# Original references for the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$?

For revision of a paper (http://arxiv.org/abs/1008.1189), I'd like to correct my references to the original work on aspects of the homotopy groups $\pi_5(SU(3))$ and $\pi_4(SU(2))$. I'm not a mathematician. I can barely read the introductions of math papers. So I'd appreciate advice. I realize that mathematicians don't bother with the original references for such things. I'm just being a bit compulsive.

There are four points I'd like to reference correctly.

(1) $\pi_5(SU(3)) = \mathbb{Z}$

For this, I referenced Beno Eckmann's thesis: B. Eckmann. Zur Homotopietheorie Gefaserter Raume. Comm. Math. Helv., 14:141–192, 1942.

The thread that led to this reference was:

• J. Diedonne, A history of algebraic and differential topology. 1900--1960, MR995842, page 411.

• A. Borel, Collected Papers Volume 1, page 426

• B. Eckmann, Espaces fibres et homotopie, Colloque de topologie (espaces fibres), Bruxelles, 1950, MR0042705

I couldn't find a copy of the last. But I did find B. Eckmann, Mathematical survey lectures 1943--2004, MR2269092. which contained B. Eckmann, "Is Algebraic Topology a Respectable Field", page 255:

"It was known in the thesis of the author already (1942) that the homotopy groups $\pi_i(U(n))$ are constant for $n \ge (i+2)/2$ for even $i$ and $(i+1)/2$ for odd $i$: these "stable" groups were known to be $0$ for $i=0,2,4$ and $\mathbb{Z}$ for all odd $i$."

Eckmann's thesis is available on-line from Comm.Math.Helv. As far as I can tell given my limited ability to decipher topology written in German, it does contain this result. I have no idea if the proof is correct, though I'd guess it is, considering the respectability of the author.

(2) $SU(3) \to G_2 \to S^6$ represents a generator of $\pi_5(SU(3)) = \mathbb{Z}$

I referenced

• Lucas M. Chaves and A. Rigas. Complex reflections and polynomial generators of homotopy groups. J. Lie Theory, 6(1):19–22, 1996. MR1406003

from which I learned the fact. It seems hard to believe that there isn't an early reference.

(3) A map $S^5 \to SU(3)$ generates $\pi_5(SU(3))$ iff its composition with $SU(3) \to SU(3)/SU(2) = S^5$ is a map $S^5 \to S^5$ of degree $1$ or $-1$.

I thought I could see this in Eckmann's 1942 thesis, so I that's what I referenced.

(4) The nontrivial element in $\pi_4(SU(2)) = \mathbb{Z}_2$ is represented by the suspension of the Hopf fibration:

• Freudenthal. Uber die Klassen der Spharenabbildungen. I. Große Dimensionen. Compos. Math., 5:299–314, 1937.

A presentation of the suspension of the Hopf fibration is given by $[0, \pi]\times SU(2) \to SU(2)$, $(\theta, g) \mapsto g^{-1}\exp(\mu_3\theta)g$ where $\mu_3$ is the diagonal generator of $\mathfrak{su}(2)$ with $\exp(\mu_3\pi) = -1$.

I learned this from:

• Thomas Puettmann and A. Rigas. Presentations of the first homotopy groups of the unitary groups. Comment. Math. Helv., 78:648–662, 2003. math/0301192

but there must be something earlier. Or is it too obvious?

Thanks for any help.

• Concerning (4): the group $SU(2)$ is diffeomorphic to $S^3,$ which has the same homotopy groups $\pi_n, n\geq 3$ as $S^2$ by the Hopf fibration argument, and $\pi_4(S^3)=Z_2$ follows from Pontryagin's computations. Thus Freudenthal's suspension theorem may be unnecessary. Besides Dieudonne, Toda's book and Ravenel's book (quoted at en.wikipedia.org/wiki/Homotopy_groups_of_spheres) may contain historical references. – Victor Protsak Aug 10 '10 at 17:26
• I don't know where you got the impression that «that mathematicians don't bother with the original references»... – Mariano Suárez-Álvarez May 16 '14 at 23:59