Algebraic Topology of Manifolds

Abstract

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 Poincaré complex) has even Euler characteristic unless the dimension is a multiple of 2 k+1 , where we call a manifold k-orientable if the i th Stiefel-Whitney class vanishes for all 0 < i < 2 k (k ≥ 0). For k = 0, 1, 2, 3, k-orientable manifolds with odd Euler characteristic exist in all dimensions 2 k+1 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 2 is 2-orientable and (O ⊗ H)P 2 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 n has the homotopy type of a linearised model T k<d , which can be thought of as a space of offset d-planes inside R n with a (potentially empty) offset 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 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 .