Inertia group I (Ex)

Let $M/var/www/vhost/map.mpim-bonn.mpg.de/tmp/AppWikiTex/tex_e1EjDR$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}M$ be a closed smooth oriented $n$$n$-manifold. The inertia group of $M$$M$ is defined to be the following subgroup of $\Theta_n$$\Theta_n$:

$\displaystyle I(M) : = \{ [\Sigma] \in \Theta_n | M \sharp \Sigma \cong M \}$ $$\displaystyle I(M) : = \{ [\Sigma] \in \Theta_n | M \sharp \Sigma \cong M \}$$

where $\cong$$\cong$ denotes orientation preserving isomorphism.

Recall that by the $h$$h$-cobordism theorem every exotic sphere in dimension $6$$6$ and higher is a twisted double

$\displaystyle \Sigma_f \cong D^n \cup_f D^n$ $$\displaystyle \Sigma_f \cong D^n \cup_f D^n$$

for some orientation preserving diffeomorphism $f \colon S^{n-1} \cong S^{n-1}$$f \colon S^{n-1} \cong S^{n-1}$. (This is also true in dimension $5$$5$ since there are no exotic $5$$5$-spheres). Set

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$M^\bullet := M - {\rm int}(D^n)$ and identify $\partial M^\bullet = S^{n-1}$$\partial M^\bullet = S^{n-1}$. Show that $\Sigma_f \in I(M)$$\Sigma_f \in I(M)$ if and only if there is an orientation preserving diffeomorphism $F \colon M^\bullet \cong M^\bullet$$F \colon M^\bullet \cong M^\bullet$ with $F|_{S^{n-1}} = f$$F|_{S^{n-1}} = f$.

Hint: You may assume a theorem of Cerf which states that all orientation preserving embeddings $D^n \to M^n$$D^n \to M^n$ of the $n$$n$-disc into an oriented $n$$n$-manifold are ambient isotopic.