G-torsor - Knowlpedia

Definition

Let $G$ be a group. A right $G$ -torsor is a nonempty set $T$ with a right action of $G$ that is free and transitive. Equivalently, for every $t,t'\in T$ , there is a unique $g\in G$ such that $t'=tg$ . In the smooth setting, $G$ is a Lie group, $T$ is a smooth manifold, and the action is smooth.

A copy of a group without an origin

Choosing $t_0\in T$ gives a $G$ -equivariant bijection

G\longrightarrow T, \qquad g\longmapsto t_0g.

Thus a torsor becomes a copy of $G$ after choosing a basepoint, but it has no preferred identity element. If $t_1=t_0h$ is chosen instead, the resulting coordinates differ by left multiplication by $h^{-1}$ .

Relation to principal bundles

For a principal $G$ -bundle $\pi:P\to M$ , every fiber $P_x=\pi^{-1}(x)$ is canonically a right $G$ -torsor. A local section chooses one point in each fiber over an open set and therefore identifies those torsors smoothly with copies of $G$ . The absence of a global compatible choice is one way to express the twisting of a nontrivial principal bundle.

Morphisms and automorphisms

A morphism of right $G$ -torsors is a $G$ -equivariant map. Every such map between nonempty torsors is an isomorphism. After choosing a point of a right torsor, its $G$ -equivariant automorphisms identify with left translations by $G$ ; changing the chosen point changes this identification by conjugation.

References

Dale Husemoller, Fibre Bundles, 3rd ed., Springer, 1994. DOI record. Relevant: principal bundles and principal homogeneous spaces.
Norman Steenrod, The Topology of Fibre Bundles, Princeton University Press, 1951. Publisher record. Relevant: principal bundles and coordinate bundles.