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On the Hopf index and the Conley index. (English) Zbl 0712.34062

The following generalization of the Poincaré Hopf index theorem is proved: Suppose \({\mathcal H}\) is a smooth vector field on a manifold M, with \(S\subseteq M\) an isolated invariant set of the induced flow. Then the intersection number of \({\mathcal H}\) on a neighborhood of S is (up to sign) the Euler characteristic of the homology Conley index of S in M, the Conley-Euler number \(\chi\) (M;S). In particular, if all zeros of \({\mathcal H}\) on S are nondegenerate, then the Conley index and the Hopf index are related by the formula \((-1)^{\eta}\chi (M;S)=\Sigma \iota_ x({\mathcal H}).\)
Two aspects of this result are worth mentioning. First, it gives the Hopf index and intersection numbers a homological interpretation in a broad setting. Second, it shows that the properties of the Hopf index, such as duality, stability, additivity, computability, etc., are also properties of Conley-Euler numbers.
Reviewer: B.Aulbach

MSC:

55M20 Fixed points and coincidences in algebraic topology
54H20 Topological dynamics (MSC2010)
37C10 Dynamics induced by flows and semiflows
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