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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

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