Part I of this thesis concerns the question in which dimensions manifolds with higher orientability properties can have an odd Euler characteristic. In chapter 1 I prove that a k-orientable manifold (or more generally Poincare complex) has even Euler characteristic unless the dimension is a multiple of 2<sup>k+1</sup>, where we call a manifold k-orientable if the i<sup>th</sup> Stiefel-Whitney class vanishes for all 0 < i < 2<sup>k</sup> (k ≥ 0). For k = 0, 1, 2, 3, k-orientable manifolds with odd Euler characteristic exist in all dimensions 2<sup>k+1</sup>m, but whether there exist a 4-orientable manifold with an odd Euler characteristic is an open problem. In Chapter 2 I present calculations on the cohomology of the first two Rosenfeld planes, revealing that (O ⊗ C)P<sup>2</sup> is 2-orientable and (O ⊗ H)P<sup>2</sup> is at least 3-orientable. Part II discusses the homotopy type of spaces of nested manifolds. I prove that the space of d-dimensional manifolds with k-dimensional submanifolds inside R<sup>n</sup> has the homotopy type of a linearised model T<sub>k<d</sub>, which can be thought of as a space of off-set d-planes inside R<sup>n</sup> with a (potentially empty) off-set k-plane inside of it, compactified with a point at infinity representing the empty set. Applying an induction I generalise this result to the case of higher nestings, establishing that the space Ψ<sub>I</sub> (R<sup>n</sup>) of nested manifolds inside R<sup>n</sup>, for I a finite list of strictly increasing dimensions between 0 and n - 1, has the homotopy type of a linearised model space T<sub>I</sub>.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:757804 |
| Date | January 2018 |
| Creators | Hoekzema, Renee |
| Contributors | Tillmann, Ulrike |
| Publisher | University of Oxford |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
| Source | http://ora.ox.ac.uk/objects/uuid:17a72743-a2a8-4a84-b1e5-624a8e17e803 |
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