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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 \(p\).

55R50 Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
55N15 Topological \(K\)-theory