Thanks to the fibrations

\begin{align*} SO(n) \to SO(n+1) &\to S^n\\ SU(n) \to SU(n+1) &\to S^{2n+1}\\ Sp(n) \to Sp(n+1) &\to S^{4n+3} \end{align*}

we know that

\begin{align*} \pi_i(SO(n)) \cong \pi_i(SO(n+1)) \cong \pi_i(SO), \quad i &\leq n-2\\ \pi_i(SU(n)) \cong \pi_i(SU(n+1)) \cong \pi_i(SU), \quad i &\leq 2n - 1 = (2n+1) - 2\\ \pi_i(Sp(n)) \cong \pi_i(Sp(n+1)) \cong \pi_i(Sp), \quad i &\leq 4n+1 = (4n + 3) - 2. \end{align*}

These values of $i$ are known as the stable range. So the first unstable groups are $\pi_{n-1}(SO(n))$, $\pi_{2n}(SU(n))$, and $\pi_{4n+2}(Sp(n))$ respectively.

I was able to find $\pi_{n-1}(SO(n))$ for $1 \leq n \leq 16$ by combining the tables on the nLab page for the orthogonal group and appendix A, section 6, part VII of the *Encyclopedic Dictionary of Mathematics*. The groups are

$$0, \mathbb{Z}, 0, \mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}_2, \mathbb{Z}, 0, \mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}_2\oplus\mathbb{Z}_2, \mathbb{Z}\oplus\mathbb{Z}_2, \mathbb{Z}_2, \mathbb{Z}\oplus\mathbb{Z}, \mathbb{Z}_2, \mathbb{Z}, \mathbb{Z}_2, \mathbb{Z}\oplus\mathbb{Z}.$$

There doesn't seem to be any pattern here, so I guess that there is no general result for $\pi_{n-1}(SO(n))$. (*Feel free to correct me if I'm wrong*.) I just noticed that every second term contains a copy of $\mathbb{Z}$, while every fourth term contains two copies.

The case of $SU(n)$ is completely different: in *The space of loops on a Lie group*, Bott proved, among other things, that $\pi_{2n}(SU(n)) \cong \mathbb{Z}_{n!}$, see Theorem 5.

Again consulting the Encyclopedic Dictionary of Mathematics, I was able to find $\pi_{4n+2}(Sp(n))$ for $n = 1, 2, 3$. The groups are $\mathbb{Z}_{12}$, $\mathbb{Z}_{120}$, and $\mathbb{Z}_{10080}$. This seems to suggest that this case is more similar to $SU(n)$ than $SO(n)$, so one might hope there is a Bott-type result.

Is there an analogue of Bott's result for $Sp(n)$? That is, is there some increasing function $f : \mathbb{N} \to \mathbb{N}$ such that $\pi_{4n+2}(Sp(n)) \cong \mathbb{Z}_{f(n)}$?

OEIS has no sequences beginning $12, 120, 10080$, so I have no guess what $f(n)$ could be. It is interesting to note that $12 \mid 120$ and $120 \mid 10080$ which is another similarity with the $SU(n)$ case.

Of course, three groups is not much to go on, so this may be a completely misguided guess. Some questions that would be nice to answer before seriously hoping for such a result are:

- Is $\pi_{4n+2}(Sp(n))$ always cyclic?
- Is $\pi_{4n+2}(Sp(n))$ always finite?
- Is $|\pi_{4n+2}(Sp(n))|$ increasing in $n$?

Any information regarding these three questions would also be interesting to know.

Falling short of answering any of these questions, have any more of these groups (namely $\pi_{18}(Sp(4)), \pi_{22}(Sp(5)), \dots$) been computed?

**Update:** I added the sequence $|\pi_{4n+2}(Sp(n))|$ to the OEIS: A301898.

Also, the answer to the question I asked was also in the Encyclopedic Dictionary of Mathematics on page 1746.

abovethe metastable range, giving (pieces of) information up to $\pi_{4n+15}Sp(n)$ and $\pi_{8n+4}Sp(n)$. $\endgroup$ – Tyrone Mar 27 '18 at 12:10