I have reduced a problem to $\pi(Y)^n/G^n$ where Y is a manifold and G is a group acting on the manifold. Can I "factor out," the $n$? i.e. $(\pi(Y)/G)^n$. Note that $\pi(X)$ is the fundamental group of X.
I can't seem to find if $G^n/H^n = (G/H)^n$ in general. Intuitively, it seems right, but I'm not sure if there is some edge case I need to be looking out for. Thanks.
Yes, it definitely holds in general. Define $f : G^n \to (G/H)^n$ to be reduction mod $H$ on each factor. It's clearly surjective and its kernel is clearly $H^n$ (exercise: fill in the missing details if you're not convinced), so it follows that $G^n/H^n \to (G/H)^n$ induced by $f$ is an isomorphism.