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Stable extendibility of the tangent bundles over lens spaces. (English) Zbl 1140.55011
A real vector bundle over a subspace $$A$$ of a topological space $$X$$ is called stably extendible if there is a vector bundle over $$X$$ whose restriction to $$A$$ is stably equivalent to the given one. In the paper under review the authors consider the inclusions of lens spaces $$L^{2n+1}(p) \subset L^{2m+1}(p)$$ for $$m\geq n$$ and a given prime $$p$$, and they ask for the maximal number $$m$$ for which the tangent bundle $$TL^{2n+1}(p)$$ is stably extendible. For $$p=2$$ the question was studied in [R. L. E. Schwarzenberger, Quart. J. Math. Oxford (2) 17, 19–21 (1966; Zbl 0141.40502)], and for $$p=3$$ T. Kobayashi and K. Komatsu proved in [Hiroshima Math. J. 35, 403–412 (2005; Zbl 1096.55010)] that the maximal number is infinity if and only if $$0\leq n \leq 3$$.
One of the main results of the present paper states that for an odd prime $$p$$ the maximal number is $$2n+1$$ if $$n \geq 2p-2$$, which together with the result of Kobayashi and Komatsu yields a complete answer for the case $$p=3$$. For $$p=5$$ and $$p=7$$ the authors also achieve complete answers: they obtain that the number is infinity if $$n \leq p$$, and that it is $$2n+1$$ in case $$n>p$$. The latter yields an improvement of their first result, and conjecturally reflects the general case. In their more recent article [Topology Appl. 154 (18), 3145–3155 (2007; Zbl 1134.55012)] the authors in fact could verify major parts of the conjectural statement. In particular they show that the second assertion concerning the case $$n>p$$ does hold for all odd primes $$p$$.

##### MSC:
 55R50 Stable classes of vector space bundles in algebraic topology and relations to $$K$$-theory 55N15 Topological $$K$$-theory