Talk:Fibre homotopy trivial bundles (Ex)
Line 9: | Line 9: | ||
From [[Tangent_bundles_of_bundles_(Ex)|another exercise]] we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. | 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 $48k$. | + | 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 $S(\xi_k)$ | It follows that the first Pontryagin class of $S(\xi_k)$ |
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 .