On the Hopf index and the Conley index

Author:
Christopher K. McCord

Journal:
Trans. Amer. Math. Soc. **313** (1989), 853-860

MSC:
Primary 58F25; Secondary 55M20

DOI:
https://doi.org/10.1090/S0002-9947-1989-0961594-0

MathSciNet review:
961594

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The following generalization of the Poincaré-Hopf index theorem is proved: If is an isolated invariant set of a flow on a manifold , then the sum of the Hopf indices on is equal (up to a sign) to the Euler characteristic of the homology Conley index of .

**[1]**R. Armstrong and R. McGehee,*Some mathematical problems concerning the ecological problem of competitive exclusion*, J. Differential Equations**23**(1977), 30-52. MR**0432239 (55:5228)****[2]**C. Conley,*Isolated invariant sets and the Morse index*, CBMS Regional Conf. Ser. in Math., no. 38, Amer. Math. Soc., Providence, R.I., 1978. MR**511133 (80c:58009)****[3]**C. Conley and E. Zehnder,*Morse-type index theory for flows and periodic solutions for Hamiltonian equations*, Comm Pure Appl. Math.**37**(1984), 207-253. MR**733717 (86b:58021)****[4]**M. Hirsch,*Differential topology*, Springer-Verlag, Berlin, 1976. MR**0448362 (56:6669)****[5]**C. McCord,*Mappings and homological properties in the Conley index theory*, Ergodic Theory Dynamical Systems**8**(1988), 175-198. MR**967637 (89m:58177)****[6]**-,*Poincaré-Lefschetz duality for the homology Conley index*, preprint.**[7]**J. Milnor,*Topology from the differentiable viewpoint*, Univ. Press of Virginia, Charlottesville, 1965. MR**0226651 (37:2239)****[8]**C. Pugh,*A generalized Poincaré index formula*, Topology**7**(1968), 217-226. MR**0229254 (37:4828)****[9]**J. Robbin and D. Salamon,*Dynamical systems, shape theory and the Conley index*, Ergodic Theory Dynamical Systems**8**(1988), 375-394. MR**967645 (89h:58094)****[10]**D. Salamon,*Connected simple systems and the Conley index of isolated invariant sets*, Trans. Amer. Math. Soc.**291**(1985), 1-41. MR**797044 (87e:58182)**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
58F25,
55M20

Retrieve articles in all journals with MSC: 58F25, 55M20

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1989-0961594-0

Article copyright:
© Copyright 1989
American Mathematical Society