# Talk:Fibre homotopy trivial bundles (Ex)

(Difference between revisions)
We consider 5-dimensional real vector bundles over $S^4$$ We consider 5-dimensional real vector bundles over S^4. Isomorphism classes of these are given by their clutching function in \pi_3(O_5) Given that \pi_3(O_5)\to \pi_3(G_5) is isomorphic to the surjection \mathbb Z\to \mathbb Z/24, we see that the vector bundle \xi_k corresponding to k times the generator has a sphere bundle S(xik) which is fiber homotopically trivial, so in particular we have homotopy equivalences S^4$. Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$$\pi_3(O_5)$ Given that $\pi_3(O_5)\to \pi_3(G_5)$$\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$$\mathbb Z\to \mathbb Z/24$, we see that the vector bundle $\xi_k$$\xi_k$ corresponding to $24k$$24k$ times the generator has a sphere bundle $S(xik)$$S(xik)$ which is fiber homotopically trivial, so in particular we have homotopy equivalences