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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| Format | Journal Article |
| Language | English |
| Published |
28.06.2024
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| 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. |
|---|---|
| AbstractList | 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. |
| Author | Vandervorst, Robert Spendlove, Kelly |
| Author_xml | – sequence: 1 givenname: Kelly surname: Spendlove fullname: Spendlove, Kelly – sequence: 2 givenname: Robert surname: Vandervorst fullname: Vandervorst, Robert |
| BackLink | https://doi.org/10.48550/arXiv.2406.19977$$DView paper in arXiv |
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| Snippet | Cartan-Eilenberg systems play an prominent role in the homological algebra of
filtered and graded differential groups and (co)chain complexes in particular.
We... |
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| SubjectTerms | Mathematics - Algebraic Topology Mathematics - Dynamical Systems |
| Title | Graded differential groups, Cartan-Eilenberg systems and conjectures in Conley index theory |
| URI | https://arxiv.org/abs/2406.19977 |
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