Extendible and stably extendible vector bundles over real projective spaces. (English) Zbl 1041.55012

Kobayashi, Teiichi; Yoshida, Toshio

J. Math. Soc. Japan 55, No. 4, 1053-1059 (2003).

Let \(P^n\) denote the real projective \(n\)-space. Let \(\zeta\) denote the complexification of the tangent bundle of \(P^n\), the normal bundle associated to an immersion of \(P^n\) in \(\mathbb R^{n+k}\) or its complexification. This paper gives a necessary and sufficient condition that \(\zeta\) is extendible (or stably extendible) to \(P^m\) for every \(m > n\), that is, there is a vector bundle over \(P^m\) whose restriction to \(P^n\) is equivalent (or stably equivalent) to \(\zeta\) as a real or complex vector bundle according to whether \(\zeta\) is so or not. The answer for the case where \(\zeta\) is the normal bundle is particularly stated as follows: \(\zeta\) is stably extendible to \(P^m\) for every \(m > n\) if and only if \(\zeta\) is stably equivalent to a direct sum of \(s\) copies of the canonical real line bundle over \(P^n\) for some integer \(s\) with \(0 \leq s \leq k\). In general, it is important in algebraic topology to find that a vector bundle can be stably decomposed as a direct sum of line bundles.
This paper also contains examples of the normal bundle which is extendible to \(P^N\) but is not stably extendible to \(P^{N+1}\). Many other results of interest have been obtained by the authors et al. [see, e.g., T. Kobayashi, H. Maki and T. Yoshida, Osaka J. Math. 39, 315–324 (2002; Zbl 1007.55012)] concerning the subject treated here.

Reviewer: Haruo Minami (Nara)

MSC:

55R50	Stable classes of vector space bundles in algebraic topology and relations to \(K\)-theory
57R25	Vector fields, frame fields in differential topology
55R25	Sphere bundles and vector bundles in algebraic topology

Keywords:

vector bundle; extendible; stably extendible; tangent bundle; span; immersion; normal bundle; K-theory; KO-theory, real projective space

Citations:

Zbl 1007.55012