## 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