Smooth G-valued Čech 1-cocycle

Definition

Let $M$ be a smooth manifold, let $\mathcal U=\{U_i\}_{i\in I}$ be an open cover, and let $G$ be a Lie group. A smooth $G$ -valued Čech $1$ -cocycle on $\mathcal U$ is a family of smooth maps

g_{ij}:U_i\cap U_j\longrightarrow G

satisfying

g_{ii}=e, \qquad g_{ji}=g_{ij}^{-1}, \qquad g_{ik}=g_{ij}g_{jk}

on every overlap on which the expressions are defined. The last equality is the cocycle condition on triple intersections.

From local sections

Choose local sections $s_i:U_i\to P$ of a principal $G$ -bundle. There is a unique map $g_{ij}$ on each double overlap such that

s_j(x)=s_i(x)g_{ij}(x).

Associativity of the action gives $g_{ik}=g_{ij}g_{jk}$ . Replacing the local sections by $s_i'=s_i h_i$ , where $h_i:U_i\to G$ , changes the cocycle by

g_{ij}'=h_i^{-1}g_{ij}h_j.

This is equivalence of cocycles, or a change of local trivialization.

Gluing a principal bundle

Conversely, a cocycle glues the disjoint union $\bigsqcup_i(U_i\times G)$ by

(x,a)_i\sim(x,g_{ij}(x)^{-1}a)_j.

The cocycle law makes this relation transitive. The quotient is a smooth principal $G$ -bundle locally trivialized over the $U_i$ . Equivalent cocycles give isomorphic bundles.

Abelian and nonabelian behavior

If $G$ is abelian, cocycles multiply pointwise and form an abelian group. For general $G$ , pointwise multiplication need not preserve the cocycle law; the natural classification is therefore a pointed set and, before taking isomorphism classes, a groupoid. See nonabelian Čech $H^1$ .

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: bundles described by coordinate functions and their equivalence.
Jean-Luc Brylinski, Loop Spaces, Characteristic Classes and Geometric Quantization, Birkhäuser, 1993. DOI record. Relevant: Čech cocycles and bundles.