### The Conley index and the critical groups via an extension of Gromoll-Meyer theory

DOI: http://dx.doi.org/10.12775/TMNA.1996.003

#### Abstract

We investigate, in a variational setting, the

relationship

between

the Gromoll-Meyer pairs of {\it a dynamically isolated critical set}\/ and

the Conley index pairs

of its {\it isolating invariant neighbourhoods}. We show that the information

given by the critical groups of such a set is equivalent to

that given by the Conley index. This allows us to derive - in a non-compact

setting - various invariance properties for the Conley index from those

of the critical

groups, as well as a formula relating the degree of a gradient vector

field in an

isolating neighbourhood to the Conley index pair associated with

it.

relationship

between

the Gromoll-Meyer pairs of {\it a dynamically isolated critical set}\/ and

the Conley index pairs

of its {\it isolating invariant neighbourhoods}. We show that the information

given by the critical groups of such a set is equivalent to

that given by the Conley index. This allows us to derive - in a non-compact

setting - various invariance properties for the Conley index from those

of the critical

groups, as well as a formula relating the degree of a gradient vector

field in an

isolating neighbourhood to the Conley index pair associated with

it.

#### Keywords

Gromoll-Meyer pairs; Conley index pairs; dynamically isolated set; isolating invariant neighbourhoods; invariance properties of the Conley index

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