# Talk:Fibre homotopy trivial bundles (Ex)

Line 2: | Line 2: | ||

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 $S(\xi_k)$ which is fiber homotopically trivial, so in particular we have | + | 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 another 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 $48k$. 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> |

## Revision as of 19:16, 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 J-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 .