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On the codegree of negative multiples of the Hopf bundle. (English) Zbl 0633.55014
Let H be the Hopf line bundle over \({\mathbb{C}}P^{k-1}\) and \(cd({\mathbb{C}}^ n-nH, {\mathbb{C}}P^{k-1})\) the codegree of \({\mathbb{C}}^ n-nH.\) Consider a fixed prime p. Denoting by \(\nu_ p(m)\) the power of p in m, set \(cd_ p=\nu_ p(cd)\). To determine \(cd_ p({\mathbb{C}}^ n-nH, {\mathbb{C}}P^{k- 1})\) the authors use the cohomology theory \(j^*\) which fits into the exact sequence \[ \to \quad j^*(-)\quad \to \quad kO^*(- )_{(p)}\quad \to \quad k Spin^*(-)_{(p)}\quad \to \quad. \] Theorem: Let \(1\leq n\leq k\). Then the codegree \(cd_ p({\mathbb{C}}^ n-nH, {\mathbb{C}}P^{k-1})\) is equal to the j-theory codegree \(cd^ j_ p({\mathbb{C}}^ n-nH, {\mathbb{C}}P^{k-1}).\) From this result one obtains explicit formulas for \(n=1\) and \(n=2\). Finally the authors discuss various quaternionic analogues of the above theorem.
Reviewer: R.Kultze

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
55Q50 \(J\)-morphism
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