Stability of vector bundles (Ex)

Latest revision as of 01:45, 6 January 2019

Let $<wikitex>; Let $E \to X$ and $F \to X$ be rank $k$ vector bundles over an $n$-dimensional CW-complex $X$ with $k > n$. 1) Prove that $E$ is isomorphism to $F$ if and only if $E$ is stably isomorphic to $F$; i.e.~there is an isomorphism $E \oplus \underline{\R^i} \cong F \oplus \underline{\R^i}$ for some $i \geq 0$, where $\underline{\R^i}$ denotes the trivial rank $i$ bundle over $X$. 2) If $k > n+1$, prove that bundle isomorphisms $\theta_1, \theta_2 \colon F \to E$ are fibrewise homotopic if and only if $\theta_1$ and $\theta_2$ are stably homotopic; i.e.~this is a fibrewise homotopy between $\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$ and $\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$ for some $i \geq 0$. </wikitex> == References == {{#RefList:}} [[Category:Exercises]] [[Category:Exercises without solution]]E \to X$ and $F \to X$ be rank $k$ vector bundles over an $n$-dimensional CW-complex $X$ with $k > n$.

1) Prove that $E$ is isomorphic to $F$ if and only if $E$ is stably isomorphic to $F$; i.e. there is an isomorphism $E \oplus \underline{\R^i} \cong F \oplus \underline{\R^i}$ for some $i \geq 0$, where $\underline{\R^i}$ denotes the trivial rank $i$ bundle over $X$.

2) If $k > n+1$, prove that bundle isomorphisms $\theta_1, \theta_2 \colon F \to E$ are fibrewise homotopic if and only if $\theta_1$ and $\theta_2$ are stably homotopic; i.e. there is a fibrewise homotopy between $\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$ and $\theta_1 \oplus \mathrm{id}_{\underline{\R^i}}$ for some $i \geq 0$.

References