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

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

Definition (equivalent cocycles)

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

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

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

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

1. 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

   $$g_{ij}=h_i\,h_j^{-1}.$$

   This is the transition-function version of the global section criterion for triviality .

2. 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.

3. 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 (the integer $k$). 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.