Talk:Fibre homotopy trivial bundles (Ex)

Revision as of 20:07, 29 May 2012

We consider 5-dimensional real vector bundles over $<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 $\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$, we see that the vector bundle $\xi_k$ corresponding to k$ times the generator has a sphere bundle $S(\xi_k)$ which is fiber homotopically trivial, so in particular we have homotopy equivalences </wikitex>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 $S(\xi_k)$ which is fiber homotopically trivial, so in particular we have homotopy equivalences