# Talk:Fibre homotopy trivial bundles (Ex)

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(Created page with "<wikitex> We consider 5-dimensional real vector bundles over $S^4$. Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$ Given that $\pi_3(O_5)\t...") |
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We consider 5-dimensional real vector bundles over $S^4$. | We consider 5-dimensional real vector bundles over $S^4$. | ||

Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$ | Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$ | ||

− | Given that $\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\Z\to \Z/24$, | + | Given that $\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$, |

we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator | we see that the vector bundle $\xi_k$ corresponding to $24k$ times the generator | ||

− | has a sphere bundle $S(xik$ | + | has a sphere bundle $S(xik)$ which is fiber homotopically trivial, so in particular we have |

+ | homotopy equivalences | ||

</wikitex> | </wikitex> |

## Revision as of 19:01, 29 May 2012

We consider 5-dimensional real vector bundles over .
Isomorphism classes of these are given by their clutching function in
Given that is isomorphic to the surjection ,
we see that the vector bundle corresponding to times the generator
has a sphere bundle which is fiber homotopically trivial, so in particular we have
homotopy equivalences