All Questions
20 questions with no upvoted or accepted answers
10
votes
0
answers
898
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
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
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
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
$$\mathcal \Phi_u^{-1}(...
- 791
4
votes
0
answers
217
views
What can be said about compact embedded exact Lagrangians in the generalized pair of pants?
What can be said about compact embedded exact Lagrangians in the $n$-dimensional generalized pair of pants e.g. the hypersurface in $(\mathbb{C}^*)^{n+1}$ defined by the equation:
$$ 1+\Sigma_i z_i = ...
- 1,331
3
votes
0
answers
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
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
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
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
164
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
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
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
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
1
vote
0
answers
94
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 ...
- 791
1
vote
0
answers
79
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 ...
- 791
1
vote
0
answers
96
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 $x(1)\in T_{q_1}^*M$ ...
- 791