Witten’s complex and infinite dimensional Morse theory. (English) Zbl 0678.58012

The author provides a new version of Morse inequalities which is suitable for Morse functions having a gradient flow of Morse-Smale type. Let M be a locally compact smooth Riemannian manifold and f: \(M\to {\mathbb{R}}^ a \)smooth Morse function having a gradient flow of Morse-Smale type. Let S be an isolated compact invariant set, let \(C^{\mu}\) be the set of critical points of f in S of Morse index \(\mu\) and let \(C^{\mu}_{{\mathbb{F}}}\) be the free \({\mathbb{F}}\)-modulus generated by \(C^{\mu}\), where \({\mathbb{F}}\) is an assigned ring. Then there exists a coboundary operator \(\delta_ 0: C^{\mu}_{{\mathbb{F}}}\to C_{{\mathbb{F}}}^{\mu +1}\) such that \(I^*(S,{\mathbb{F}})\simeq \ker \delta_ 0/im \delta_ 0\), where \(I^*(S,{\mathbb{F}})\) denotes the cohomological Conley index of S. The result is then applied to a problem of Lagrangian intersections in symplectic geometry.
Reviewer: M.Degiovanni


58E05 Abstract critical point theory (Morse theory, Lyusternik-Shnirel’man theory, etc.) in infinite-dimensional spaces
53C20 Global Riemannian geometry, including pinching
Full Text: DOI