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

概要

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In this paper, we prove that the tangent bundle $\tau(L^{n}(3))$ of the $(2n+1)$-dimensional mod 3 standard lens space $L^{n}(3)$ is stably extendible to $L^{m}(3)$ for every $m \geq n$ if and only if $0 \leq n \leq 3$. Combining this fact with the results obtained in [6],we see that $\tau(L^{2}(3))$ is stably extendible to $L^{3}(3)$, but is not extendible to $L^{3}(3)$. Furthermore, we prove that the $t$-fold power of $\tau(L^{n}(3))$ and its complexification are extendible to $L^{m}(3)$ for every $m \geq n$ if $t \geq 2$, and have a necessary and sufficient condition that the square $\nu^{2}$ of the normal bundle $\nu$ associated to an immersion of $L^{n}(3)$ in the Euclidean $(4n+3)$-space is extendible to $L^{m}(3)$ for every $m \geq n$.
広島大学の論文