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For $G$ an abelian group, the Eilenberg-Maclane space $K(G, n)$ represents singular cohomology $H^n(-; G)$ with coefficients in $G$ on the homotopy category of CW-complexes. If $n > 1$, then $G$ must be abelian, but for $n = 1$ there are also Eilenberg-Maclane spaces for $G$ nonabelian, so one might imagine that they represent some kind of nonabelian cohomology $H^1(-; G)$. Is this functor known by a better name, and what's known about it? In particular,

  • Does it have an alternate definition along the lines of the usual definition of singular cohomology?
  • Is there a universal coefficient theorem for it?
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What happened you deleted a Galois theory question of yours which you just now posted. –  user9413 May 9 '11 at 8:21
    
I figured it out. –  Qiaochu Yuan May 9 '11 at 8:58
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1 Answer 1

up vote 13 down vote accepted

K(G,1) aka BG classifies G-bundles — i.e. G-coverings, if G is discrete. (Details can be found e.g. in May's Concise Course in Algebraic Topology.)

Usual definition of Cech cohomology works for $H^1(X;G)$ even in non-abelian case (but it's just the usual cocycle definition of G-bundle).

As for universal coefficient theorem, even if $H_1(X;\mathbb Z)$ is trivial, $H^1(X;G)$ needn't be; but (if G is discrete) $H^1(X;G)=\operatorname{Hom}(\pi_1(X);G)/\text{conjugation}$ (reference: Hatcher, 1B.9). (But if one wishes to consider BG for general G, things get worse — "$H^1$" is no longer defined by 2-skeleton of X. Perhaps, AHSS from cohomology to K-theory can be viewed as kind of "universal coefficient spectral sequence" for $G=U=\lim U(n)$.)

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Thanks. Do you have a reference? This material doesn't seem to be covered in Hatcher, for example. –  Qiaochu Yuan Apr 30 '11 at 11:45
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Also, I googled www-math.mit.edu/~mbehrens/18.906/prin.pdf -- which looks good –  Grigory M Apr 30 '11 at 12:07
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