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## Degenerate critical points and the Conley index

The thesis has two main themes: some homological results on the Conley index are put into a more natural homotopical context; and degenerate isolated critical points are studied from the point of view of the Conley index theory. A critical point of a smooth function is a rest point of the induced gradient flow so, if isolated, has a Conley index; this is the <I>k</I>-sphere, <I>S<SUP>k</SUP></I>, if the point is non-degenerate with Morse index <I>k</I>. The question as to which spaces can occur as the Conley index of a (degenerate) critical point is addressed. It is shown that the Lusternik Schnirelmann category of an invariant set (in general) is at least that of its Conley index less one. Consequently, the Conley index of a critical point can have Lusternik Schnirelmann category at most two. Conversely, the suspension of any finite <I>CW</I>-complex is shown to be the Conley index of a critical point of some function. A degenerate critical point may be broken up into a collection of non-degenerate points by perturbing the function in a neighbourhood of the point. The Conley index of the degenerate point is used to study this collection - homotopy invariants are introduced that give lower bounds on the number of critical values obtained in this manner. Despite its homotopical definition, much of the previous work using methods of algebraic topology with the Conley index concentrates on the homological properties of the index. This thesis, exploiting the definition of the Conley index as the homotopy type of a pointed space, studies the implications a flow on a space has on the homotopy of that space. It is shown that <I>S</I>-duality relates the forward and reverse flow Conley indices, generalising and clarifying a known PoincarĂ© duality theorem on the homology of the indices.

Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:660480 |

Date | January 1994 |

Creators | Pears, J. R. |

Publisher | University of Edinburgh |

Source Sets | Ethos UK |

Detected Language | English |

Type | Electronic Thesis or Dissertation |

Source | http://hdl.handle.net/1842/15617 |

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