For revision of a paper (http://arxiv.org/abs/1008.1189), I'd like to correct correct my references to the original work on aspects of the homotopy groups pi_5 of SU(3) groups $\pi_5(SU(3))$ and pi_4 of SU(2)$\pi_4(SU(2))$. I'mI'm not a mathematician. I canI can barely read the introductions of math papers. SoSo I'd appreciate advice advice. I I realize that mathematicians don't bother with the original references references for such things. I'mI'm just being a bit compulsive.
There are 4four points I'd like to reference correctly.
(1) pi_5 of SU(3) = Z $\pi_5(SU(3)) = \mathbb{Z}$
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
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 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
"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. AsAs far as I I can tell given my limited ability to decipher topology written in German German, it does contain this result. II have no idea if the proof is correct correct, though I'd guess it is, considering the respectability of the the author.
(2) SU(3) -> G_2 -> S^6$SU(3) \to G_2 \to S^6$ represents a generator of pi_5 of SU(3) = Z$\pi_5(SU(3)) = \mathbb{Z}$
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
- 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. ItIt seems hard to believe that there isn't an early reference.
(3) A map S^5 -> SU(3)$S^5 \to SU(3)$ generates pi_5$\pi_5(SU(3))$ iff its composition with SU(3) -> SU(3)/SU(2)= S^{5}$SU(3) \to SU(3)/SU(2) = S^5$ is a map S^5 -> S^5$S^5 \to S^5$ of degree +1$1$ or -1$-1$.
(4) The nontrivial element in pi_4 of SU(2) = Z_2$\pi_4(SU(2)) = \mathbb{Z}_2$ is represented by the 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.
- Freudenthal. Uber die Klassen der Spharenabbildungen. I. Große Dimensionen. Compos. Math., 5:299–314, 1937.
I learned this from: Thomas Puettmann and A. Rigas. Presentations presentation of the first homotopy groupssuspension of the unitary groups. Comment. Math. Helv., 78:648–662Hopf fibration is given by $[0, \pi]\times SU(2) \to SU(2)$, 2003$(\theta, g) \mapsto g^{-1}\exp(\mu_3\theta)g$ where $\mu_3$ is the diagonal generator of $\mathfrak{su}(2)$ with $\exp(\mu_3\pi) = -1$. math/0301192 but
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
