Graded Differential Vector Spaces, Cartan–Eilenberg Systems and Conjectures in Conley Index Theory

Abstract Cartan–Eilenberg systems play a prominent role in the homological algebra of filtered and graded differential vector spaces and (co)chain complexes in particular. We define the concept of Cartan–Eilenberg systems of vector spaces over a poset. Our main result states that a filtered chain isomorphism between $${{\textsf{P}}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>P</mml:mi> </mml:math> -graded differential vector spaces 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, as detailed in Robbin and Salamon (Ergod Theory Dyn Syst 12(1):153–183, 1992. https://doi.org/10.1017/S0143385700006647 ) and Franzosa and Mischaikow (J Differ Equ 71(2):270–287, 1988. https://doi.org/10.1016/0022-0396(88)90028-9 ). The main result of this paper also proves that the connection matrix theories in Franzosa (Trans Am Math Soc 311(2):561–592, 1989. https://doi.org/10.2307/2001142 ), Robbin and Salamon (1992) and Harker et al. (J Appl Comput Topol 5(3):459–529, 2021. https://doi.org/10.1007/s41468-021-00073-3 ) are equivalent in the setting of vector spaces, as well as uniqueness of connection matrices for Morse–Smale gradient systems, cf. Reineck (Trans Am Math Soc 322(2):523–545, 1990. https://doi.org/10.2307/2001713 ).