The Conley Index Over the Circle

Abstract

We study the Conley index over a base in the case when the base is the circle. Such an index arises in a natural way when the considered flow admits a Poincare section. In that case the fiberwise pointed spaces over the circle generated by index pairs are semibundles, i.e., admit a special structure similar to locally trivial bundles. We define a homotopy invariant of semibundles, the monodromy class. We use the monodromy class to prove that the Conley index of the Poincare map may be expressed in terms of the Conley index over the circle.