Stable geometric dimension of vector bundles over odd-dimensional real projective spaces (Q2467007)

The geometric dimension of a stable vector bundle \(\theta\) over a space \(X\) is the smallest integer \(m\) such that \(\theta\) is stably equivalence to an \(m\)-plane bundle. Let \(\widetilde{KO}(P)\) be the group of equivalent classes of stable vector bundles over the real projective space. It is a finite cyclic 2-group generated by the Hopf line bundle. In collaboration with M. Mahowald the authors have shown that, for sufficiently large even \(n\), the geometric dimension of a stable vector bundle over \(P^n\) depends only on its order in \(\widetilde{KO}(P^n)\) and the \(\text{mod\,}8\) value of \(n\). In the present paper the authors perform a similar determination when \(n\) is odd and the order of \(\theta\) is \(2^e\) with \(e> 6\). This improvement is more delicate and there are a few extreme cases that remain unsettled precisely.