Talk:Bott periodicity theorem

Significance of Bott periodicity

Without the Bott periodicity theorems, we would not know that real and complex K-theories are periodic extraordinary cohomology theories. As these are subjects classified of high importance, one has to regard the Bott periodicity theorem as being of high importance, too. -- Chuck 05:28, 30 May 2007 (UTC)[reply]

And the actual homotopy groups are ... OMITTED

It is very strange that, although the periodicity results are expressed clearly in this article, the actual homotopy groups π_k(G) for G = U, O, Sp and k = 0, 1, 2, ... are left unmentioned.

I could copy these homotopy groups, but ideally the groups themselves would be be introduced into the article by someone who is an expert in this material.Daqu (talk) 04:47, 27 October 2015 (UTC)[reply]

It is strange. But waiting for an expert to fly in and do the work for you is maybe an unreasonable expectation. I've added the homotopy groups for the orthogonal groups, but I don't know what they are in the unitary case. Woscafrench (talk, contribs) 11:27, 27 December 2017 (UTC)[reply]

Discovery of Bott periodicity

John Baez says (citing a paper by Cartan): "This 'period-8' behavior was discovered by Cartan in 1908 ..."

Baez, John C. (2002). "The Octonions". Bulletin of the American Mathematical Society. 39 (2): 145–205. arXiv:math/0105155. doi:10.1090/S0273-0979-01-00934-X. ISSN 0273-0979. MR 1886087. {{cite journal}}: Invalid |ref=harv (help)
Élie Cartan (1908). J. Molk (ed.). "Nombres complexes". Encyclopédie des sciences mathématiques. 1: 329–468.

This seems relevant. Perhaps someone familiar with the subject matter can add this? —Quondum 16:49, 9 August 2017 (UTC)[reply]

Utterly unclear sentence

"The context of Bott periodicity is that the homotopy groups of spheres, which would be expected to play the basic part in algebraic topology by analogy with homology theory, have proved elusive (and the theory is complicated)."

Can someone please make the meaning of this sentence clear? I cannot imagine what it is intended to mean.

That "the homotopy groups of sphere" (by which is meant the computation of tables of the homotopy groups of spheres) is "elusive" is clear.

What is utterly unclear is "the basic part in algebraic topology by analogy with homology theory".

Homology theory is of course part of algebraic topology. So: What is this supposed to mean???2600:1700:E1C0:F340:7967:7502:8C03:5EC0 (talk) 20:05, 30 September 2018 (UTC)[reply]