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stub for metaplectic structure
If we have
πn(O) = Z/2, Z/2, 0, Z, 0, 0, 0, Z for n = 0, …, 7 mod 8,
and Sp is half a cycle out, don’t we have
πn(Sp) = 0, 0, 0, Z, Z/2, Z/2, 0, Z for n = 0, …, 7 mod 8?
So where does the double cover come in? Why isn’t n=3 the first level to co-kill?
Hmm, but elsewhere I see π1(Sp(2n))=Z, so a double cover is fine. Is it a questioning of stabilizing the Sp(2n)? Is the universal cover of Sp(2n) of any interest?
(By the way, any reason for using \mathcal{B} on the spin structure page?)
Sp is half a cycle out,
Wait, what do you mean by that? Maybe there is some wrong assumption going into this statement. But maybe I am missing something, let me know what you have in mind.
but elsewhere I see π1(Sp(n))=ℤ
Yes, the maximal compact subgroup of Sp(n) is U(n), and so both have the same homotopy groups.
By half a cycle, I meant this result:
πk(O)=πk+4(Sp)
πk(Sp)=πk+4(O),k=0,1,…
What am I getting wrong here? p. 38 of this has π1(Sp(n))=0 in a table called “Homotopy groups of symplectic groups”
Oh, am I getting confused between Sp(2n,C), Sp(2n,R) and Sp(n)? But according to this Wikipedia page, only Sp(2n,R) has nontrivial fundamental group.
OK, so you were talking about the latter.
Right, I should better disentangle my notation. I have been speaking about Sp(2n,ℝ) here throughout. It certainly has π1=ℤ.
Let me try to make the notation in the entries more consistent…
So I went through a couple of entries and wrote out
Sp(2n,\mathbb{R})
and
Mp(2n\mathbb{R})
and
Ml(n,\mathbb{R})
and so on everywhere. I hope I caught them all, not to leave a mess.
Urs, I know some references on metaplectic representation, which I would be glad to add to a relevant entry. I am not sure what is your big plan: to consider it under metaplectic group, under metaplectic structure or to create a new entry metaplectic representation. I would be glad to do create it if this fits your picture.
I’d think that deserves an entry of its own!
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