# Original references for the homotopy groups pi_5 of SU(3) and pi_4 of 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 of SU(3) and pi_4 of 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 4 points I'd like to reference correctly.

(1) pi_5 of SU(3) = 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 =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) -> G_2 -> S^6 represents a generator of pi_5 of SU(3) = 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 -> SU(3) generates pi_5 iff its composition with SU(3) -> SU(3)/SU(2)= S^{5} is a map S^5 -> 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 of SU(2) = 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] x SU(2) -> SU(2)
(\theta, g)     -> g^(-1) exp(\mu_3\theta) g
where \mu_3 is the diagonal generator of 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.

Daniel Friedan

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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-Alvarez May 16 '14 at 23:59