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Stable extendibility of vector bundles over lens spaces. (English) Zbl 1419.55016

Summary: Firstly, we obtain conditions for stable extendibility and extendibility of complex vector bundles over the \((2n + 1)\)-dimensional standard lens space \(L^n(p) \pmod p\), where \(p\) is a prime. Secondly, we prove that the complexification \(c(\tau_n (p))\) of the tangent bundle \(\tau_n (p) (=\tau(L^n(p)))\) of \(L^n(p)\) is extendible to \(L^{2n+1}(p)\) if \(p\) is a prime, and is not stably extendible to \(L^{2n+2}(p)\) if \(p\) is an odd prime and \(n \geq 2p-2\). Thirdly, we show, for some odd prime \(p\) and positive integers \(n\) and \(m\) with \(m > n\), that \(\tau(L^n(p))\) is stably extendible to \(L^m(p)\) but is not extendible to \(L^m(p)\).

MSC:

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