Lie groups as framed boundaries (Q917725)

This paper is concerned with the problem of representing a compact Lie group, G, as the boundary of a compact, parallelizable manifold, V. In particular, here G is required to have a closed subgroup, S, isomorphic either to the group of unit complex numbers or unit quaternions. A candidate for V is then the disc bundle, D(\(\lambda\)), of the canonical line bundle, \(\lambda\), over G/S. The parallelizability of the resultant V is then reduced to the stable triviality of \(\lambda\), which in turn leads to the computation of the Stiefel-Whitney and Pontrjagin classes for \(\lambda\). This is considered in some detail for various classical groups: Spin(n), SU(n), Sp(n), SO(n), PSO(8). A motivation for this study is provided by the fact that framed Lie groups are related to the stable homotopy groups of spheres.