# Talk:Fibre homotopy trivial bundles (Ex)

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homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$. | homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$. | ||

− | From another exercise we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. | + | From another [[Tangent_bundles_of_bundles_(Ex)|exercise]] we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. |

− | From a third exercise we know that the first Pontryagin class of $\xi_k$ is $48k$. | + | From a third [[Obstruction_classes_and_Pontrjagin_classes_(Ex)|exercise]] we know that the first Pontryagin class of $\xi_k$ is $48k$. |

It follows that the first Pontryagin class of $S(\xi_k)$ | It follows that the first Pontryagin class of $S(\xi_k)$ |

## Revision as of 19:21, 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 .

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 .