# Morse Theory for Invariant Functions and its Application to the n-Body Problem

LLoyd, Gemma (2009) Morse Theory for Invariant Functions and its Application to the n-Body Problem. Doctoral thesis, Manchester Institute for Mathematical Sciences, The University of Manchester. PDF thesis_final.pdf Download (919kB)

## Abstract

We study various topological properties of G-manifolds and G-complexes where G is a finite group. We achieve this by first observing the work of others, T.F. Banchoff, Milnor and R. Bott, and then extending these ideas and concepts to the situation of G-manifolds and complexes. We start with embedded polyhedra and a critical point theorem, linking numbers of critical points to the Euler characteristic, which we adapt to encompass G-complexes and G-invariant functions to form a critical orbit theorem. We then move on to the Lefschetz fixed point theorem. We describe the concepts G-trace and G-Euler characteristics which have a direct link to their non-G counterparts. We use these to prove what we call the G-Lefschetz fixed orbit theorem; an extension of the Lefschetz fixed point theorem. We develop G-Morse and G-Morse-Bott theory from their usual respective theories. We define the notion of orientation representation and G-Morse and G-Poincare polynomials. The main result of this work is MG_t(f) - PG_t (M) = (1 + t)QG_t (f); where f is a non-degenerate (in the sense of Morse-Bott theory) function on the manifold M, MG_t(f) is the G-Morse polynomial of f, PG_t(M) is the G-Poincare polynomial of M and QG_t(f) is a polynomial with non-negative coefficients. These polynomials have coefficients in the representation ring. Finally we apply G-Morse theory to the n-body problem and fully describe the relative equilibria solutions up to and including five particles.

Item Type: Thesis (Doctoral) Invariant functions, finite group actions, Morse theory, Lefschetz fixed point theorem, equivariant maps MSC 2010, the AMS's Mathematics Subject Classification > 37 Dynamical systems and ergodic theoryMSC 2010, the AMS's Mathematics Subject Classification > 58 Global analysis, analysis on manifolds Dr James Montaldi 18 Dec 2009 20 Oct 2017 14:12 http://eprints.maths.manchester.ac.uk/id/eprint/1374 View Item