Talk:Homotopy spheres II (Ex)

Exercise 3.

Suppose that $f:\Sigma\times S^k\rightarrow S^n\times S^k$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}f:\Sigma\times S^k\rightarrow S^n\times S^k$ is a diffeomorphism. Then $f|_{\Sigma\times\cdot}$$f|_{\Sigma\times\cdot}$ is an embedding of $\Sigma$$\Sigma$ into $S^n\times S^k$$S^n\times S^k$ with trivial normal bundle. Therefore the composition of $f|_{\Sigma\times\cdot}$$f|_{\Sigma\times\cdot}$ with the standard embedding $S^n\times S^k\rightarrow S^{n+k+1}$$S^n\times S^k\rightarrow S^{n+k+1}$ is an embedding $\Sigma\rightarrow S^{n+k+1}$$\Sigma\rightarrow S^{n+k+1}$ with trivial normal bundle.

Now assume that there is an embedding of $\Sigma$$\Sigma$ into $S^{n+k+1}$$S^{n+k+1}$ with trivial normal bundle. Consider the tubular neighborhood of $\Sigma$$\Sigma$ in $S^{n+k+1}$$S^{n+k+1}$. Its boundary is $\Sigma\times S^k$$\Sigma\times S^k$ embedded into $S^{n+k+1}$$S^{n+k+1}$.

Denote by $A$$A$ the "interior" connected component of $S^{n+k+1}-\Sigma\times S^k$$S^{n+k+1}-\Sigma\times S^k$, i.e. the component homeomorphic to $\Sigma\times D^{k+1}$$\Sigma\times D^{k+1}$. Let $S^n\times S^k$$S^n\times S^k$ be standardly embedded into $S^{n+k+1}$$S^{n+k+1}$ and disjoint with $\Sigma\times S^k$$\Sigma\times S^k$. Denote by $B$$B$ the "exterior" connected component of $S^{n+k+1}-S^n\times S^k$$S^{n+k+1}-S^n\times S^k$, i.e. the component homeomorphic to $D^{n+1}\times S^k$$D^{n+1}\times S^k$.

Then $A\cup B=S^{n+k+1}$$A\cup B=S^{n+k+1}$ and $W:=A\cap B$$W:=A\cap B$ is a manifold with boundary $\Sigma\times S^k \sqcup S^n\times S^k$$\Sigma\times S^k \sqcup S^n\times S^k$. Using Mayer–Vietoris sequence for $S^{n+k+1}, A, B$$S^{n+k+1}, A, B$ we obtain that the homologies of $W$$W$ equal to the homologies of $\Sigma\times S^k$$\Sigma\times S^k$ (or $S^n\times S^k$$S^n\times S^k$) and that inclusions $\Sigma\times S^k\hookrightarrow \partial W$$\Sigma\times S^k\hookrightarrow \partial W$ and $S^n\times S^k\hookrightarrow \partial W$$S^n\times S^k\hookrightarrow \partial W$ induce isomorphism in homologies. Since both $\Sigma\times S^k$$\Sigma\times S^k$ and $S^n\times S^k$$S^n\times S^k$ are simply-connected then these inclusions are homotopy equivalences.

So, $W$$W$ is an $h$$h$-cobordism. Therefore since $n\geq 5$$n\geq 5$ then $h$$h$-cobordism Theorem implies that $\Sigma\times S^k$$\Sigma\times S^k$ is diffeomorphic to $S^n\times S^k$$S^n\times S^k$.