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added a table with some homotopy groups in the unstable range to orthogonal group – Homotopy groups
Extended by two rows. $SO(6)$ is the highest rank in which peculiar things happen below dimension 10. I think it would be good to extend it up to $\pi_11$ or $\pi_12$. Mimura and Mimura-Toda have this information for the last two rows. I’m looking around to see if I can find it for the other groups.
Ah, the venerable Encyclopedic Dictionary of Mathematics has a nice big table of these homotopy groups, for all the groups of interest. Also, the Abanov document has evaporated off the web.
Is there a way to see this online? In GoogleBooks somewhere?
Not that I’m aware of, which is a pity.
I’m sticking the table in, at least up to $\pi_12(SO(12))$, since this probably covers all the orthogonal/homotopy groups of interest to string theory. I can do more if requested; the table goes up to $\pi_15(SO(17))$.
In fact, in writing this out, it becomes clear that something different is going to happen for Ninebrane, since stably ($SO(11)$ and beyond, hence in the right range for 11d supergravity) we have nontrivial torsion homotopy groups in dimension 8 and 9. $SO(11)$ also has a nonstable $\mathbb{Z}/2$ in $\pi_10$, and its $\pi_11$, though $\mathbb{Z}$, is not in the stable range. Hmm
One can see a copy of the table in the EDM in page 970 of the Handbook of Algebraic Topology This chapter, number 19, was written by Mimura.
Thanks for expanding the table so much!
Yes, ninebrane structure is meant to kill just the fractional $p_3$ of a Fivebrane structure, not the two $\mathbb{Z}/2\mathbb{Z}$-copies in between.
The way this appears in string theory is not actually that the structure group needs to be highly connected as such (that happens to be a byproduct), it’s that the polynomials in the $p_i$ appear as anomalies, and these need to be removed.
But how does one get the fractional $p_3$? Does this exist even if the structure group doesn’t lift through those two copies?
There is a $\mathbb{Z}/2\mathbb[p_3$ on $B \mathrm{Fivebrane}$ pulled back from $B \mathrm{String}$ and it will be a multiple of something. Do you mean how to pick the specific fraction?
The way I suggested to do it is just to let the process of higher Lie integration run. There is that $L_\infty$-cocycle $\mu_{12}$ on $\mathfrak{fivebrane}$ and just integrating that as in Cech Cocycles for Differential characteristic Classes (schreiber) gives some $\mathbf{B} \mathrm{Ninebrane}$ whose geometric realization is a candidate $B \mathrm{Ninebrane}$. I don’t know what the fraction comes out as, but the way the Lie integration works one is guaranteed that one kills a minimal element.
Ah, so the fact it’s a Lie algebra theoretic fact means you don’t see the $\mathbb{Z}/2$? I worry though that in doing the Lie integration one needs to pick at some point a filler of some simplex. But let me remind myself how it works, first, before complaining it doesn’t! :-)
(reads…)
OK, I see. It’s just a formal thing, but one can then ’go down’ the Whitehead tower and construct the quotients by various to get the intermediate stages. The Lie integration just skips them altogether.
One point: under Lie integration#integrating_the_string_lie_2algebra_to_the_string_lie_2group there is what I think is a typo: $\mathbf{B}^n\mathbb[D^2_\ast \to D$ should probably by $D^2_\ast \to G$, but I hesitate to change in case I’ve missed something.
Yes! Absolutely.
And yes, that should be a typo. If you have a second, please fix it. Myself I need to rush off..
Will do.
Fixed two errors in the table in the $SO(6)$ row.
thanks. I wonder what happened there. Did I just make a misprint while copying? Probably.
@Urs, no it was probably me, since I added the citation for the $SO(6)$ row via the exceptional isomorphism.
I have added to orthogonal group:
statement and proof that $O(n) \hookrightarrow O(k)$ is an $(n-1)$-equivalence (here)
statement and proof that the Stiefel manifold $V_n(k)$ is $(n-1)$-connected (here).
The latter I am also copying over to Stiefel manifold
I have now added the analogous statements with their analogous proofs also to unitary group.
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