Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. 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

I have some manifold $M$ and am wondering what kind of Principal Bundles I am allowed to construct on it.

To be more precise, what are the restrictions when trying to construct principal Bundles over some Manifold? I imagine the topological properties give some quite strict restrictions, but I couldn't find anything in the literature I own.

I am specifically looking for restrictions found on the Torus $T^2$. Any pointers are greatly appreciated!

share|cite|improve this question
If you are specifically interested in flat principal $G$-bundles, then there is a one to one correspondance between isomorphism classes of those and $\mathrm{Hom}(\pi_1X, G)/G$. This is achieved by lifting paths horizontally, the flatness of the bundle assures that homotopic loops in $X$ lift to homotopic paths in $P$ with the same endpoint, and the.right action of $G$ (by conjugation) erases the dependency on the origin of those lifts. For general principal bundles, they are classified by homotopy classes of maps from $X$ to the classifying space of $G$. – Olivier Bégassat May 8 '12 at 9:43
As for restrictions, I don't know how to answer you. What do you have in mind? There may be some characteristic classes to consider associated to principal bundles, like (I might be mistaken) Chern-Simons and of course the curvature, so if the cohomology is simple you might get restrictions... I apologize if you know this already ^^ – Olivier Bégassat May 8 '12 at 9:50
This might need some fleshing out, but I think $\pi_1X$ (and maybe $\pi_1G$) might be the only obstruction to constructing principal bundles over a surface, if you construct the bundle step by step over a skeleton of the CW structure. – Olivier Bégassat May 8 '12 at 10:02
up vote 1 down vote accepted

Isomorphism classes of principal $G$-bundles over $T^2$ are labeled by $\pi_1(G)$. Please, see the following article by: Klimek-Chudy and Kondracki: "The topology of the Yang-Mills theory over torus", where the isomorphism classes are computed for a few low dimensional base manifolds.

For an alternative method of computation, please see the following two articles by: Yu. A. Kubyshin: arXiv:math/9911217: "A classification of fibre bundles over 2-dimensional spaces", and arXiv:math-ph/0309059: "Geometrical formalism in gauge theories".

share|cite|improve this answer

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.