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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeSep 30th 2014
    • (edited Sep 30th 2014)

    Want to warm up with something that is really elementary, but I need to get back to speed here.

    There is a canonical OO-action (“\infty-action” really) on any infinite loop space, via the canonical O(n)O(n)-action on any nn-fold loop space.

    Let’s produce some list of examples of these.

    To start with the simplest case: what’s the SOSO-action on K(,k)K(\mathbb{Z},k)?

    I suppose it’s trivial. Like so: the action factors through the J-homomorphism

    SO×Ω XΩ S ×Ω XprecompΩ X SO \times \Omega^\infty X \stackrel{}{\longrightarrow} \Omega^\infty S^\infty \times \Omega^\infty X \stackrel{precomp}{\longrightarrow} \Omega^\infty X

    but the image of J on homotopy groups is pure torsion, and so there is no non-trivial action of the homotopy groups on the \mathbb{Z} in K(,k)K(\mathbb{Z},k), I suppose.

    By this reasoning then the action on K(U(1),k)K(U(1), k) may in principle have a nontrivial bit for k=1,3,7,8,9,11,...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?

    • CommentRowNumber2.
    • CommentAuthorCharles Rezk
    • CommentTimeOct 1st 2014

    What is K(U(1),k)K(U(1),k), if not K(Z,k+1)K(Z,k+1)?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2014
    • (edited Oct 1st 2014)

    I really mean the discrete group U(1)U(1) here, whence K()K(-) as opposed to B()B(-).

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2014

    But generally, for AA any (discrete) abelian group, I am wondering what’s the SOSO-action on K(A,n)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=A = \mathbb{Z}.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeOct 1st 2014
    • (edited Oct 1st 2014)

    [wait]

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeOct 18th 2014

    I have added the statement that the canonical SO-\infty-action on B nB^n \mathbb{Z} 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.)