Questions tagged [symplectic-topology]

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On exotic symplectic structures of smooth closed 4-manifolds

What are some known techniques and examples of exotic symplectic structures on a fixed smooth closed 4-manifolds [by exotic I mean two symplectic structures that are not symplectomorphic]. This sounds ...
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-1 votes
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Conformal projection of ellipsoid in cartesian coordinates

Has anyone ever seen a conformal projection map of the ellipsoid (triaxial, oblate, prolate) into the plane, in cartesian coordinates?. I'm aware of the existence of Jacobi's map and the Mercator ...
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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 $\...
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2 votes
1 answer
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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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Is diffeomorphism of symplectic manifolds which preserves Lagrangian submanifolds and $J$-holomorphic curves necessarily symplectomorphism?

Consider a diffeomorphism of two symplectic manifolds $M_1$ and $M_2$ with compatible J structures, which sends Lagrangian submanifolds of $M_1$ to Lagrangian submanifolds of $M_2$. Assume moreover ...
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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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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 ...
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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 ...
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Multisymplectic connections and topological invariants

I was wondering if there exists any notion of multisymplectic connection that naturally generalizes symplectic (Fedosov) connections in symplectic geometry. From symplectic connections, it is well ...
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5 votes
1 answer
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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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If two symplectic toric manifolds are diffeomorphic, are they necessarily equivariantly diffeomorphic?

Suppose $M$ and $N$ are two symplectic toric manifolds. If $M$ is diffeomorphic to $N$, can we deduce that $M$ is equivariantly diffeomorphic to $N$ with respect to their torus actions?
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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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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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6 votes
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Kahler property and finite covering

Let $(M,\omega)$ be a compact symplectic manifold and $\pi:\tilde M\to M$ a finite covering. Clearly $(\tilde M,\pi^*\omega)$ is a compact symplectic manifold. Suppose we know that $(\tilde M,\pi^*\...
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2 answers
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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
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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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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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Removal of singularities theorem for J-holomorphic curves in non-compact manifolds

Following the book by Mcduff and Salamon, J-holomorphic curves and Symplectic Topology, we know that every $J-$holomorphic curve on the punctured disk with values in a compact symplectic manifold ...
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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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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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Linking number of specific Reeb orbits in a toric domain ($S^3$ diffeomorphic)

Consider a toric domain defined by the region bounded on the first quadrant by a function $f:[0,a]\mapsto [0,b]$ with $a,b>0;f(0)=b,f(a)=0,f(x)>0 \hspace{2mm} \forall x\in [0,a)$. We know that $\...
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2 votes
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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^...
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1 vote
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387 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 ...
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7 votes
1 answer
239 views

Cotangent bundles of surfaces as varieties

As far as I understand, it is easy to see (and find in the literature) that the affine variety $$z_1^2+z_2^2+z_3^2=1$$ with the restriction of the standard $\omega_{std}$ of $\mathbb{C}^3$ is ...
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2 votes
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Are there paths of non-degenerate $2$-forms joining two symplectic structures on open $4$-manifolds?

There is a Theorem of Conolly, L$\hat{\text{e}}$ and Ono which states that on a closed simply-connected $4$-manifold two cohomologous symplectic forms can be joined by a path of non-degenerate $2$-...
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3 votes
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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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2 votes
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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 \...
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10 votes
1 answer
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A theorem about the symplectic geometry of projective bundles

I am trying to understand the following theorem about symplectomorphisms of projective bundles. Theorem 1.5 of "Characteristic Classes in Symplectic Topology" A.G. Reznikov. Selecta ...
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2 votes
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How does the Maslov index of a loop `project’ to the rotation number?

I’m trying to learn some Legendrian contact homology and the grading of the generators of the DGA are given by computing a fractional rotation number. In the symplectisation, this number is the Conley-...
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4 votes
1 answer
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(Contradiction) All symplectic manifolds are holomorphic

I’m studying symplectic manifolds and almost complex structures. This lead to two propositions: Proposition 1 (from da Silva’s Lectures on Symplectic Geometry): If $J_0$ and $J_1$ are almost complex ...
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2 votes
0 answers
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Is the forgetful map a submersion away from the nodal points?

Let $\overline{\mathcal{M}}_{0,n}$ be the moduli space of stable curves of genus zero with $n$ marked points. For $n \geq 4$ we have a forgetful map $\pi \colon \overline{\mathcal{M}}_{0,n}\rightarrow ...
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3 votes
0 answers
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Displacing a conormal Lagrangian from the zero section

I was told that the conormal bundle $\nu^*K$ of a knot $K\subset S^3$ can be displaced from the zero section $0_{S^3}$ in $T^*S^3.$ Having no intuition about whether/how often this happens in general, ...
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1 vote
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On prequantization bundles over integral symplectic manifolds

I am trying to clarify certain subtleties regarding prequantization bundles over symplectic manifolds, for which I haven't found any clear explanation so far. Let me fix some definitions first. ...
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3 votes
0 answers
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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 ...
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6 votes
0 answers
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Hamiltonian dynamics on cotangent bundle

I'm stuck with the following claim made in Section 13.1 of Y-G. Oh's book "Symplectic topology and Floer homology". Assume that $N$ is a differential manifold and $S_0 ,S_1\subseteq N$ two ...
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3 votes
1 answer
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On some prerequisites for J-holomorphic curves and Gromov-Witten invariants

I'm currently reading through J-holomorphic curves and Quantum cohomology by McDuff and Salamon, and I've been facing some unfamiliarity issues with respect to the PDE and functional analysis tools ...
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2 votes
0 answers
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Two closed lagrangians embedded in $\mathbb{R}^{2n}$ can be shifted and perturbed to intersect at two points

In Polterovich paper "The surgery of Lagrange manifolds". It is stated in page 204 - in proof of Theorem 7a - the following, If $L_1$ and $L_2$ are two Lagrangian submaifolds of $\mathbb{C}^...
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1 vote
0 answers
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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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3 votes
0 answers
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Moment map of $\mathrm{O}(n)$-action on $\mathbb{C}^n$

Let $(\mathbb{C}^n, \omega_0)$ be the complex Euclidean space of dimension $n$ with the standard Kähler structure $\omega_0$. I am looking for a Hamiltonian $\mathrm{O}(n)$-action on $(\mathbb{C}^n, \...
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8 votes
2 answers
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Which curves are boundary of pseudoholomorphic curves?

I have posted it on Mathstackexchange but nobody replied. Consider a loop $\gamma:\mathbb{S}^1\to M^{2n}$ in a symplectic manifold $(M^{2n},\omega)$. Let $J$ be an $\omega$-compatible almost complex ...
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9 votes
0 answers
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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. ...
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2 answers
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Terminology for exact symplectomorphism

Let $(M,\omega = d\alpha)$ be an exact symplectic manifold. Then a symplectomorphism $\varphi \colon M \to M$ is said to be exact, iff $\varphi^*\alpha - \alpha$ is exact. Is there a terminology for ...
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13 votes
1 answer
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How not to use J-holomorphic curves [closed]

The field of symplectic topology is filled with subtle traps for the unwary, particularly when it comes to the analysis of $J$-holomorphic curves. So that the next generation of symplectic topologists ...
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2 votes
0 answers
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Local contractibility of group of symplectomorphisms for open manifolds

It is well know that for a closed symplectic manifold $(M, \omega)$ the group of symplectomorphisms in locally contractible. The gist of this proof goes as follows. Given a $\psi \in \operatorname{...
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6 votes
1 answer
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Is the symplectic quotient $\mu^{-1}(0)/G$ unique up to something?

Given a Hamiltonian action of a compact Lie group $G$ on a symplectic manifold $(M,\omega)$, we may choose a moment map $\mu \colon M\to \mathfrak{g}^* $ and obtain the symplectic reduction $M/\!\!/G =...
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2 votes
0 answers
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Augmentations of wrapped Floer cochains

Let $M$ be a closed, simply-connected spin manifold and let $F_b \subset T^*M$ be the cotangent fiber over a point $b \in M$. Let $CW^*(L,L)$ be the $A_{\infty}$-algebra of wrapped Floer cochains over ...
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3 votes
1 answer
309 views

Viterbo restriction map surjective on Weinstein neighbourhood

In a Liouville manifold $M$ having a Liouville subdomain $i: N \hookrightarrow M$, there is the so-called Viterbo restriction map in symplectic cohomology $$SH^*(i): SH^*(M)\rightarrow SH^*(N).$$ In ...
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2 votes
1 answer
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Superlevel sets of a parametrized quadratic forms

Let $N$ be an odd integer, $n\in\mathbb{N}$, and $-\frac{2T}{NR^2}\leq a\leq0$ for given $R,T\in\mathbb{R}$ with $\frac{T}{NR^2}\leq\frac{\pi}{2}$. Now consider the quadratic form $\Omega(a)=\sum_{l\...
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4 votes
2 answers
244 views

Contactomorphisms have in general no fixed points

Let $(V, \xi = \ker \alpha)$ be a cooriented contact manifold. A contactomorphism of $(V, \xi)$ is a diffeomorphism $\phi$ which preserves the contact structure $\xi$ and its coorientation. In other ...
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  • 1,157
2 votes
1 answer
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Comparing the minimal Chern number and the cup-length of a symplectic manifold

Let $(M,\omega)$ be a symplectic manifold. One can define its minimal Chern number $N_M$ as: $$ N_M := \text{inf} \lbrace k > 0 \ |\ \exists A \in H_2(M; \mathbb{Z}), \langle c_1, A \rangle = k \...
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