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Want to warm up with something that is really elementary, but I need to get back to speed here.
There is a canonical O-action (“∞-action” really) on any infinite loop space, via the canonical O(n)-action on any n-fold loop space.
Let’s produce some list of examples of these.
To start with the simplest case: what’s the SO-action on K(ℤ,k)?
I suppose it’s trivial. Like so: the action factors through the J-homomorphism
SO×Ω∞X⟶Ω∞S∞×Ω∞Xprecomp⟶Ω∞Xbut the image of J on homotopy groups is pure torsion, and so there is no non-trivial action of the homotopy groups on the ℤ in K(ℤ,k), I suppose.
By this reasoning then the action on K(U(1),k) may in principle have a nontrivial bit for k=1,3,7,8,9,11,.... But what is it exactly.
This will be super-basic. What’s a standard reference that would list some such examples?
What is K(U(1),k), if not K(Z,k+1)?
I really mean the discrete group U(1) here, whence K(−) as opposed to B(−).
But generally, for A any (discrete) abelian group, I am wondering what’s the SO-action on K(A,n)?
I am suspecting these actions are all trivial. Maybe I should use some universal coefficient argument to extrapolate from the case A=ℤ.
[wait]
I have added the statement that the canonical SO-∞-action on Bnℤ is trivial to cobordism hypothesis in this proposition with a brief indication of the proof. (A writeup spelling out more details is in section 3.2.2 of Local prequantum field theory (schreiber). I am still thinking somebody should tell me that this is a basic textbook excercise, please with a pointer to the textbook.)
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