Stable indecomposability of certain summands in Miller's splitting of the special unitary and symplectic groups
In 1985, Miller showed that Stiefel Manifolds could be split, in stable homotopy, into a number of wedge summands, each being the Thom Space of certain bundles over Grassmannian manifolds. The question arises as to whether these summands may themselves decompose into further non-trivial summands. The thesis aims to answer this question in the negative for certain cases. Using natural operations, we show that the homology and K-homology of certain Miller summands of SU(n) do not split algebraically; the calculations are performed unstably and we provide an argument which allows an interpretation of the results in the stable homotopy category. We then deduce that the summands are stably indecomposable. Similarly for Sp(n), we use operations dual to those above on the cohomology and K-Theory of certain Miller summands. Again we obtain an indecomposability theorem and the method of proof leads to a further result on the atomicity of the spaces concerned and their skeleta.