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 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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