The Conley index over a base

Mrozek, Marian; Reineck, James; Srzednicki, Roman

doi:10.1090/S0002-9947-00-02163-2

The Conley index over a base
HTML articles powered by AMS MathViewer

by Marian Mrozek, James F. Reineck and Roman Srzednicki PDF

Trans. Amer. Math. Soc. 352 (2000), 4171-4194 Request permission

Abstract:

We construct a generalization of the Conley index for flows. The new index preserves information which in the classical case is lost in the process of collapsing the exit set to a point. The new index has most of the properties of the classical index. As examples, we study a flow with a knotted orbit in $\mathrm {R}^3$, and the problem of continuing two periodic orbits which are not homotopic as loops.

References

Thomas Bartsch, The Conley index over a space, Math. Z. 209 (1992), no. 2, 167–177. MR 1147812, DOI 10.1007/BF02570828
Karol Borsuk, Theory of retracts, Monografie Matematyczne, Tom 44, Państwowe Wydawnictwo Naukowe, Warsaw, 1967. MR 0216473
Richard C. Churchill, Isolated invariant sets in compact metric spaces, J. Differential Equations 12 (1972), 330–352. MR 336763, DOI 10.1016/0022-0396(72)90036-8
Charles Conley, Isolated invariant sets and the Morse index, CBMS Regional Conference Series in Mathematics, vol. 38, American Mathematical Society, Providence, R.I., 1978. MR 511133
Saunders MacLane, Steinitz field towers for modular fields, Trans. Amer. Math. Soc. 46 (1939), 23–45. MR 17, DOI 10.1090/S0002-9947-1939-0000017-3
Leo F. Epstein, A function related to the series for $e^{e^x}$, J. Math. Phys. Mass. Inst. Tech. 18 (1939), 153–173. MR 58, DOI 10.1002/sapm1939181153
I. M. James, General topology over a base, Aspects of topology, London Math. Soc. Lecture Note Ser., vol. 93, Cambridge Univ. Press, Cambridge, 1985, pp. 141–174. MR 787827
I. M. James, General topology and homotopy theory, Springer-Verlag, New York, 1984. MR 762634, DOI 10.1007/978-1-4613-8283-6
I. M. James, Fibrewise topology, Cambridge Tracts in Mathematics, vol. 91, Cambridge University Press, Cambridge, 1989. MR 1010230, DOI 10.1017/CBO9780511896835
Tomasz Kaczyński and Marian Mrozek, Connected simple systems and the Conley functor, Topol. Methods Nonlinear Anal. 10 (1997), no. 1, 183–193. MR 1646643, DOI 10.12775/TMNA.1997.028
Dale Rolfsen, Knots and links, Mathematics Lecture Series, vol. 7, Publish or Perish, Inc., Houston, TX, 1990. Corrected reprint of the 1976 original. MR 1277811
Krzysztof P. Rybakowski, The homotopy index and partial differential equations, Universitext, Springer-Verlag, Berlin, 1987. MR 910097, DOI 10.1007/978-3-642-72833-4
Rudolph E. Langer, The boundary problem of an ordinary linear differential system in the complex domain, Trans. Amer. Math. Soc. 46 (1939), 151–190 and Correction, 467 (1939). MR 84, DOI 10.1090/S0002-9947-1939-0000084-7
Robert M. Switzer, Algebraic topology—homotopy and homology, Die Grundlehren der mathematischen Wissenschaften, Band 212, Springer-Verlag, New York-Heidelberg, 1975. MR 0385836
Cahit Arf, Untersuchungen über reinverzweigte Erweiterungen diskret bewerteter perfekter Körper, J. Reine Angew. Math. 181 (1939), 1–44 (German). MR 18, DOI 10.1515/crll.1940.181.1
George W. Whitehead, Elements of homotopy theory, Graduate Texts in Mathematics, vol. 61, Springer-Verlag, New York-Berlin, 1978. MR 516508