Talk:Fibre homotopy trivial bundles (Ex)

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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 $\pi_3(O_5)$

Revision as of 19:03, 29 May 2012

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)\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 has a sphere bundle S(xik) which is fiber homotopically trivial, so in particular we have homotopy equivalences

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