Okole, Francis Iheanacho
(1977)
A general theory of exact fillings.
*University of Southampton, Doctoral Thesis*.

## Abstract

Broadly speaking, the theory of exact fillings uses the theory of fibre bundles to study certain geometrical phenomena in which a fibre bundle 'collapses' in the sense that the fibres are preserved intact, but may intersect one another.To got a first idea of what this involves, consider a continuous map p:R~B. Now consider another space X, and a continuous map f:EtX. We study the effect of f on the fibres p-1(b), bcB. In many geometrical structures, such a pair of maps p,f is found satisfying certain additional conditions. For example, p is usually a fibre-bundle projection. The conditions on f are of two kinds, to which we give the names 'filling' and 'exactness'.The 'filling' condition says no more than that f is surjectivo. The 'exactness' condition is designed to ensure that none of the fibres p-1(b) is redundant.The concept of an exact filling is due to ProfessorS.A. Robertson and the main contribution we are making in this thesis is to provide a uniform account of the subject, to make a number of improvements in the linear and projective theories, to present the theory of maximal tori in a compact Lie group in the language of exact fillings and to show that the linear theory of exact fillings belongs to the much wider context of Lie group exact fillings.Throughout, we have worked in the topological category,U. And Chapters One and Two are technical, being mainly devoted to a development of the category 3• of exact fillings, S-fillings and C -fillings. Chapter Three is devoted mainly to the linear theory and the main result here concerns a necessary and sufficient condition for an .t -filling over a compact manifold to admit a section. We have also proved a 'splitting' theorem for x -fillings which is exploited in Chapter Four to ive a relatively simple proof of a theorem of Robertson, [11], namely a necessary and sufficient condition for a 6'-filling over a compact manifold to admit a section.Chapter Five deals entirely with Lie group exact fillings. And we have obtained three main results: (1) The family of maximal tori of a compact connnected Lie group G fills G exactly. (2) This filling determines an ;6-filling of L(G). (3) The natural inclusion of linear fillings over compact manifolds in the category of Lie group exact fillings over compact manifolds has a left inverse.

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