Principal bundles and transition functions

A principal right $G$-bundle consists of a smooth projection $\pi:P\to M$ and a smooth right action preserving its fibers. The action is free and transitive on each fiber, and every point of the base has a neighborhood with an equivariant product trivialization. Local triviality is an additional requirement: freeness alone does not make a quotient a principal bundle. A local section $s_i:U_i\to P$ identifies a point in the fiber uniquely as $s_i(x)g$.

On an overlap, define $g_{ij}$ by $s_j=s_i g_{ij}$. Substituting the expression for a third section gives an ordered cocycle. The order matters for a noncommutative group: reversing the factors generally describes a different relation. We have $g_{ii}=e$ and $g_{ji}=g_{ij}^{-1}$. These maps describe how origins change inside the same torsor, not how a physical point in the total space moves. A single point therefore has different group coordinates in different local products.

Conversely, smooth transition functions satisfying these identities glue the products $U_i\times G$ by identifying $(x,a_j)$ in chart $j$ with $(x,g_{ij}a_j)$ in chart $i$. The cocycle makes the identification transitive on triple intersections. Standard bundle construction over the stated manifold base yields local products with a compatible right action. One should check both the relation and its inverse when implementing a gluing calculation; an inverse misplaced here propagates into every later connection formula.

Replace each section by $s'_i=s_i h_i$, where $h_i:U_i\to G$ is smooth. Comparing the two expressions for $s'_j$ gives the new cocycle below. This is a change of local description of the same bundle. A global section would supply an equivariant isomorphism $M\times G\to P$ by $(x,g)\mapsto s(x)g$, and conversely a trivialization supplies a global section. Local sections always exist; their compatibility is the obstruction to making that choice globally.

For a concrete test, glue trivial circle bundles over two hemispheres of an oriented sphere. A transition map on an equatorial annulus retracts to a map from the equator into $U(1)$. Its winding number distinguishes clutching classes once the equator and hemisphere convention are fixed. Changes extending across the two disks have zero boundary winding, so they cannot remove a nonzero clutching degree. This argument concerns the bundle topology; it needs neither curvature nor a chosen connection.

The integer sign depends on whether one calls the transition $g_{NS}$ or its inverse and on the boundary orientation. For now record the explicitly chosen winding, without silently equating it to a signed Chern number. A contractible patch sees none of this information. This is why an atlas with nonsingular local formulas does not prove global triviality, and why refining an open cover cannot by itself eliminate a genuine obstruction.