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  • Suppose with have a topological manifold $X$ and a group $G$, is there a way to compute the fundamental group of $X/G$ in function of $\pi(X)$ and $\pi(G)$?

  • are there any settings on X that can simplify the computing of $\pi(X/G)$ in function of $\pi(X)$ and $\pi(G)$?

could you please suggest to me a reference where can I find related topics?

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See Proposition 8.10 in Chapter I of Transformation Groups and Algebraic K-Theory by Lück. When a Lie group $G$ acts properly on a connected manifold $X$, the proposition provides an exact sequence $$ \pi_1(X,x_0)\rightarrow\pi_1(X/G,x_0G)\rightarrow\pi_0(G)/M\rightarrow 0 $$ where $M$ is the normal subgroup of $\pi_0(G)$ generated by $\pi_0$ of the stabilizers. A description of the kernel of the first morphism $\pi_1(X,x_0)\rightarrow\pi_1(X/G,x_0G)$ can also be found there.

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    $\begingroup$ thank you so much for your answer $\endgroup$
    – Yushi MuGiwara
    Commented Jan 11, 2023 at 15:37

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