Title:

Algebraic topology of manifolds : higher orientability and spaces of nested manifolds

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 korientable manifold (or more generally Poincare complex) has even Euler characteristic unless the dimension is a multiple of 2^{k+1}, where we call a manifold korientable if the i^{th} StiefelWhitney class vanishes for all 0 < i < 2^{k} (k ≥ 0). For k = 0, 1, 2, 3, korientable manifolds with odd Euler characteristic exist in all dimensions 2^{k+1}m, but whether there exist a 4orientable 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^{2} is 2orientable and (O ⊗ H)P^{2} is at least 3orientable. Part II discusses the homotopy type of spaces of nested manifolds. I prove that the space of ddimensional manifolds with kdimensional submanifolds inside R^{n} has the homotopy type of a linearised model T_{k<d}, which can be thought of as a space of offset dplanes inside R^{n} with a (potentially empty) offset kplane 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_{I} (R^{n}) of nested manifolds inside R^{n}, for I a finite list of strictly increasing dimensions between 0 and n  1, has the homotopy type of a linearised model space T_{I}.
