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 therein12. 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’,3 which makes a nice short topic of discussion, whence this post.

We briefly4 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

  1. $G$ acts freely;
  2. $G$ acts transitively on the orbits; and
  3. $G$ 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.

Footnotes

  1. 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. 

  2. (which I won’t talk about here because (a) I think there are many other places to read about this that are much better than something that I could write; and (b) I should be working on my thesis but I’m sort of using this post as a method of procrastination/searching for inspiration). 

  3. An earlier version of this post incorrectly said ‘$\mathrm{GL}_r$-torsor’; thanks to Barbara Fantechi for pointing this out! 

  4. 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?)