On the Hopf index and the Conley index

Shift Equivalence and the Conley Index

In this paper we introduce filtration pairs for isolated invariant sets of continuous maps. We prove the existence of filtration pairs and show that, up to shift equivalence, the induced map on the corresponding pointed space is an invariant of the isolated invariant set. Moreover, the maps defining the shift equivalence can be chosen canonically. Lastly, we define partially ordered Morse decom...

The Conley Index and Rigorous Numerics

The Conley Index, Gauge Theory, and Triangulations

This is an expository paper about Seiberg-Witten Floer stable homotopy types. We outline their construction, which is based on the Conley index and finite dimensional approximation. We then describe several applications, including the disproof of the high-dimensional triangulation conjecture.

The Conley Index and Symbolic Dynamics

Suppose that N is an isolating neighborhood for S with respect to a continuous map f which decomposes into a disjoint union of nitely many compact sets. We establish a relationship between the Conley indices of subsets of S which appear in a natural way when studying the behavior of f from the viewpoint of symbolic dynamics. The relationship is then used to prove a simple criterion for chaos in...

On the Hopf Index Theorem and the Hopf Invariant

Let ƒ: N —• M be a C°° map of oriented compact manifolds, and let L be an oriented closed submanifold of codimension q > 1 in M. If w is a closed form Poincaré dual to L, we show that f~L, with multiplicities counted, is Poincaré dual to ƒ *w in N and is even meaningful on a "secondary" level. This leads to generalized versions of the Hopf invariant, the Hopf index theorem and the Bezout theore...