Use this URL to cite or link to this record in EThOS:
Title: Algebraic topology of manifolds : higher orientability and spaces of nested manifolds
Author: Hoekzema, Renee
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2018
Availability of Full Text:
Access from EThOS:
Full text unavailable from EThOS. Please try the link below.
Access from Institution:
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 2k+1, where we call a manifold k-orientable if the ith Stiefel-Whitney class vanishes for all 0 < i < 2k (k ≥ 0). For k = 0, 1, 2, 3, k-orientable manifolds with odd Euler characteristic exist in all dimensions 2k+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 Tk<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 ΨI (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 TI.
Supervisor: Tillmann, Ulrike Sponsor: Hendrik Mullerfonds
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Manifolds (Mathematics) ; Nested manifold ; Orientability ; Cobordism category ; Steenrod squares ; Euler characteristic