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I have split off spin^c from spin^c structure
in the section As the homotopy fiber of smooth W3 I have added some more comments and a proposition to make more explicit why the “determinant line” map is given on the canonical $U(1)$-components by multiplication by 2.
cross-linked the definition with central product group.
Moved exceptional example to under the definition, and hence before all the abstract higher-topos style description. This isomorphism is not mentioned on Wikipedia, and for some reason, when looking for it more widely, it’s not easy to find; I even missed seeing it here in my scrambling around.
Added the exceptional isomorphism $Spin^c(4)\simeq U(2)\times_{U(1)}U(2)$, where the fibre product is via the determinant homomorphism on both sides. A bit more minor clarifying details as well in the definition section.
Here’s a question: what should $Spin^c(2)$ be? The underlying manifold is a 2-torus, but what’s the group structure?
The page currently claims $Spin^c(n)$ is defined for all natural numbers, but I’m struggling to imagine $Spin^c(1)$ being useful :-) Is it just isomorphic to $U(1)$ itself?
Reminder to self: add the citation
for the exceptional examples.
Hmm, isn’t $Spin^c(5)$ “just” $Sp(2)\cdot U(1)$ inside Sp(2).Sp(1)? Probably there’s not much more to say past that ….
EDIT: but maybe? What is the image of $Sp(2)\times U(1) \to U(4)$, where we take the embedding of $Sp(2)$ inside $U(4)$ and then multiply? Can we see it as the intersection of $U(4)$ and a subgroup of $GL(4,\mathbb{C})$ larger than $Sp(4,\mathbb{C})$? [Edit of edit: fixed the dimension]
In discussion with John Baez on Mathstodon, I thought of the following:
@johncarlosbaez OK, how’s this. Consider the 4x4 matrix Omega as at https://en.wikipedia.org/wiki/Symplectic_group#Sp(2n,_F), Then define a subgroup G of 4x4 invertible complex matrices M satisfying M^t Omega M = z Omega for some unit complex number z. Then I think U(4) \cap G is double covered by Spin^c(5), as a factorisation of the isomorphism Sp(2)xU(1) = [Sp(4,C) \cap U(4)]xU(1) followed by the multiplication map [Sp(4,C) \cap U(4)]xU(1) -> G \cap U(4). Because M appears twice in the LHS of the equation defining G, I think G\cap U(4) is a little too small, since the condition squares the extra factor coming from U(1)…
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