Questions tagged [floer-homology]

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6 votes
1 answer
279 views

Relation between symplectic (co)homology and Hochschild (co)homology and deformations

A very fluffy question in which I'm ignorant of homology/cohomology, grading etc: The open-closed and closed-open string maps relating the symplectic (co)homology and Hochschild (co)homology of the ...
3 votes
0 answers
83 views

Why is this special symplectic loop contractible? (Floer Homology)

In the construction of Floer homology, one shows a formula that connects the Maslov index/ Conley–Zehnder index $\mu$ with the dimension of the moduli spaces of connecting gradient flow lines: $$\dim \...
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4 votes
1 answer
121 views

Choice of a family of almost complex structures when defining Floer Homology

Consider a $1$-periodic Hamiltonian $H:S^{1}\times M\rightarrow \mathbb{R}$ defined on a compact symplectic manifold $M$. Let's suppose $M$ is nice enough so that we can develop Floer theory on it. ...
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3 votes
1 answer
94 views

Perturbation of vector fields in Morse Homology

Recently I have been reading on Morse Homology. Suppose we have a compact manifold $M$ and a smooth function $f:M \rightarrow \mathbb{R}$ and a Morse vector field $X$ such that we can do Morse ...
  • 589
2 votes
1 answer
78 views

Generic choice of non-degenerate Hamiltonians $H$ in Floer theory

When developing floer theory for an Hamiltonian $H:M\times S^{1}\rightarrow \mathbb{R}$ we will want $H$ to satisfy a non-degenerancy condition, that is, for every $x\in \mathcal{P}(H)$, a periodic ...
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2 votes
0 answers
90 views

Positive symplectic homology with different Liouville filling

Assume $(Y, \alpha)$ has an exact Liouville filling, i.e. there exists $(W, d\lambda)$ with $Y=\partial W, \alpha=\lambda|_{Y}$. Then for two different exact Liouville fillings $(W_1,d\lambda_1), (W_2,...
1 vote
0 answers
165 views

Definition of Floer complex in Floer's "Morse theory for Lagrangian intersections"

I am moving the first steps into Lagrangian Floer theory and I am trying to understand the construction, as in the original paper, for the field $\mathbb{Z}_2$ (no orientations) and $\pi_2(P,L) = 0$. ...
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1 vote
0 answers
47 views

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 $\...
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2 votes
1 answer
154 views

Linearization of the Floer equation

In Floer Homology we want to prove that the Moduli spaces $\mathcal{M}(x^{-},x^{+})$ are finite dimensional manifolds. This is done by expressing them as the zero set of a Fredholm map. First one ...
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3 votes
1 answer
162 views

Moduli space of flat connection over homology 3-sphere

I'm trying to understand the space of flat connections of the trivial $\mathrm{SU}(2)$-bundle over a closed, oriented homology three-sphere (for the purpose of understanding the instanton Floer ...
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1 vote
0 answers
76 views

Gluing of Morse-type trajectories in "Floer Homology of Cotangent bundles"

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 ...
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1 vote
0 answers
46 views

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 ...
  • 589
1 vote
0 answers
60 views

Banach manifold structure on the moduli space of hybrid trajectories

I am reading the paper "On the Floer homology of cotangent bundles", (arXiv link) , by Abbondandolo and Schwarz and in page $35$ to define the isomorphism between the Morse complex and the ...
  • 589
2 votes
0 answers
92 views

Motivation behind the usual setting of the (weak) Arnold conjecture for fixed points of an hamiltonian diffeomorphism

I'm trying to find out the motivations that led V. Arnold to formulate his famous conjecture (I guess theorem by now) in the following form: Let $(M,\omega)$ be a closed symplectic manifold (add ...
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5 votes
1 answer
125 views

Transversality and $C^l$, $C^{\infty}$ spaces of almost complex structures

Recently I have been trying to get a grip on transversality results in Floer homology. That is suppose we the section $\partial_{J,H}: W^{1,p}(u^*(TM))\rightarrow L^{p}(u^*(TM))$ and we want to prove ...
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2 votes
1 answer
119 views

Influence of symplectic invariants of the complement on being superheavy

Let $(M,\omega)$ be a symplectic manifold. I'm trying to show that a compact subset $K\subset M$ is $1$-superheavy [1, Definition 1.3] where $1=PD([M])$ is the unit in $QH^0(M)$. My question is: How ...
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1 vote
0 answers
73 views

Cup product and PSS map

Let $(M,\omega)$ be a symplectic manifold and let $H$ a Hamiltonian function. If $M$ is not closed we consider $H$ to be linear at infinity to ensure that $HF^*(H)$ is well-defined (I'm particularly ...
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2 votes
0 answers
74 views

Continuation principle for solutions of Floer's equation in $\mathbb{R}\times [0,1]$ and transversality

Consider $(M,\omega)$ a symplectic manifold and $J$ a compatible almost complex structure. For me it's well known that if we consider 2 solutions $u,v:\mathbb{R}\times S^1\rightarrow M$ of Floer's ...
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4 votes
0 answers
170 views

Writting the Floer map in local coordinates using the exponential chart

Following Salamon's Notes in Floer Homology , consider the Floer equation $$\mathcal{F}(u):=\partial_su+J_t(\partial_tu+\nabla H_t(u))=0$$ Then we can write in local coordinates $$\mathcal \Phi_u^{-1}(...
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4 votes
2 answers
316 views

Dismissing pseudoholomorphic curves in embedded contact homology

In the papers The periodic Floer homology of a Dehn twist, Rounding corners of polygons and the embedded contact homology of $T^3$, and Combinatorial embedded contact homology for toric contact ...
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2 votes
1 answer
190 views

Associativity of orientations of determinant bundles 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 ...
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2 votes
0 answers
94 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 ...
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5 votes
0 answers
228 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 ...
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3 votes
0 answers
194 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 ...
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3 votes
1 answer
174 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})\...
  • 589
1 vote
0 answers
88 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 $x(1)\in T_{q_1}^*M$ ...
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1 vote
0 answers
108 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 $(\...
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1 vote
0 answers
114 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 ...
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2 votes
0 answers
69 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^...
user avatar
1 vote
1 answer
168 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 $L$ ...
user avatar
2 votes
0 answers
91 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 ...
user avatar
2 votes
1 answer
156 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 $\mathbb{C}\mathbb{P}^n$...
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6 votes
0 answers
143 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 ...
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5 votes
1 answer
409 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 votes
0 answers
125 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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1 vote
0 answers
118 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 ...
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3 votes
0 answers
98 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 ...
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3 votes
1 answer
251 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 ...
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2 votes
0 answers
98 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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15 votes
1 answer
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 votes
0 answers
84 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 ...
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3 votes
0 answers
84 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 votes
2 answers
409 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 ...
1 vote
0 answers
53 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 votes
0 answers
616 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 votes
0 answers
257 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 votes
0 answers
158 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-...
  • 1,838
6 votes
0 answers
235 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,\...
  • 1,838
5 votes
1 answer
533 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 ...
  • 1,838
7 votes
1 answer
476 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 $L$...
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