2009 Topology Seminar - 06

Consider a vector bundle E of rank k over a finite CW complex X. Assume k>dimX for simplicity. If E is not trivial, then it will contain various trivial sub-bundles of maximal rank r. The geometric dimension of E is simply defined to be gd(E)=k-r. It provides a (crude) measure of E's deviation from triviality. In this talk I'll present some new observations concerning gd(E) when X is a sphere or projective space. These will culminate in a new proof that for X=S^8m+1 or S^8m+2, gd(E) is always equal to 6. [Joint work with D. Randall]