# Restrictions for Principal Bundles on Manifolds

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!

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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

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.