1
vote
1answer
45 views

Symplectic submanifolds in $\mathbb{R}^{4}$

Which symplectic submanifolds can be realized in $\mathbb{R}^{4}$? It easy to show that such submanifolds aren't compact. So, they are spheres with some handles and holes. Which relations between the ...
2
votes
1answer
68 views

self-intersection of lagrangian submanifold

Let's consider lagrangian submanifold $X$ in symplectic manifold $M$. Is it true that self-intersection index of $X$ is equal to the Euler characteristic $\chi(X)$? Can we construct (not canonical) ...
2
votes
1answer
79 views

Inclusion $O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $

How to show, that an inclusion of homogenious spaces $$O(2n)/U(n)\to GL(2n,\mathbb{R})/GL(n,\mathbb{C}) $$ is homotopy equivalence? The big space is the space of complex structures on ...
0
votes
0answers
76 views

Obstruction to construct the set of vectors in lattice

Lets consider a lattice $\mathbb{Z}^n$ with some unimodular scalar product $\mathbb{Z}^n \times \mathbb{Z}^n \mapsto \mathbb{Z}$ and the set of vectors $e_0,\ldots,e_k$ with conditions: $$ ...
3
votes
1answer
144 views

Homology of symplectic manifolds

Could you show me some example of compact symplectic 4-manifold $M$ with the torsion in $H_{2}(M;\mathbb{Z})$
5
votes
1answer
121 views

Symplectic geometry as a prequisite for Heegaard Floer homology

I would like to study Heegaard Floer homology in the future in the connection to knot theory. I read a wikipedia article and it seems that I need to first learn a symplectic geometry (topology?). I ...
3
votes
0answers
73 views

A proof of simply connectedness of a symplectic quotient

Let $\rho$ be a unitary representation of a torus $G$ on $\mathbb{C}^n$. The action of $\rho$ is Hamiltonian with a moment map $\mu:\mathbb{C}^n \to \mathfrak{g}^*$. Here $\mathfrak{g}^*$ is the dual ...
9
votes
2answers
342 views

Floer theory or Floer homology, an introduction for physicists needed

I need an introduction to Floer theory that's suitable for perhaps a beginning math grad student or a 2nd year physics grad student. The wiki article is sufficiently over my head that it reads as ...