K-invariant for G/PL (Ex)

In talk 16 the $2$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}2$-local homotopy type of $G/PL$$G/PL$ was described as being

$\displaystyle (G/PL)[2]\cong X^4 \times \prod_{n> 1} K(\mathbb{Z},4n)\times K(\mathbb{Z}_2,4n-2)~,$ $$\displaystyle (G/PL)[2]\cong X^4 \times \prod_{n> 1} K(\mathbb{Z},4n)\times K(\mathbb{Z}_2,4n-2)~,$$

where $X^4$$X^4$ is a two stage Postnikov system with nontrivial homotopy groups $\pi_2(X)=\mathbb{Z}_2$$\pi_2(X)=\mathbb{Z}_2$, $\pi_4(X)=\mathbb{Z}[2]$$\pi_4(X)=\mathbb{Z}[2]$ and (non-trivial) Postnikov invariant $k^4\in H^5(K(\mathbb{Z},2),\mathbb{Z}[2])$$k^4\in H^5(K(\mathbb{Z},2),\mathbb{Z}[2])$. The goal of this exercise is to show that $k^4$$k^4$ is nonzero.

Exercise 0.1. Let $\sigma_4:\pi_4(G/PL)\to \mathbb{Z}$$\sigma_4:\pi_4(G/PL)\to \mathbb{Z}$ be the surgery obstruction map, and $\sigma_4[2]$$\sigma_4[2]$ its factorization through the localized Hurewicz map $h_{MSO[2]}$$h_{MSO[2]}$, so that $\sigma_4=\sigma_4[2]\circ h_{MSO[2]}$$\sigma_4=\sigma_4[2]\circ h_{MSO[2]}$. Show that if $\sigma_4[2]$$\sigma_4[2]$ is surjective, then $k^4$$k^4$ is nonzero.

Exercise 0.2. Show that $\sigma_4[2]$$\sigma_4[2]$ is surjective by constructing a $PL$$PL$-manifold $M^4$$M^4$ and a map $f:M\to G/PL$$f:M\to G/PL$ with surgery invariant 1.

Hint 0.3. Try to construct a degree one normal map $g:\mathbb{C}P^2 \,\sharp \,8 \overline{\mathbb{C}P^2} \to \mathbb{C}P^2$$g:\mathbb{C}P^2 \,\sharp \,8 \overline{\mathbb{C}P^2} \to \mathbb{C}P^2$, where $\sharp$$\sharp$ denotes connected sum and $\overline{\mathbb{C}P^2}$$\overline{\mathbb{C}P^2}$ denotes $\mathbb{C}P^2$$\mathbb{C}P^2$ with the conjugate complex struture.