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

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 betw...

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Bibliographic Details
Main Authors Spendlove, Kelly, Vandervorst, Robert
Format Journal Article
LanguageEnglish
Published 28.06.2024
Subjects
Mathematics - Algebraic Topology
Mathematics - Dynamical Systems
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Summary: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.
DOI:10.48550/arxiv.2406.19977