| Summary: | <p>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 &LT; i &LT; 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. </p> <p>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&LT;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> .</p>
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