lifting a flat $G/H$ connection to a $G$ connection

Question

Let $G$ be a simply connected Lie group, and $G/H$ a quotient by a subgroup, $H$, of the center. Now, a flat $G/H$ connection on a manifold $M$ is determined by a homomorphism, $\phi$, from $\pi_1(M)$ into $G/H$, modulo conjugation. This may live on some non-trivial $G/H$ bundle over $M$, and there can be an obstruction to lifting this $G/H$ bundle to a $G$ bundle, which can be measured by a class in $H^2(M,H)$ (generalizing the second Stiefel Whitney class for the case $G=\text{Spin}(N)$ and $H=\mathbb{Z}_2$). Is there a simple way to obtain this cohomology class from the map $\phi$? Maybe there is a purely algebraic formulation of this question in terms of lifting the homomorphism, $\phi$, itself.

Answer 1

Let $p:G\rightarrow G/H$ the quotient map, consider any section $s:G/H\rightarrow G$, that is a map $s$ such that $p\circ s=Id_{G/H}$.

The map defined on $\pi_1(M)\times\pi_1(M)$ by $c(u,v)=s(\phi(u))s(\phi(v))s(\phi(uv))^{-1}$ is a $2$-cocycle whose cohomology class is the obstruction to lift $\phi$ to $\tilde\phi:\pi_1(M)\rightarrow G$.