English

Lagrangian Intersections and the Serre Spectral Sequence

Differential Geometry 2007-07-23 v2 Symplectic Geometry

Abstract

For a transversal pair of closed Lagrangian submanifolds L, L' of a symplectic manifold M so that π1(L)=π1(L)=0=c1π2(M)=ωπ2(M)\pi_{1}(L)=\pi_{1}(L')=0=c_{1}|_{\pi_{2}(M)}=\omega|_{\pi_{2}(M)} and a generic almost complex structure J we construct an invariant with a high homotopical content which consists in the pages of order 2\geq 2 of a spectral sequence whose differentials provide an algebraic measure of the high-dimensional moduli spaces of pseudo-holomorpic strips of finite energy that join L and L'. When L and L' are hamiltonian isotopic, these pages coincide (up to a horizontal translation) with the terms of the Serre-spectral sequence of the path-loop fibration ΩLPLL\Omega L\to PL\to L. Among other applications we prove that, in this case, each point xL\Lx\in L\backslash L' belongs to some pseudo-holomorpic strip of symplectic area less than the Hofer distance between L and L'.

Keywords

Cite

@article{arxiv.math/0401094,
  title  = {Lagrangian Intersections and the Serre Spectral Sequence},
  author = {J. F. Barraud and O. Cornea},
  journal= {arXiv preprint arXiv:math/0401094},
  year   = {2007}
}

Comments

55 pages. LaTeX. fixed some imprecisions in the appendix