I encountered the following problem:
Let $X$ be a closed manifold, $S^n$ be the $n$-dimensional standard sphere and denote $\Omega(S^n,X)$ be the space of base point-preserving maps from $S^n$ to $X$. Then $\pi_{k+n}(X)=\pi_k(\Omega(S^n,X))$.
Where $\pi_k$ are the $k$-th homotopy group.
Did there any exact reference I can refer? Thanks!
First convince yourself that if $X, Y, Z$ are pointed sets then the set $[Y, Z]$ of pointed maps $Y \to Z$ is itself pointed by the map which sends all of $Y$ to the basepoint of $Z$, and that there is a natural isomorphism
$$\text{Hom}(X \wedge Y, Z) \cong \text{Hom}(X, [Y, Z])$$
where $X \wedge Y$ denotes the smash product, the quotient of $X \times Y$ obtained by identifying $x_0 \times Y$ and $X \times y_0$ with the basepoint $x_0 \times y_0$ of $X \times Y$. The point is that both sides are naturally isomorphic to the set of all maps $X \times Y \to Z$ which are "pointed in each argument"; this is very much like the tensor-hom adjunction in linear algebra, and implies that the category of pointed sets has a natural closed monoidal structure.
Now it shouldn't be hard to believe that the same is true of spaces. Modulo some technical conditions, if $X, Y, Z$ are pointed spaces it is still true that
$$\text{Hom}(X \wedge Y, Z) \cong \text{Hom}(X, [Y, Z])$$
where $[Y, Z]$ now denotes the space of pointed maps $Y \to Z$. In particular,
$$\text{Hom}(S^k \wedge S^n, Z) \cong \text{Hom}(S^k, [S^n, Z]) \cong \pi_k([S^n, Z]).$$
Now the only remaining observation is that $S^k \wedge S^n \cong S^{k+n}$ (exercise!), so that the LHS is $\pi_{k+n}(Z)$ as desired.
But even more fundamentally, because we can write $S^n \cong S^1 \wedge ... \wedge S^1$, we see that $[S^n, Z]$ is nothing more than the $n$-fold based loop space $\Omega^n Z$, where $\Omega Z \cong [S^1, Z]$. So we have
$$\pi_{k+n}(Z) \cong \pi_0(\Omega^{k+n}(Z)) \cong \pi_0(\Omega^k(\Omega^n Z)) \cong \pi_k(\Omega^n Z )$$
which is another way of thinking about this result.