Talk:Fibre homotopy trivial bundles (Ex)

We consider 5-dimensional real vector bundles over $S^4$$<wikitex>; We consider 5-dimensional real vector bundles over $S^4$. Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$. Given that $J:\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$, we see that the vector bundle $\xi_k$ corresponding to k$ times the generator has a sphere bundle $\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$. From [[Tangent_bundles_of_bundles_(Ex)|another exercise]] we know that stably $TS(\xi_k)\cong \pi^*\xi_k$. From [[Obstruction_classes_and_Pontrjagin_classes_(Ex)|a third exercise]] we know that the first Pontryagin class of $\xi_k$ is k$. It follows that the first Pontryagin class of $S(\xi_k)$ is non-trivial, since under $f_k$ the map $\pi$ just corresponds to projection to one factor. Hence $f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$ has stably trivial tangent bundle, hence trivial $p_1$. Similarly one can argue with $(4n+1)$-dimensional vector bundles over $S^{4n}$; the $J$-homomorphism has always a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$. </wikitex>S^4$. Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$$\pi_3(O_5)$.

Given that $J:\pi_3(O_5)\to \pi_3(G_5)$$J:\pi_3(O_5)\to \pi_3(G_5)$ is isomorphic to the surjection $\mathbb Z\to \mathbb Z/24$$\mathbb Z\to \mathbb Z/24$, we see that the vector bundle $\xi_k$$\xi_k$ corresponding to $24k$$24k$ times the generator has a sphere bundle $\pi:S(\xi_k)\to S^4$$\pi:S(\xi_k)\to S^4$ which is fiber homotopically trivial, so in particular we have homotopy equivalences $f_k:S(\xi_k)\to S^4\times S^4$$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$$TS(\xi_k)\cong \pi^*\xi_k$. From a third exercise we know that the first Pontryagin class of $\xi_k$$\xi_k$ is $48k$$48k$.

It follows that the first Pontryagin class of $S(\xi_k)$$S(\xi_k)$ is non-trivial, since under $f_k$$f_k$ the map $\pi$$\pi$ just corresponds to projection to one factor. Hence $f_k$$f_k$ is a homotopy equivalence which doesn't preserve the first Pontryagin class, as $S^4\times S^4$$S^4\times S^4$ has stably trivial tangent bundle, hence trivial $p_1$$p_1$.

Similarly one can argue with $(4n+1)$$(4n+1)$-dimensional vector bundles over $S^{4n}$$S^{4n}$; the $J$$J$-homomorphism has always a non-trivial kernel, and the top Pontryagin class of the corresponding bundles are non-zero. This produces homotopy equivalences $S(\xi_k)\to S^{4n}\times S^{4n}$$S(\xi_k)\to S^{4n}\times S^{4n}$ which do not preserve $p_n$$p_n$.