Which Groups are Infinite Loop Spaces? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-25T03:55:41Z http://mathoverflow.net/feeds/question/42875 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/42875/which-groups-are-infinite-loop-spaces Which Groups are Infinite Loop Spaces? Ma Ming 2010-10-20T06:11:07Z 2010-10-20T13:12:52Z <p>At first, if a group G is an infinite loop space (all are based), then <code>\pi_0(G)</code> 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? </p> http://mathoverflow.net/questions/42875/which-groups-are-infinite-loop-spaces/42877#42877 Answer by David Roberts for Which Groups are Infinite Loop Spaces? David Roberts 2010-10-20T06:38:55Z 2010-10-20T09:02:11Z <p>Edit: I just reread the question, and it says "<em>if</em> a group is an infinite loop space..." I realise the first paragraph of my answer is incorrect. The rest still stands.</p> <hr> <p>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.</p> <p>The coefficients of cohomology do not <em>have</em> 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 <a href="http://mathoverflow.net/questions/36466/do-we-have-non-abelian-sheaf-cohomology/36508#36508" rel="nofollow">this MO answer</a>.</p> <p>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)</p> http://mathoverflow.net/questions/42875/which-groups-are-infinite-loop-spaces/42880#42880 Answer by Mark Grant for Which Groups are Infinite Loop Spaces? Mark Grant 2010-10-20T07:22:44Z 2010-10-20T07:22:44Z <p>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".</p>