On the Hopf index and the Conley index
On the Hopf index and the Conley index
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关于 Hopf 指数和 Conley 指数
DOI：
10.1090/s0002-9947-1989-0961594-0
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发表时间：
1989
影响因子：
1.3
通讯作者：
C. McCord
中科院分区：
文献类型：
--
作者：
C. McCord
The following generalization of the Poincare-Hopf index theorem is proved: If S is an isolated invariant set of a flow on a manifold M , then the sum of the Hopf indices on S is equal (up to a sign) to the Euler characteristic of the homology Conley index of 5 . One of the classical results in dynamical systems relating local and global information is the Poincare-Hopf index theorem. If Sf is a vectorfield on a compact manifold M with isolated zeros, then the intersection number of with the zero section of TM is defined to be the sum of the Hopf indices of the zeros: I ist f) = Yhlx(s^) If Л/ is without boundary or if af is outward on dM , the Poincare-Hopf theorem asserts that the intersection number of Sf is equal to the homological Euler characteristic of M : х{Щ = I (Я?) • See [4, 7]. This has the restriction that the local information (the Hopf indices) is only related to the global information (the Betti numbers) when the vectorfield is outward on the boundary. Pugh [8] removes this restriction in the case of generic contact of with д M by relating the intersection number to a sum of Euler characteristics: /(^) = ,dM) + J') , where R'_ is the "/-codimensional exit region" and Г' is its boundary. In this paper, we further develop this result by relating the Hopf index and intersection numbers to the Conley index for dynamical systems. Suppose that X is a smooth vectorfield on M , with S c M an isolated invariant set of the induced flow. Then the intersection number of a? on a neighborhood of S is (up to a sign) the Euler characteristic of the homology Conley index of S in M , the Conley-Euler number /(Л/;5) . In particular, if all zeros on af on S are nondegenerate, then the Conley index and the Hopf index are related by the formula (-')nx(M;S) = ^2ix(^). There are two aspects to this result. 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 additivity, stability, etc., are also properties of Conley-Euler numbers. To develop the theorem, we first Received by the editors February 8, 1988. 1980 Mathematics Subject Classification (1985 Revision ). Primary 34C35, 55M20; Secondary 54H20, 58F25. Work supported in part by OARPA Applied and Computational Mathematics Program. ©1989 American Mathematical Society 0002-9947/89 $1.00 + 5.25 per page
