Talk:Integral homology 3-spheres embed in the 4-sphere (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. By Freedman, there exists a (unique) contractible topological $4$$4$-manifold $\Omega^4$$\Omega^4$ with $\partial Y=\Omega$$\partial Y=\Omega$.

Let $W:=\Omega\cup_{Y\times I}(-\Omega)$$W:=\Omega\cup_{Y\times I}(-\Omega)$, where we identify the two copies of $Y$$Y$ in the boundary by the identity. Since $\pi_1(\Omega)=0$$\pi_1(\Omega)=0$, Seifert van Kampen says that $\pi_1(W)=0$$\pi_1(W)=0$. Moreover, $\tilde{H}_*(\Omega)=0$$\tilde{H}_*(\Omega)=0$ And $\tilde{H}_*(\Omega\cap -\Omega)=\tilde{H}_*(Y)=0$$\tilde{H}_*(\Omega\cap -\Omega)=\tilde{H}_*(Y)=0$, so the Mayer-Vietoris sequence says that $\tilde{H}_*(W)=0$$\tilde{H}_*(W)=0$. In particular, $H_2(W)=0$$H_2(W)=0$, so by Freedman’s theorem $W$$W$ is homeomorphic to $S^4$$S^4$. Therefore,

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$Y\into W^4\cong S^4$ is a (topological) embedding of $Y$$Y$ into $S^4$$S^4$.