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

Kung Ching Chang, N. Ghoussoub

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.

Keywords


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

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