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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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