## Hiroshima Mathematical Journal

- Hiroshima Math. J.
- Volume 48, Number 1 (2018), 57-66.

### Stable extendibility and extendibility of vector bundles over lens spaces

Mitsunori Imaoka and Teiichi Kobayashi

#### Abstract

Firstly, we obtain conditions for stable extendibility and extendibility of complex vector bundles over the $(2n+1)$-dimensional standard lens space $L^n(p)$ mod $p$, 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 $n \ge 2p-2$. 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 $L^m(p)$.

#### Article information

**Source**

Hiroshima Math. J., Volume 48, Number 1 (2018), 57-66.

**Dates**

Received: 14 September 2016

Revised: 16 June 2017

First available in Project Euclid: 8 March 2018

**Permanent link to this document**

https://projecteuclid.org/euclid.hmj/1520478023

**Digital Object Identifier**

doi:10.32917/hmj/1520478023

**Mathematical Reviews number (MathSciNet)**

MR3772000

**Zentralblatt MATH identifier**

06329508

**Subjects**

Primary: 55R50: Stable classes of vector space bundles, $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX}

Secondary: 57R25: Vector fields, frame fields

**Keywords**

vector bundle tangent bundle lens space extendible stably extendible K-theory

#### Citation

Imaoka, Mitsunori; Kobayashi, Teiichi. Stable extendibility and extendibility of vector bundles over lens spaces. Hiroshima Math. J. 48 (2018), no. 1, 57--66. doi:10.32917/hmj/1520478023. https://projecteuclid.org/euclid.hmj/1520478023