Let $H(n)$ be the group of self-homotopy-equivalences of $S^n$ preserving the basepoint. I read that $H(n)$ may be identified with ''two components of $\Omega^nS^n$''.
What does this mean and how can I see it?
Let $H(n)$ be the group of self-homotopy-equivalences of $S^n$ preserving the basepoint. I read that $H(n)$ may be identified with ''two components of $\Omega^nS^n$''.
What does this mean and how can I see it?
Maps $[S^n,S^n]$ are classified by degree. For each $z\in\mathbb{Z}$ there is a component of $\Omega^nS^n$. The only maps that are homotopy equivalences have degree $1$ and $-1$.
EDIT: In answer to the comment below from JasonDevito: For any finite CW complex $X$, $[S^1,X]$ is the same thing as $[S^0,\Omega X]$, since $\Omega$ and $\Sigma$ (reduced suspension) are adjoints. But, based maps from $S^0$ to $Y$ is the same as $Y$ for any finite CW complex. Thus
$ [S^1,X]=[S^0,\Omega X]=\Omega X. $
Iterate this and you get
$ [S^n,S^n]=[\Sigma S^{n-1},S^n]=[S^{n-1},\Omega S^n]=\cdots =\Omega^n S^n. $