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 $\pi_3(O_5)$. |
− | Given that $\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$, | + | |
+ | Given that $J:\pi_3(O_5)\to \pi_3(G_5)$ 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( | + | has a sphere bundle $\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have |
− | homotopy equivalences | + | homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$. |
+ | |||
+ | From [[Tangent_bundles_of_bundles_(Ex)|another exercise]] we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. | ||
+ | From [[Obstruction_classes_and_Pontrjagin_classes_(Ex)|a third exercise]] we know that the first Pontryagin class of $\xi_k$ is $48k$ times the generator of $H^4(S^4)$. | ||
+ | |||
+ | It follows that the first Pontryagin class of $S(\xi_k)$ | ||
+ | is non-trivial, since under $f_k$ the map $\pi$ just corresponds to projection to one factor. | ||
+ | Hence $f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$ has | ||
+ | stably trivial tangent bundle, hence trivial $p_1$. | ||
+ | |||
+ | Similarly one can argue with $(4n+1)$-dimensional vector bundles over $S^{4n}$; the $J$-homomorphism has always | ||
+ | a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces | ||
+ | homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$. | ||
</wikitex> | </wikitex> |
Latest revision as of 20:32, 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 .
From another exercise we know that stably . From a third exercise we know that the first Pontryagin class of is times the generator of .
It follows that the first Pontryagin class of is non-trivial, since under the map just corresponds to projection to one factor. Hence is a homotopy equivalence which doesn't preserve the first Pontryagin class, as has stably trivial tangent bundle, hence trivial .
Similarly one can argue with -dimensional vector bundles over ; the -homomorphism has always a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces homotopy equivalences which do not preserve .