Isotopy classes of maps from the $3$-sphere to the $6$-sphere

Question

I read somewhere that the group of isotopy classes of continuous maps from the three-sphere to the six-sphere is infinite cyclic. Why is that?

A reference would also be appreciated.

Answer 1

This is done in the following paper by André Haefliger:

Differential embeddings of $S^ n$ in $S ^{n + q}$ for $q ≥ 2$, Ann. of Math. 83 (1966) 402–436

From the introduction of the article:

This paper can be considered as a complement to the fundamental paper of J. Levine [same Ann. (2) 82 (1965), 15–50; MR0180981]. Instead of studying the group $θ_n^q$ of isotopy classes of embedded homotopy $n$-spheres in $S^{n+q}$, we are interested here in the group $C_n^q$ of isotopy classes of embeddings of the usual $n$-sphere $S^n$ in $S^{n+q}$. Our main result is the isomorphism of $C_n^q$ with the triad homotopy group $π_{n+1}(G;SO,G_q)$ for $q>2$, where $G_q$ is the space of maps of degree one of $S_{q−1}$ onto itself, and $G$ its stable suspension.