Equivalence of cocycles

Let $M$ be a smooth manifold and let $\{U_i\}$ be an open cover. A $G$ -valued cocycle on this cover is a collection of smooth maps $g_{ij}:U_i\cap U_j\to G$ satisfying the cocycle condition

g_{ij}\,g_{jk}=g_{ik}\quad\text{on }U_i\cap U_j\cap U_k, \qquad g_{ii}=e,\quad g_{ji}=g_{ij}^{-1}.

Such data defines a principal bundle by gluing trivial bundles, producing the usual notion of transition functions and a corresponding bundle atlas.

Two cocycles $\{g_{ij}\}$ and $\{g'_{ij}\}$ on the same cover are equivalent if there exist smooth maps

h_i:U_i\to G

such that on each overlap $U_i\cap U_j$ ,

g'_{ij}=h_i^{-1}\,g_{ij}\,h_j.

Change of local sections

Equivalently, if $\{g_{ij}\}$ arises from local sections $s_i:U_i\to P$ via $s_j=s_i\,g_{ij}$ (as in transition functions from local sections), then replacing the sections by

s'_i:=s_i\,h_i

changes the transition functions to $g'_{ij}=h_i^{-1}g_{ij}h_j$ . Thus, equivalence of cocycles is precisely “the same gluing data after a change of local trivializations.”

Equivalent cocycles define isomorphic principal bundles; the corresponding isomorphism is a principal bundle isomorphism and, from the atlas perspective, this is the same relation as equivalence of bundle atlases.

Examples

Trivial cocycle and coboundaries. The trivial bundle $M\times G$ has cocycle $g_{ij}=e$ . A cocycle $\{g_{ij}\}$ is equivalent to the trivial cocycle exactly when there exist $h_i$ with $g_{ij}=h_i^{-1}h_j.$ This is the transition-function version of the global section criterion for triviality.

Changing local sections on a trivial bundle. On $P=M\times G$ , start with the canonical local sections $s_i(x)=(x,e)$ giving $g_{ij}=e$ . If you pick arbitrary smooth maps $h_i:U_i\to G$ and set $s'_i(x)=(x,h_i(x))$ , then the new transition functions are $g'_{ij}=h_i^{-1}h_j,$ and the cocycles $\{e\}$ and $\{g'_{ij}\}$ are equivalent by construction.

Hopf line bundles of different degree are not equivalent. Cover $S^2$ by the standard northern and southern charts $U_N,U_S$ with overlap $U_N\cap U_S\simeq S^1$ . For $G=U(1)$ , transition functions $g_{NS}(\theta)=e^{ik\theta}$ define complex line bundles with different first Chern class (with sign determined by the chosen orientation and transition convention). If $k\neq k'$ , there are no $h_N,h_S$ making $e^{ik'\theta}=h_N^{-1}e^{ik\theta}h_S$ , so the cocycles are not equivalent.