Stability of vector bundles (Ex)
From Manifold Atlas
Let and be rank vector bundles over an -dimensional CW-complex with .
1) Prove that is isomorphic to if and only if is stably isomorphic to ; i.e. there is an isomorphism for some , where denotes the trivial rank bundle over .
2) If , prove that bundle isomorphisms are fibrewise homotopic if and only if and are stably homotopic; i.e. there is a fibrewise homotopy between and for some .