MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Fix a group $G$. Recall a space $X$ is a $G$-torsor if $G$ acts freely and transitively on $X$ (if $G$ is non-abelian one has to talk about left or right torsors).

A choice of a point $x \in X$ induces a bijection between $X$ and $G$, and hence $X$ is non-canonically isomorphic to $G$.

Is there a universally accepted name for such a choice, as in:

"Let $X$ be a $G$-torsor, and choose a [insert name here] between $X$ and $G$..."

The word that comes naturally to mind is affine map, but this somehow isn't appropriate when $G$ has torsion, say...

I note that the wikipedia entry:

does not give a name to such a map; neither does the lovely little expository article by Baer:

Apologies if this question isn't deemed appropriate for mathoverflow - it's hardly a research level question I know, but it's not the sort of thing I could really ask anywhere else!

share|cite|improve this question
With Dan Petersen, I would probably say "choose a trivialization of $X$". But if you want to fit it into your sentence, "Let $X$ be a $G$-torsor, and choose a $G$-torsor isomorphism between $X$ and $G$" works fine. Note also that in other categories, and notably in the category of bundles over $Y$, one can easily talk about torsors, but it is not true that every torsor has a global element = trivialization. Anyway, "global element" or "global section" are fine words to adopt. – Theo Johnson-Freyd Apr 5 '11 at 19:31

I would call it a trivialization of the torsor. This terminology comes from the relative situation when one deals with principal bundles rather than principal homogeneous spaces. In this case, a trivialization would be a global section of the principal bundle $X \to Y$ (rather than a point of X), so a trivialization is the same thing as an isomorphism between X and the trivial bundle $G \times Y \to Y$.

share|cite|improve this answer
I agree. Or, in the case of a torsor over one point, I would call the map of the OP a "choice of identity" or a "choice of identity section". – Qfwfq Apr 5 '11 at 17:20

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.