Torsors and principal bundles
31st of October, 2018
In my thesis, switching between vector bundles and principal \mathrm{GL}_r-bundles has often made certain problems easier (or harder) to understand. Due to my innate fear of all things differentially geometric, I often prefer working with principal bundles, and since reading Stephen Sontz’s (absolutely fantastic) book Principal Bundles — The Classical Case, I’ve really grown quite fond of bundles, especially when you start talking about all the lovely \mathbb{B}G and \mathbb{E}G things therein1 Point is, I haven’t posted anything in forever, and one of my supervisor’s strong pedagogical beliefs is that ‘affine vector spaces should be understood as G-torsors, where G is the underlying vector space acting via translation’,2 which makes a nice short topic of discussion, whence this post.
We briefly3 recall the definition of a principal G-bundle over a space X, where G is some topological group.
Definition. A principal G-bundle over X is a fibre bundle P\xrightarrow{\pi}X with a continuous right action P\times G\to P such that
- G acts freely;
- G acts transitively on the orbits; and
- G acts properly.
It is maybe helpful to think of the following ‘dictionary’:
- free = injective (i.e. \exists x s.t. gx=hx\implies g=h)
- transitively = surjective (i.e. \forall x,y \exists g s.t. gx=y)
- properly = something that you care about if you care about infinite sequences or Hausdorffness or things like that (i.e. the inverse image of G\times X\to X\times X given by (g,x)\mapsto(gx,x) preserves compactness)
Thus the fibres F are homeomorphic to G, and also give the orbits, and the orbit space P/G is homeomorphic to X.
Another definition is now useful.
Definition. A G-torsor is a space upon which G acts freely and transitively.
Motto. G-torsors are principal G-bundles over a point are affine versions of G.
What do we mean by this last ‘equivalence’? Just that G-torsors retain all the structure of G, but don’t have some specified point that acts as the identity. Here are some nice examples.
- \mathrm{GL}_r-torsors are vector spaces; \mathrm{GL}_r-bundles are vector bundles.
- O(r)-torsors are vector spaces with an inner product.
- \mathrm{GL}_r^+-torsors are oriented vector spaces (where \mathrm{GL}_r^+ is the connected component of \mathrm{GL}_r consisting of matrices with determinant strictly positive).
- \mathrm{SL}_r-torsors are vector spaces with a specified isomorphism \det V\xrightarrow{\sim} k, where \det V:=\wedge_{i=1}^r V, and k is our base field. Note that this is weaker than a choice of basis: it is a choice of an \mathrm{SL}_r-conjugacy class of bases.
About which I recently had a nice little Twitter conversation with John Baez; his replies starting here are really quite nice. P.S. if you are not on Twitter then I would highly recommend it: the maths community is really friendly and interesting, and if you have a little question to ask then chances are you’ll get a bunch of nice responses. Also a chance to talk to people across the globe in a completely different time zone. Don’t get me wrong, Twitter has many problems, but you can ignore most of them and just follow the people that you like.↩︎
An earlier version of this post incorrectly said ‘\mathrm{GL}_r-torsor’; thanks to Barbara Fantechi for pointing this out!↩︎
In particular we really sort of assume that the reader already knows what one of these is and are just writing this for some mild effort towards self-containedness. (Bonus question for anybody actually reading this: what is the word/phrase I can’t think of that means ‘self-containedness’ but is actually a real word/phrase?)↩︎