Inertia group II (Ex)

Let $M$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a closed smooth oriented $n$$n$-manifold. The homotopy inertia group of $M$$M$, $I_H(M)$$I_H(M)$, is the subgroup of homotopy $n$$n$-spheres $\Theta_n$$\Theta_n$ defined by

Tex syntax error

$$\displaystyle I_H(M) : = \{ \Sigma | \exists f \colon \Sigma \sharp M \cong M, f \simeq {\rm Id}_M \},$$

where for the statement

Tex syntax error

$f \simeq {\rm Id}_M$ we regard $M$$M$ and $\Sigma \sharp M$$\Sigma \sharp M$ as the same topological space (with different smooth structures).

Exercise 0.1.

1. Show that $\Sigma \in I_H(M)$$\Sigma \in I_H(M)$ if and only if

   Tex syntax error

   ${\rm Id}_M \colon \Sigma \sharp M \to M$ is equivalent to

   Tex syntax error

   ${\rm Id}_M \colon M \to M$ in $\mathcal{S}(M)$$\mathcal{S}(M)$.
2. Given that $\Theta_8 \cong \textup{Coker}(J_8) \cong \Zz/2$$\Theta_8 \cong \textup{Coker}(J_8) \cong \Zz/2$, determine $I_H(\Hh P^2)$$I_H(\Hh P^2)$.
3. Assuming that $I(\Hh P^2) = \Theta_8$$I(\Hh P^2) = \Theta_8$, deduce that $\Hh P^2$$\Hh P^2$ admits a self-homotopy equivalence which is not homotopic to a diffeomorphism.