Nonabelian Čech H1 and principal bundles

Statement

Let $M$ be a smooth manifold and $G$ a Lie group. Regard $U\mapsto C^\infty(U,G)$ as a sheaf of groups. Then the nonabelian Čech set

\check H^1\!\left(M,C^\infty(-,G)\right)

is naturally identified with the set of isomorphism classes of smooth principal $G$ -bundles over $M$ . Its distinguished point is the class of the trivial bundle $M\times G$ .

A fixed cover

On an open cover $\mathcal U=\{U_i\}$ , let $Z^1(\mathcal U,G)$ be the set of smooth $G$ -valued Čech $1$ -cocycles. The group of $0$ -cochains

C^0(\mathcal U,G)=\prod_i C^\infty(U_i,G)

acts by

(h\cdot g)_{ij}=h_i^{-1}g_{ij}h_j.

The orbit set $Z^1(\mathcal U,G)/C^0(\mathcal U,G)$ classifies principal $G$ -bundles equipped with trivializations over that cover. Passing through common refinements removes the choice of cover and gives the displayed Čech set.

Why it is usually not a group

For nonabelian $G$ , multiplying two cocycles componentwise need not produce a cocycle, so the orbit set has no natural group law in general. It is a pointed set, with the trivial cocycle as basepoint.

The cocycles and $0$ -cochains retain more structure as a Čech cocycle groupoid: cocycles are objects, and a family $h_i$ satisfying $g'_{ij}=h_i^{-1}g_{ij}h_j$ is a morphism $g\to g'$ . Automorphisms in this groupoid recover the gauge transformations of the glued bundle.

The abelian case

If $G$ is abelian, cocycles and coboundaries form abelian groups, so $\check H^1(M,C^\infty(-,G))$ is an abelian group. For $G=U(1)$ , principal $U(1)$ -bundles correspond to complex line bundles, the group law is tensor product, and the first Chern class gives, for a smooth manifold,

\{\text{complex line bundles over }M\}/\cong \;\cong\; H^2(M;\mathbb Z).

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: classification by transition functions and classifying spaces.
Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser, 1993. DOI record. Relevant: Čech descriptions of bundles and line bundles.