Fixed point theory of finite polyhedra

Singh, Gauri Shanker (1982) Fixed point theory of finite polyhedra. Masters thesis, Memorial University of Newfoundland.

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Abstract

In this thesis the behavior of the fixed point property (f.p.p.) in the category of finite polyhedra is studied. The f.p.p. behaves very badly with respect to many geometric constructions, for example, it is not invariant under topological products, products with the unit interval, suspensions, smash products, join products, nor is it a homotopy invariant. The only construction under which the f.p.p. trivially behaves nicely is the wedge (i.e. one point union) of two spaces. Even if one restricts attention to very nice spaces (e.g. simply connected polyhedra) the f.p.p. is not preserved under topological products and many other geometric constructions. This can be seen from the classical counter-examples due to Fadell-Lopez and Bredon which, together with many of their consequences, are described and discussed in detail. In a more restrictive setting, for example for simply connected polyhedra satisfying the so-called Shi condition, the f.p.p. behaves more nicely, and it's invariance under topological products can also be proved under specific assumptions on the cohomology of the spaces involved.

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