This study was motivated by the observation that most smooth bundles do not admit a smooth function that is Morse when restricted to every fibre. The complexity <I>c</I> of a critical point of a smooth map is measured by an appropriate codimension of its germ. The subset of smooth maps from a bundle to a manifold with complexity on fibres not exceeding <I>c</I> is studied. Bounds for <I>c</I> are established such that this subset is open and dense in the set of all smooth maps, where sets of smooth maps are always given the Whitney <I>C</I><SUP>∞</SUP> topology. The bounds are calculated in terms of the dimensions of the base space, the fibre and the manifold into which the bundle is mapped and are proved using the theory of finite germs and a suitable adaptation of the Thom Transversality Theorem. Recent work of Vasil'ev is used to investigate real-valued functions on compact principal <I>S</I><SUP>1</SUP>-bundles. The existence is established of a function with complexity on fibres no more than roughly half of the minimum value for <I>c</I> for the open and dense subsets mentioned above. For certain bundles with fibre of dimension one, the set of smooth real-valued functions that are Morse when restricted to every fibre is shown to be <I>C</I><SUP>0</SUP> dense but not, in general, <I>C</I><SUP>1</SUP> dense. For all <I>n</I>-sphere bundles over the circle the set is shown to be <I>C</I><SUP>0</SUP> dense. The homotopy type of the space of smooth Morse functions on the circle is derived. Arnold's determination of the fundamental group of the generalised Morse functions on the circle is included.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:652223 |
| Date | January 1993 |
| Creators | Hassell Sweatman, Catherine Zoe Wollaston |
| Publisher | University of Edinburgh |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://hdl.handle.net/1842/10947 |
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