This question is inspired by a statement of Atiyah's in "Geometry and Physics of Knots" on page 24 (chapter 3 – Non-abelian moduli spaces).
Here he says that for a Riemann surface $\Sigma$ the first cohomology $H^1(\Sigma,U(1))$, where $U(1)$ is just complex numbers of norm 1, parametrizes homomorphisms $$\pi_{1}(\Sigma)\to U(1).$$
This is fine by me, after all by Brown Representability we know $$H^1(\Sigma,G)\cong [\Sigma,BG]=[B\pi_{1}\Sigma,BG]$$ since we know that Riemann surfaces are $K(\pi_{1},1)$s. We then use the handy fact from Hatcher Prop. 1B.9 (pg 90) that shows that…
For $X$ a connected CW complex and $Y$ a $K(G,1)$ every homomorphism $\pi_1(X,x_0)\to\pi_1(Y,y_0)$ is induced by a map $(X,x_0)\to(Y,y_0)$ that is unique up to homotopy fixing $x_0$.
So group homomorphisms $\pi_{1}(\Sigma)\to U(1)$ correspond on the nose with first cohomology of the $\Sigma$ with coefficients in $U(1)$.
EDIT: To be accurate the Hatcher result shows we have an injection of group homomorphisms into homotopy classes of maps. Does anyone know if every homotopy class of maps is realized by a group homomorphism?
What bothers me is what comes next. Now replace $U(1)$ with $G$ – any compact simply connected Lie group, take $G=SU(n)$ for example. Now Atiyah claims that $H^1(\Sigma,G)$ parametrizes conjugacy classes of homomorphisms $\pi_{1}(\Sigma)\to G$.
Now if the fundamental group or the Lie group were abelian this would be the same statement, but higher genus Riemann surfaces (genus greater than 1) have non-abelian fundamental groups and Atiyah is looking specifically at non-abelian $G$. Also, it seems that the statement of Brown Representability requires abelian coefficient groups, so I am stuck.
Does anyone know a clean way of proving Atiyah's claim?
EDIT2: I renamed the question to draw in the "right" people. I think the answer has to do with the fact that principal G-bundles over a Riemann surface are determined by maps of $\pi_1(\Sigma)\to G$ (can someone explain why?). This is related to local systems and/or flat connections, which I don't understand well. Thanks!
Best Answer
Since it is not always clear what $H^1(X;G)$ means for a non-abelian group, Atiyah might have meant this loosely, perhaps using Cech 1-cocycles to construct flat $G$ bundles, with homologous cycles giving isomorphic bundles. The moduli space of flat bundles is homeomorphic (and real analytically isomorphic) to the space of conjugacy classes of $G$ representations via the holonomy. I suspect this is what Ben is hinting at in his answer above.
But another interpretation in terms of homotopy classes of maps to $BG$ is as follows. If you give $G$ the discrete topology, then $BG$ is a pointed space with a fixed choice of isomorphism $\pi_1(BG)\cong G$. So
$Hom(\pi_1(X),G)$ is in bijective correspondence with the pointed homotopy classes of maps $[X,BG]_0$. The free homotopy classes $[X,BG]$ are then in bijective correspondence with the conjugacy classes $Hom(\pi_1(X),G)/conj$, since the action of $G=\pi_1(BG)$ on the pointed homotopy classes corresponds to conjugation, with quotient the free homotopy classes. So if you define $H^1(X;G)=[X,BG]$ (unbased), you get Atiyah's statement.
In the case of Atiyah's book, though, more important that the notation for the space of conjugacy classes of reps is the fact that at a representation $r:\pi_1(\Sigma)\to G$, the (usual, twisted by the adjoint rep) cohomology $H^1(\Sigma; g_r)$ ($g$ the lie algebra of $G$)is the Zariski tangent space to the variety of conjugacy classes of representations at $r$, i.e. to the moduli space of flat $G$ connections on $\Sigma$. The cup product $H^1(\Sigma; g_r)\times H^1(\Sigma; g_r)\to R$ determines a symplectic form on this variety, and a holomorphic structure on $\Sigma$ induces a complex structure on this variety, which reflects itself in the Zariski tangent space as an almost complex structure $J:H^1(\Sigma; g_r)\to H^1(\Sigma; g_r)$ which coincides with the Hodge $*$ operator on harmonic forms.
Incidentally, all principal $SU(2)$ bundles over a Riemann surface are topologically trivializable.