Hiroshima Mathematical Journal

Extendibility, stable extendibility and span of some vector bundles over lens spaces mod 3

Teiichi Kobayashi and Kazushi Komatsu

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Abstract

Let $L^{n}(3)$ be the $(2n+1)$-dimensional standard lens space mod 3 and let $\nu$ denote the normal bundle associated to an immersion of $L^{n}(3)$ in the Euclidean $(4n+3)$-space. In this paper we obtain a theorem on stable unextendibility of $R$-vector bundles over $L^{n}(3)$ improving some results in Extendibility and stable extendibility of vector bundles over lens spaces and Stable extendibility of normal bundles associated to immersions of real projective spaces and lens spaces, and study relations between stable extendibility and span of vector bundles over $L^{n}(3)$. Furtheremore, we prove that $c\nu$ is extendible to $L^{m}(3)$ for every $m > n$ if and only if $0 \leq n \leq 5$, and prove that $c(\nu \otimes \nu)$ is extendible to $L^{m}(3)$ for every $m > n$ if and only if $0 \leq n \leq 13$ or $n = 15$, where $c$ stands for the complexification and $\otimes$ denotes the tensor product.

Article information

Source
Hiroshima Math. J., Volume 37, Number 1 (2007), 45-60.

Dates
First available in Project Euclid: 11 April 2007

Permanent link to this document
https://projecteuclid.org/euclid.hmj/1176324094

Digital Object Identifier
doi:10.32917/hmj/1176324094

Mathematical Reviews number (MathSciNet)
MR2308523

Zentralblatt MATH identifier
1141.55010

Subjects
Primary: 55R50: Stable classes of vector space bundles, $K$-theory [See also 19Lxx] {For algebraic $K$-theory, see 18F25, 19-XX}
Secondary: 57R42: Immersions

Keywords
Extendible stably extendible span tensor product immersion normal bundle $KO$-theory $K$-theory lens space

Citation

Kobayashi, Teiichi; Komatsu, Kazushi. Extendibility, stable extendibility and span of some vector bundles over lens spaces mod 3. Hiroshima Math. J. 37 (2007), no. 1, 45--60. doi:10.32917/hmj/1176324094. https://projecteuclid.org/euclid.hmj/1176324094


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