Questions tagged [floer-homology]
The floer-homology tag has no usage guidance.
63 questions with no upvoted or accepted answers
11
votes
0
answers
527
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 ...
- 928
10
votes
0
answers
897
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. ...
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
8
votes
0
answers
694
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 ...
- 211
7
votes
0
answers
211
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 ...
- 745
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
0
answers
208
views
Does this pseudo-holomorphic triangle contribute to the product $\mu_2$ in Lagrangian Floer cohomology?
I'm computing the product map $$\mu_2 : CF(L_0,V)\otimes CF(V,L_1)\to CF(L_0,L_1)$$ in Seidel's exact triangle for this specific case:
This is a genus 2 surface, and I color-coded the three (...
- 2,018
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
6
votes
0
answers
268
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 ...
- 791
6
votes
0
answers
175
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 ...
- 2,018
6
votes
0
answers
332
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 ...
- 171
6
votes
0
answers
276
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,\...
- 2,018
6
votes
0
answers
270
views
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 H^1(...
- 1,628
5
votes
0
answers
98
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 ...
- 2,356
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
5
votes
0
answers
90
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 $\mathbb{Z}_N$-...
user129710
5
votes
0
answers
308
views
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
5
votes
0
answers
381
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 ?
5
votes
0
answers
514
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 ...
- 51
4
votes
0
answers
198
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
$$\mathcal \Phi_u^{-1}(...
- 791
4
votes
0
answers
385
views
Some clarifications on the PSS isomorphism in Hamiltonian Floer cohomology
I'm looking for some help in understanding the PSS isomorphism map in the context of Hamiltonian Floer cohomology and Morse cohomology with universal Novikov coefficients $\Lambda_{\omega}$ (à la ...
- 2,018
4
votes
0
answers
135
views
Choice of almost complex structure in Seidel's Symplectic Floer Homology of a Dehn twist
I'm looking for a clarification of a construction done in Seidel's Symplectic Floer Homology of a Dehn twist: I don't get why his choice of almost complex structure on $\Sigma$ is a valid one for ...
- 2,018
4
votes
0
answers
105
views
How is the instanton Floer homology of Seifert fibrations related to that of a trivial fibration
My question centers around the relationship of the Chern-Simons theories of a Seifert fibration and the trivial product space $\Sigma_g \times S^1$, and its implication for instanton Floer homology. ...
- 1,155
4
votes
0
answers
217
views
What can be said about compact embedded exact Lagrangians in the generalized pair of pants?
What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation:
$$ 1+\Sigma_i z_i = ...
- 1,331
3
votes
0
answers
320
views
Bubbling off a sphere in a splitting/stretching manifold
This question is related to my old question Bubbling off of a pseudo holomorphic sphere on surface with cylindrical ends about the bubbling off argument in Seidel's paper The symplectic Floer homology ...
- 2,018
3
votes
0
answers
98
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 \...
- 169
3
votes
0
answers
114
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 ...
- 791
3
votes
0
answers
115
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 ...
- 791
3
votes
0
answers
102
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 ...
- 461
3
votes
0
answers
200
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-...
- 2,018
3
votes
0
answers
233
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 '...
- 31
3
votes
0
answers
67
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{...
- 101
3
votes
0
answers
124
views
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
3
votes
0
answers
392
views
$C^0$ 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 ...
- 1,331
2
votes
0
answers
82
views
Why should we restrict the multiplicitiy of hyperbolic orbit to be one in Embedded contact homology?
Embedded contact homology(abbreviated by ECH) is a Floer type theory specially designed for three dimensional contanct manifolds(or generally, manifold with stable Hamiltonian structure) invented by ...
- 285
2
votes
0
answers
60
views
Confusion about proof of $C^0$ bounds for Floer curves on cotangent bundles
I have trouble understanding the proof of theorem 5.4 from Cielibak's article "Pseudo-holomorphic curves and periodic orbits on cotangent bundles". At the bottom of page 267 he defines a ...
- 21
2
votes
0
answers
107
views
Product structures in Rabinowitz Floer homology
Let $(M,d\lambda)$ be a compact exact symplectic manifold and $\overline{M}$ its symplectic completion. For simplicity we can think of $\overline{M}$ has a cotangent bundle and $\partial M$ the sphere ...
- 791
2
votes
0
answers
154
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 ...
- 2,018
2
votes
0
answers
163
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 ...
- 791
2
votes
0
answers
71
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^...
user174565
2
votes
0
answers
105
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 ...
user174565
2
votes
0
answers
115
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 \...
- 113
2
votes
0
answers
69
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 ...
- 281
2
votes
0
answers
136
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,\...
- 548
2
votes
0
answers
237
views
Parametric Sard-Smale theorem - when is the generic set open?
I am learning Morse/Floer Theory and in my work on my master's thesis I want to apply the parametric Sard-Smale theorem. I.e. I consider a Banach bundle $\mathcal{L} \to \mathcal{M}\times \mathcal{G}$ ...
- 529
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
92
views
Lagrangian Floer theory for negative monotone symplectic manifolds and Lagrangians
In the paper "Floer cohomology of Lagrangian intersections and pseudo-holomorphic Disks I", Oh shows that for a compact monotone Lagrangian $L$ in a closed monotone symplectic manifold $M$ ...
- 791
1
vote
0
answers
90
views
Index of Floer operator for Hamiltonian vs Lagrangian Floer Homology
I am trying to see if there is a way to translate the computation of the index of the Floer operator for Hamiltonian Floer to Lagrangian Floer. Hamiltonian Floer homology is a theory that counts (...
- 111
1
vote
0
answers
119
views
Seidel's calculation of the Floer cohomology of a cotangent fibre and its Dehn twist
I am reading Seidel's paper on exact Lagrangian submanifolds in $T^*S^n$ and the graded Kronecker quiver, and in Lemma 2 (2) he claims the following fact: if $F_0$ is a cotangent fibre and $F_1$ is $\...
- 143
1
vote
0
answers
241
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 $\pi_2(P,L) = 0$.
...
- 178