The Conley index over a base

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 <inline-formula content-type="math/mathml"> <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" alttext="normal upper R cubed"> <mml:semantics> <mml:msup> <mml:mrow class="MJX-TeXAtom-ORD"> <mml:mi mathvariant="normal">R</mml:mi> </mml:mrow> <mml:mn>3</mml:mn> </mml:msup> <mml:annotation encoding="application/x-tex">\mathrm {R}^3</mml:annotation> </mml:semantics> </mml:math> </inline-formula> , and the problem of continuing two periodic orbits which are not homotopic as loops.