Homotopy spheres I (Ex)

In the following, you should use the $h$$\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}{^}h$-cobordism Theorem, [Smale1962a, Theorem 3.1].

Exercise 0.1.

1. Show that every homotopy $n$$n$-sphere $\Sigma$$\Sigma$ is diffeomorphic to a manifold
   $\displaystyle \Sigma_f : = D^n \cup_f D^n$ $$\displaystyle \Sigma_f : = D^n \cup_f D^n$$
   where $f \colon S^{n-1} \cong S^{n-1}$$f \colon S^{n-1} \cong S^{n-1}$ is a diffeomorphism.
2. Using 1. show that the group of homotopy $n$$n$-spheres, $\Theta_n$$\Theta_n$, can be identified with the set of oriented diffeomorphism classes of smooth structures on $S^n$$S^n$, at least for $n \geq 6$$n \geq 6$.
3. What happens for $n = 5$$n = 5$?
4. If $\Sigma \in \Theta_n$$\Sigma \in \Theta_n$ is a homotopy $n$$n$-sphere,$n \geq 5$$n \geq 5$, show that $\Sigma \sharp (-\Sigma)$$\Sigma \sharp (-\Sigma)$ is diffeomorphic to $S^n$$S^n$: this is [Kervaire&Milnor1963, Lemma 2.4].

References

• [Kervaire&Milnor1963] M. A. Kervaire and J. W. Milnor, Groups of homotopy spheres. I, Ann. of Math. (2) 77 (1963), 504–537. MR0148075 (26 #5584) Zbl 0115.40505
• [Smale1962a] S. Smale, On the structure of manifolds, Amer. J. Math. 84 (1962), 387–399. MR0153022 (27 #2991) Zbl 0109.41103