Graded differential groups, Cartan-Eilenberg systems and conjectures in Conley index theory

TL;DR

This paper explores the role of Cartan-Eilenberg systems in homological algebra, establishing their equivalence with filtered chain isomorphisms and applying these results to address open conjectures in Conley index theory and dynamical systems.

Contribution

It introduces Cartan-Eilenberg systems of abelian groups over a poset and proves their equivalence with filtered chain isomorphisms, resolving conjectures in Conley index theory.

Findings

Equivalence between filtered chain isomorphisms and Cartan-Eilenberg systems.

Resolution of open conjectures in Conley index theory.

Demonstration that three connection matrix theories are equivalent in vector spaces.

Abstract

Cartan-Eilenberg systems play an prominent role in the homological algebra of filtered and graded differential groups and (co)chain complexes in particular. We define the concept of Cartan-Eilenberg systems of abelian groups over a poset. Our main result states that a filtered chain isomorphism between free, P-graded differential groups is equivalent to an isomorphism between associated Cartan-Eilenberg systems. An application of this result to the theory of dynamical systems addresses two open conjectures posed by J. Robbin and D. Salamon regarding uniqueness type questions for connection matrices. The main result of this paper also proves that three connection matrix theories in the literature are equivalent in the setting of vector spaces, as well as uniqueness of connection matrices for Morse-Smale gradient systems.