Talk:Fibre homotopy trivial bundles (Ex)

We consider 5-dimensional real vector bundles over $S^4$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}S^4$. Isomorphism classes of these are given by their clutching function in $\pi_3(O_5)$$\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$$48k$ times the generator of $H^4(S^4)$$H^4(S^4)$.

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$.