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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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