×

zbMATH — the first resource for mathematics

Floer homology, Novikov rings and clean intersections. (English) Zbl 0948.57025
Eliashberg, Ya. (ed.) et al., Northern California symplectic geometry seminar. Providence, RI: American Mathematical Society. Transl., Ser. 2, Am. Math. Soc. 196(45), 119-181 (1999).
In the first part of this paper the author describes the construction of Novikov homology, i.e., the generalization of the Morse complex to the case of a non-exact closed form. Since in general one cannot expect the existence of a suitable filtration \(\{N_i\}\), an alternative proof that \({\partial}^2=0\) is needed. The main purpose of the first part is to show that the Novikov homology is isomorphic to the homology of the manifold with coefficients in a local system (Theorem 2.2.2), this result being proved independently by V. Lê Hông and K. Ono. In the second part of the paper, the author uses Floer homology in the context of the Lagrangian intersections. These homology groups have not been computed in general. Theorem 3.4.11 provides a method for computing them when the intersection of the Lagrangian submanifolds is connected and sufficiently regular. As an example, the intersection of linear tori in \(T^{2n}\) is described. Also, the author shows that the Floer homology for a non-exact perturbation of the zero section in the cotangent bundle is isomorphic to the Novikov homology of its flux form.
For the entire collection see [Zbl 0930.00050].

MSC:
57R58 Floer homology
58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
57R70 Critical points and critical submanifolds in differential topology
PDF BibTeX XML Cite