Which Groups are Infinite Loop Spaces? - MathOverflow

Which Groups are Infinite Loop Spaces?

2010-10-20T06:11:07Z

At first, if a group G is an infinite loop space (all are based), then \pi_0(G) must be Abelian. Therefore, if G is discret, then it must be Abelian. In fact, any Abelian group does be infinite loop space, by the EM space construction. But we have non-Abelian examples, the infinite groups U and O are infinite loop spaces, by Bott periodicity. (Does this contradict with the statement that the coefficient of a cohomology must be an Abelian group?) Are there any other examples?

Answer by David Roberts for Which Groups are Infinite Loop Spaces?

2010-10-20T06:38:55Z

Edit: I just reread the question, and it says "if a group is an infinite loop space..." I realise the first paragraph of my answer is incorrect. The rest still stands.

Firstly, $\pi_0(G)$ does not have to be abelian - it is $\pi_1(G,e)$ which is abelian, as the Eckmann-Hilton argument shows.

The coefficients of cohomology do not have to be abelian groups, but I guess you are referring to the idea that the ordinary cohomology of a space (singular, or sheaf, say) only makes sense in all non-negative dimensions for abelian-group coefficients. One can define $H^1(X,G)$ ($X$ a space) for a nonabelian (topological) group, but for higher degree cohomology it is not straightforward, see this MO answer.

Now notice that one can use loop spectra as coefficients for cohomology, but one gets an extraordary cohomology theory: this is the case of $U$ and $O$, which represent spectra, and given $K$-theory and $KO$-theory respectively. (see the Wikipedia page on K-theory for example)

Answer by Mark Grant for Which Groups are Infinite Loop Spaces?

2010-10-20T07:22:44Z

There is a recognition principle for infinite loop spaces, which involves a lot of machinery and tells you whether a given space is equivalent to an infinite loop space. More specifically, $Y$ is equivalent to an infinite loop space if and only if ($\pi_0(Y)$ is a group and) $Y$ is an $E_\infty$-space, meaning $Y$ admits a product which is associative and commutative up to some coherent sequence of "higher homotopies". A good place to start reading about this is J. F. Adams' book "Infinite Loop Spaces".