Smoothing the Kervaire manifold (Ex)

Let $M_K$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M_K$ be the $PL$$PL$ Kervaire $(4k+2)$$(4k+2)$-manifold and let $\tau \colon M_K \to BPL$$\tau \colon M_K \to BPL$ and let $B\pi \colon BPL \to B(PL/O)$$B\pi \colon BPL \to B(PL/O)$ be the canonical map. In addition let $\Sigma_K \in \Theta_{4k+1}$$\Sigma_K \in \Theta_{4k+1}$ be the Kervaire sphere and let

$\displaystyle \Psi \colon \Theta_{4k+1} \cong \pi_{4k+1}(PL/O)$ $$\displaystyle \Psi \colon \Theta_{4k+1} \cong \pi_{4k+1}(PL/O)$$

be the bijection defined taking the standard sphere as the base-point for smooth structures on $S^{4k+1}$$S^{4k+1}$.

Exercise 0.1.

1. Show that the Spivak normal fibration of $M_K$$M_K$ admits a vector bundle reduction.
2. Determine the homotopy class of $B \pi \circ \tau \colon M_K \to B(PL/O)$$B \pi \circ \tau \colon M_K \to B(PL/O)$ in terms of $\Psi(\Sigma_K) \in \pi_{4k+1}(PL/O)$$\Psi(\Sigma_K) \in \pi_{4k+1}(PL/O)$.