Talk:Uniqueness of contractible coboundary (Ex)

Let $Y^3$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}Y^3$ be a homology $3$$3$-sphere bounding contractible $4$$4$-manifolds $\Omega_1^4$$\Omega_1^4$ and $\Omega_2^4$$\Omega_2^4$. Let $W:=\Omega_1\cup_{Y\times 0}(Y\times I)\cup_{Y\times 1}(-\Omega_2)$$W:=\Omega_1\cup_{Y\times 0}(Y\times I)\cup_{Y\times 1}(-\Omega_2)$. By Seifert van-Kampen and Mayer-Vietoris, $\pi_1(W)=0$$\pi_1(W)=0$ and $H_2(W;\mathbb{Z})=0$$H_2(W;\mathbb{Z})=0$, so Freedman’s theorem says that $W$$W$ is homeomorphic to $S^4$$S^4$. Therefore, $W$$W$ bounds $V^5$$V^5$, where $V$$V$ is homeomorphic to the $5$$5$-ball.

Thus, we have constructed a cobordism $V$$V$ from $\Omega_1$$\Omega_1$ to $\Omega_2$$\Omega_2$. Since $\Omega_1,\Omega_2$$\Omega_1,\Omega_2$, and $V$$V$ are all contractible, this is an $h$$h$-cobordism. By the $h$$h$-cobordism theorem for manifolds with boundary, there is a homeomorphism $\phi:V\to\Omega_2\times I$$\phi:V\to\Omega_2\times I$ which is the indentity on $Y\times I$$Y\times I$. Therefore, $\phi\vert_{\Omega_1}$$\phi\vert_{\Omega_1}$ is a homeomorphism from $\Omega_1$$\Omega_1$ to $\Omega_2$$\Omega_2$ fixing the boundary pointwise.