There is a closed curve that fills up the whole $\mathbb{S}^n$?

Question

I'm working with fundamental group of $\mathbb{S}^n$ where $n\geq 2$. Ok, we have that $\pi_{1}(\mathbb{S}^n)=0$. This means that every closed curve on $\mathbb{S}^n$ is homotopic to a point. So I was thinking about the case where we have for instance a CLOSED curve that fills up the whole $\mathbb{S}^n$, a kind of Peano curve on the sphere. Such a curve exists? If so, then it is very cool, because the way that the curve fills up the sphere allow itself to unroll continuously.

Answer 1

It seems that you know there exist surjective maps $p_{1,2} : I = [0,1] \to I^2 = I \times I$, for example the Peano curve.

Then you also get surjective maps $p_{1,n} : I \to I^n$ for arbitrary $n$ by defining recursively $$p_{1,k+1} : I \stackrel{p_{1,2}}{\to} I \times I \stackrel{p_{1,k} \times id_I}{\longrightarrow} I^k \times I = I^{k+1} .$$

We also know that $I^n \approx D^n$ = closed unit ball in $\mathbb R^n$ and that $D^n/S^{n-1} \approx S^n$. This gives us a surjective map

$$f : I \stackrel{p_{1,n}}{\to} I^n \stackrel{\approx}{\to} D^n \stackrel{q}{\to} S^n .$$

This path may not be closed, but the composition $f * f^{-1}$ of $f$ and its inverse is certainly closed.