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James quasi-periodicity for the codegree of vector bundles over complex projective spaces. (English) Zbl 0627.55013
Let E, F be an oriented vector bundle over \(P({\mathbb{C}}^ n)\) of dimension a, b, respectively. If the sphere bundle of F is stably fibre homotopy trivial then E and \(E\oplus F\) have the same codegree. The authors show that if the restriction of F to \(P({\mathbb{C}}^ n)\), \(m\leq n\leq 2m\), is stably fibre homotopy trivial then the codimensions of E and \(E\oplus F\) can be related using j-theory. As applications they compute the codegree of such a bundle F as well as the codegree of the Hopf bundle.
Reviewer: S.O.Kochman

MSC:
55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
55R25 Sphere bundles and vector bundles in algebraic topology
55Q10 Stable homotopy groups
57T20 Homotopy groups of topological groups and homogeneous spaces
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