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1. • CommentRowNumber1.
   • CommentAuthorUrs
   • CommentTimeSep 30th 2014
   • (edited Sep 30th 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexWant to warm up with something that is really elementary, but I need to get back to speed here. There is a canonical $O$-action ("$\infty$-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 $K(\mathbb{Z},k)$? I suppose it's trivial. Like so: the action factors through the J-homomorphism $$ SO \times \Omega^\infty X \stackrel{}{\longrightarrow} \Omega^\infty S^\infty \times \Omega^\infty X \stackrel{precomp}{\longrightarrow} \Omega^\infty X $$ 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 $K(\mathbb{Z},k)$, 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?

   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 (“$\infty$-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 $K(\mathbb{Z},k)$?

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

   $$SO \times \Omega^\infty X \stackrel{}{\longrightarrow} \Omega^\infty S^\infty \times \Omega^\infty X \stackrel{precomp}{\longrightarrow} \Omega^\infty X$$

   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 $K(\mathbb{Z},k)$, 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?

2. • CommentRowNumber2.
   • CommentAuthorCharles Rezk
   • CommentTimeOct 1st 2014
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   Author: Charles Rezk
   Format: MarkdownItexWhat is $K(U(1),k)$, if not $K(Z,k+1)$?

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

3. • CommentRowNumber3.
   • CommentAuthorUrs
   • CommentTimeOct 1st 2014
   • (edited Oct 1st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItexI really mean the discrete group $U(1)$ here, whence $K(-)$ as opposed to $B(-)$.

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

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeOct 1st 2014
   • PermaLink
   Author: Urs
   Format: MarkdownItexBut 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 $A = \mathbb{Z}$.

   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 $A = \mathbb{Z}$.

5. • CommentRowNumber5.
   • CommentAuthorUrs
   • CommentTimeOct 1st 2014
   • (edited Oct 1st 2014)
   • PermaLink
   Author: Urs
   Format: MarkdownItex[wait]

   [wait]

6. • CommentRowNumber6.
   • CommentAuthorUrs
   • CommentTimeOct 18th 2014
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   Author: Urs
   Format: MarkdownItexI have added the statement that the canonical SO-$\infty$-action on $B^n \mathbb{Z}$ is trivial to _[[cobordism hypothesis]]_ in [this proposition](http://ncatlab.org/nlab/show/cobordism+hypothesis#CanonicalSOActionOnBnZ) with a brief indication of the proof. (A writeup spelling out more details is in section 3.2.2 of _[[schreiber:Local prequantum field theory]]_. I am still thinking somebody should tell me that this is a basic textbook excercise, please with a pointer to the textbook.)

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