Immersing n-spheres in 2n-space (Ex)

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

$\displaystyle V_{n, n} \to V_{2n, 2n} \to V_{2n, n}.$ $$\displaystyle V_{n, n} \to V_{2n, 2n} \to V_{2n, n}.$$

Complete the following:

• Use the Hirsch-Smale immersion theorem to give an interpretation of the boundary map in the homotopy exact sequence of the above fibration:

$\displaystyle \partial \colon \pi_n(V_{2n, n}) \to \pi_{n-1}(V_{n, n})$ $$\displaystyle \partial \colon \pi_n(V_{2n, n}) \to \pi_{n-1}(V_{n, n})$$

• Hence prove the following: if $n \neq 1, 3, 7$$n \neq 1, 3, 7$, any immersion $f \colon S^n \to \Rr^{2n}$$f \colon S^n \to \Rr^{2n}$ is regularly homotopic to an embedding if and only if the normal bundle of $f$$f$ is trivial.
• Given an example of an immersion $S^1 \to \Rr^2$$S^1 \to \Rr^2$ with trivial normal bundle and which is not regularly homotopic to an embedding.