# Fibre homotopy trivial bundles (Ex)

Exercise 0.1.

1. Observe that $\pi_i(G_{k+1})$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\pi_i(G_{k+1})$ classifies $k$$k$-spherical fibrations over $S^{i+1}$$S^{i+1}$. Using the isomorphisms $\pi_3(G_5) \cong \pi_3^S \cong \Zz/24$$\pi_3(G_5) \cong \pi_3^S \cong \Zz/24$, $\pi_3(O_5) \cong \Zz$$\pi_3(O_5) \cong \Zz$ and the fact that the J-homomophism in dimension $3$$3$ is isomorphic to the surjective homomorphism $\Zz \to \Zz/24$$\Zz \to \Zz/24$, find a homotopy equivalence of manifolds $f \colon M_0 \simeq M_1$$f \colon M_0 \simeq M_1$ such that $f^*p(M_0) \neq p(M_1)$$f^*p(M_0) \neq p(M_1)$. Here $p(M_i) \in H^{4*}(M_i; \Zz)$$p(M_i) \in H^{4*}(M_i; \Zz)$ denotes the total Pontrjagin class.
2. The above exercise showed that the first Pontrjain class, $p_1$$p_1$, is not a homotopy invariant. Apply the same idea to show that $p_k$$p_k$ is not a homotopy invariant for any $k \geq 1$$k \geq 1$.

Remark 0.2. A reference to Novikov is needed here.