8.3 $\pi_{k\leq n}$ of an n-connected space and $\pi_{k

Let $(A,a)$ be a pointed type and $n:\mathbb{N}$. Recall from \autorefthm:homotopy-groups that if $n>0$ the set $\pi_{n}(A,a)$ has a group structure  , and if $n>1$ the group is abelian.

We can now say something about homotopy groups  of $n$-truncated and $n$-connected types.

Lemma 8.3.1.

If $A$ is $n$-truncated and $a:A$, then $\pi_{k}(A,a)=\mathbf{1}$ for all $k>n$.

Proof.

The loop space  of an $n$-type is an $(n-1)$-type, hence $\Omega^{k}(A,a)$ is an $(n-k)$-type, and we have $(n-k)\leq-1$ so $\Omega^{k}(A,a)$ is a mere proposition. But $\Omega^{k}(A,a)$ is inhabited, so it is actually contractible  and $\pi_{k}(A,a)=\mathopen{}\left\|\Omega^{k}(A,a)\right\|_{0}\mathclose{}=% \mathopen{}\left\|\mathbf{1}\right\|_{0}\mathclose{}=\mathbf{1}$. ∎

Lemma 8.3.2.

If $A$ is $n$-connected and $a:A$, then $\pi_{k}(A,a)=\mathbf{1}$ for all $k\leq{}n$.

Proof.
 $\pi_{k}(A,a)=\mathopen{}\left\|\Omega^{k}(A,a)\right\|_{0}\mathclose{}=\Omega^% {k}(\mathopen{}\left\|(A,a)\right\|_{k}\mathclose{})=\Omega^{k}(\mathopen{}% \left\|\mathopen{}\left\|(A,a)\right\|_{n}\mathclose{}\right\|_{k}\mathclose{}% )=\Omega^{k}(\mathopen{}\left\|\mathbf{1}\right\|_{k}\mathclose{})=\Omega^{k}(% \mathbf{1})=\mathbf{1}.$

The third equality uses the fact that $k\leq{}n$ in order to use that $\|\mathord{\hskip 1.0pt\text{--}\hskip 1.0pt}\|_{k}\circ\|\mathord{\hskip 1.0% pt\text{--}\hskip 1.0pt}\|_{n}=\|\mathord{\hskip 1.0pt\text{--}\hskip 1.0pt}\|% _{k}$ and the fourth equality uses the fact that $A$ is $n$-connected. ∎

Corollary 8.3.3.

$\pi_{k}(\mathbb{S}^{n})=\mathbf{1}$ for $k.

Proof.

The sphere $\mathbb{S}^{n}$ is $(n-1)$-connected by \autorefcor:sn-connected, so we can apply \autoreflem:pik-nconnected. ∎

Title 8.3 $\pi_{k\leq n}$ of an n-connected space and $\pi_{k