### Homology index braids in infinite-dimensional Conley index theory

DOI: http://dx.doi.org/10.12775/TMNA.2005.024

#### 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> ].

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> ].

#### Keywords

Morse-Conley index theory; homology index braid; continuation properties; singular perturbations

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