann surface $\Sigma$ and three Lagrangians $L_0,L_1,L_2$ in general position. let's assume that we can set up Lagrangian Floer (co)homology - Here I'm being vague because I don't want ...
5
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1answer
328 views

Invariance of morse homology, doubt in proof in book "Morse Theory and Floer homology"

I am reading the book "Morse theory and Floer Homology" by Michele Audin and Mihai Damian. Now I am reading the proof of the following theorem. Link to the statement of the theorem ...
4
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0answers
113 views

Choice of almost complex structure in Seidel's Symplectic Floer Homology of a Dehn twist

I'm looking for a clarification of a construction done in Seidel's Symplectic Floer Homology of a Dehn twist: I don't get why his choice of almost complex structure on $\Sigma$ is a valid one for ...
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0answers
88 views

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 ...
3
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0answers
94 views

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 ...
3
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1answer
181 views

Lagrangian Floer (co)homology, Novikov coverings and exact symplectic manifolds

I started reading the book "Lagrangian intersection Floer theory anomaly and obstruction", and there are a couple of details and assumptions in the definition of the Novikov covering that I ...
2
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0answers
84 views

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 \...
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89 views

Action functional for the definition of Lagrangian Floer homology

I have been starting to learn about Lagrangian Floer homology using notes by A. Pedroza (arXiv link). Consider $(M,\omega)$ a symplectic manifold that is symplectic aspherical and $L_0,L_1$ two ...
15
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1answer
1k views

Why is embedded contact homology so powerful?

The Embedded Contact Homology (ECH), introduced by M. Hutchings, is an invariant of (contact) three-manifolds. Since its introduction, well-known conjectures in symplectic/contact topology in ...
5
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0answers
81 views

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 ...
3
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0answers
71 views

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 ...
4
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2answers
320 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 ...
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0answers
49 views

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 ...
9
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0answers
350 views

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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0answers
223 views

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 ...
3
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0answers
136 views

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-...
6
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0answers
213 views

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,\...
5
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1answer
468 views

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 ...
7
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1answer
369 views

Lagrangian intersection Floer homology: understanding some assumptions

Let $(X,\omega)$ be a symplectic manifold and $L\subset X$ be a Lagrangian subspace. Let $\mu_L:H_2(X,L;\mathbb{Z})\to \mathbb{Z}$ be the Maslov index homomorphism. Usual hypothesis Recall that $L$...
6
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2answers
618 views

The Floer Equation is Elliptic

Let $(M,\omega)$ be a symplectic manifold and $H \in C^\infty(M \times \mathbb{S}^1)$. Furthermore, let $J$ be an $\omega$-compatible almost complex structure on $M$. The Floer equation is the ...
2
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0answers
88 views

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,\...
5
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2answers
709 views

Background needed to understand modern research on knot homology theories

I am a student of mathematics, and have some background in Algebraic Topology (Hatcher, Bott-Tu, Milnor-Stasheff), Differential Geometry (Lee, Kobayashi-Nomizu), Riemannian Geometry (Do Carmo), ...
3
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0answers
189 views

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 '...
4
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0answers
90 views

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. ...
7
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0answers
193 views

$A_{\infty}$ 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 ...
1
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1answer
279 views

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 ...
6
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1answer
410 views

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 ...
4
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1answer
655 views

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 ...
3
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0answers
61 views

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{...
5
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0answers
284 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 ...
18
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1answer
1k views

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 $$CS(A)=\frac{k}{8\pi^2}\int_M \text{Tr}\left(A\wedge dA + \frac{2}{3}A\wedge A \wedge A\right)$$ ...
5
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0answers
82 views

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 $\mathbb{Z}_N$-...
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454 views

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 ...