A neo-classical approach to sphere parallelisability

The 1950’s are often considered as the “golden age” of topology, seeing a vast number of results and new theories appear. One such result is that the n-sphere Sⁿ is parallelisable (that is, its tangent bundle is trivial) if and only if n = 0, 1, 3, or 7. In addition to being an interesting result of geometric topology in and of itself, it also has close connections to the existence of certain algebraic structures over Rⁿ, and the homotopy theory of the classical groups, and spheres. In this thesis, we provide a synthesis of the key concepts and results that are found throughout the relevant literature as they relate to the sphere parallelisability problem, in an effort to clarify the main ideas involved. In addition, we present a new variation on the proof that S^4s+1, s ≥ 1, is not parallelisable (a sub-case proved as part of the general theorem). Using the theory of framed cobordism, we are able to present a key part of the proof in a more geometric setting, simplifying the original, purely homotopical, presentation. We also briefly discuss the use of topological K-theory in the solution to some related problems in topology, such as the famous Hopf invariant one problem.