Stable extendibility of vector bundles over real projective spaces and bounds for theSchwarzenberger numbers $\beta(k)$

概要

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For a non-negative integer $k$, R. L. E. Schwarzenberger defined in [{\bf7}] aninteger $\beta(k)\ge 0$ which we call the Schwarzenberger number of $k$. Let $\zeta$ be a$k$-dimensional $\mF$-vector bundle over the real projective $n$-space $\mRP^{n}$, where $\mF$ is either thereal number field $\mR$ or the complex number field $\mC$. Then $\beta(k)$ is closely related to the problem tofind thedimension $m$ with $m\ge n$ which has the property that $\zeta$ is stably equivalent to a sum of $k$ $\mF$-linebundles if $\zeta$ is stably extendible to $\mRP^{m}$. The problem for $\mF=\mR$ has been studied in[{\bf7}], [{\bf5}] and [{\bf4}], and that for $\mF=\mC$ has been studied in [{\bf6}] and [{\bf4}]. Inthis note we obtain further results on the problem and determine bounds for the Schwarzenberger numbers$\beta(k)$.
広島大学の論文