Morse Theory on the Loop Space of Flat Tori and Symplectic Floer Theory
نویسنده
چکیده
We use closed geodesics to construct and compute Botttype Morse homology groups for the energy functional on the loop space of flat n-dimensional tori, n ≥ 1, and Bott-type Floer cohomology groups for their cotangent bundles equipped with the natural symplectic structure. Both objects are isomorpic to the singular homology of the loop space. In an appendix we perturb the equations in order to eliminate degeneracies and to get to a situation with nondegenerate critical points only. The (co)homology groups turn out to be invariant under the perturbation.
منابع مشابه
Quantum Cohomology and Morse Theory on the Loop Space of Toric Varieties
On a symplectic manifold M , the quantum product defines a complex, one parameter family of flat connections called the A-model or Dubrovin connections. Let ~ denote the parameter. Associated to them is the quantum D module D/I over the Heisenberg algebra of first order differential operators on a complex torus. An element of I gives a relation in the quantum cohomology of M by taking the limit...
متن کاملSemi-Infinite de Rham Theory
I shall start by explaining just what “semi-infinite” means and how it gives rise to interesting structures in geometry. These structures are studied as part of String Theory and are closely related to Floer Theory. Manifolds which carry a “semi-infinite” structure include loop spaces (C∞(S1, M) for M a closed manifold) and the theory seems to have particular simplicities when the original mani...
متن کاملReidemeister torsion in symplectic Floer theory and counting pseudo-holomorphic tori
The Floer homology can be trivial in many variants of the Floer theory; it is therefore interesting to consider more refined invariants of the Floer complex. We consider one such instance—the Reidemeister torsion τF of the Floer complex of (possibly non-hamiltonian) symplectomorphisms. τF turns out not to be invariant under hamiltonian isotopies, but we introduce a “correction term”—the Floerth...
متن کاملMorse Theory , Floer Theory and Closed Geodesics of S
We construct Bott-type Floer homology groups for the sym-plectic manifold (T S 1 ; can) and Bott-type Morse homology groups for the energy functional on the loop space of S 1. Both objects turn out to be isomorpic to the singular homology of the loop space of S 1. So far our objects depend on all choices involved, but the above isomorphism suggests further investigation to show independence of ...
متن کاملFloer ' S Infinite Dimensional Morse Theory and Homotopy Theory
x1 Introduction This paper is a progress report on our eeorts to understand the homotopy theory underlying Floer homology; its objectives are as follows: (A) To describe some of our ideas concerning what, exactly, the Floer homology groups compute. (B) To explain what kind of an object we think thèFloer homotopy type' of an innnite dimensional manifold should be. (C) To work out in detail how t...
متن کامل