Homology index braids in infinite-dimensional Conley index theory

Abstract

We extend the notion of a categorial Conley-Morse index, as defined in [K. P. rybakowski, < i> The Morse index, repeller-attractor pairs and the connection index for semiflows on noncompact spaces< /i> , J. Differential Equations < b> 47< /b> (1987), 66–98], to the case based on a more general concept of an index pair introduced in [R. D. Franzosa and K. Mischaikow, < i> The connection matrix theory for semiflows on (not necessarily locally compact) metric spaces< /i> , J. Differential Equations < b> 71< /b> (1988), 270–287]. We also establish a naturality result of the long exact sequence of attractor-repeller pairs with respect to the choice of index triples. In particular, these results immediately give a complete and rigorous existence result for homology index braids in infinite dimensional Conley index theory. Finally, we describe some general regular and singular continuation results for homology index braids obtained in our recent papers [M. C. Carbinatto and K. P. Rybakowski, < i> Nested sequences of index filtrations and continuation of the connection matrix< /i> , J. Differential Equations < b> 207< /b> (2004), 458–488] and [M. C. Carbinatto and K. P. Rybakowski, < i> Continuation of the connection matrix in singular perturbation problems< /i> ].