Homotopy spheres II (Ex)

In the following, $\Sigma \in \Theta_n$$\newcommand{\Z}{\mathbb{Z}} \newcommand{\R}{\mathbb{R}} \newcommand{\C}{\mathbb{C}} \newcommand{\N}{\mathbb{N}} \newcommand{\Q}{\mathbb{Q}} \newcommand{\F}{\mathbb{F}} \newcommand{\bZ}{\mathbb{Z}} \newcommand{\bR}{\mathbb{R}} \newcommand{\bC}{\mathbb{C}} \newcommand{\bH}{\mathbb{H}} \newcommand{\bQ}{\mathbb{Q}} \newcommand{\bF}{\mathbb{F}} \newcommand{\bN}{\mathbb{N}} \DeclareMathOperator\id{id} % identity map \DeclareMathOperator\Sq{Sq} % Steenrod squares \DeclareMathOperator\Homeo{Homeo} % group of homeomorphisms of a topoloical space \DeclareMathOperator\Diff{Diff} % group of diffeomorphisms of a smooth manifold \DeclareMathOperator\SDiff{SDiff} % diffeomorphism under some constraint \DeclareMathOperator\Hom{Hom} % homomrphism group \DeclareMathOperator\End{End} % endomorphism group \DeclareMathOperator\Aut{Aut} % automorphism group \DeclareMathOperator\Inn{Inn} % inner automorphisms \DeclareMathOperator\Out{Out} % outer automorphism group \DeclareMathOperator\vol{vol} % volume \newcommand{\GL}{\text{GL}} % general linear group \newcommand{\SL}{\text{SL}} % special linear group \newcommand{\SO}{\text{SO}} % special orthogonal group \newcommand{\O}{\text{O}} % orthogonal group \newcommand{\SU}{\text{SU}} % special unitary group \newcommand{\Spin}{\text{Spin}} % Spin group \newcommand{\RP}{\Rr\mathrm P} % real projective space \newcommand{\CP}{\Cc\mathrm P} % complex projective space \newcommand{\HP}{\Hh\mathrm P} % quaternionic projective space \newcommand{\Top}{\mathrm{Top}} % topological category \newcommand{\PL}{\mathrm{PL}} % piecewise linear category \newcommand{\Cat}{\mathrm{Cat}} % any category \newcommand{\KS}{\text{KS}} % Kirby-Siebenmann class \newcommand{\Hud}{\text{Hud}} % Hudson torus \newcommand{\Ker}{\text{Ker}} % Kernel \newcommand{\underbar}{\underline} %Classifying Spaces for Families of Subgroups \newcommand{\textup}{\text} \newcommand{\sp}{^}\Sigma \in \Theta_n$ is a homotopy sphere, $n \geq 5$$n \geq 5$, and $bP_{n+1} \in \Theta_n$$bP_{n+1} \in \Theta_n$ is the group of homotopy spheres bounding parallelisable manifolds.

Exercise 0.1.

1. Show that $\Sigma$$\Sigma$ embeds into $S^{n+1}$$S^{n+1}$ if and only if $\Sigma$$\Sigma$ is diffeomorphic to $S^n$$S^n$.
2. Show that $\Sigma$$\Sigma$ embeds into $S^{n+2}$$S^{n+2}$ if and only if $\Sigma \in bP_{n+1}$$\Sigma \in bP_{n+1}$.
3. For $k \geq 2$$k \geq 2$, show that $\Sigma$$\Sigma$ embeds into $S^{n+k+1}$$S^{n+k+1}$ with trivial normal bundle if and only if there is a diffeomorphism $\Sigma \times S^k \cong S^n \times S^k$$\Sigma \times S^k \cong S^n \times S^k$.
4. Show that if $\Sigma \in bP_{n+1}$$\Sigma \in bP_{n+1}$, then for all $k \geq 2$$k \geq 2$, there is a diffeomorphism $\Sigma \times S^k \cong S^n \times S^k$$\Sigma \times S^k \cong S^n \times S^k$.