Algebraic topology of manifolds

Thesis

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^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+1m, 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)P² is 2-orientable and (O ⊗ H)P² 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 Rⁿ has the homotopy type of a linearised model T_k<d, which can be thought of as a space of off-set d-planes inside Rⁿ 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_I (Rⁿ) of nested manifolds inside Rⁿ, 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 .

Authors: Hoekzema, R

• Alternative title: Higher orientability and spaces of nested manifolds
• DOI: 10.5287/ora-dqkappraa
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: Orientability; Steenrod squares; Nested manifold; Euler characteristic; Cobordism category
• Subjects: Manifolds (Mathematics)