Exercise 34, § 5

Let $X$ be a path-connected topological space, let $G$ be a discrete group acting continuously on the left in $X$. Let $M$ be an open subset of $X$ such that $G\cdot M = X$. Define the sets S, T, the group F and the group homomorphism $\varphi : F\rightarrow G$ as in Prop. 10 of I, p. 136. Let $a$ be a point of M.

a) Let $c:\mathbf{I}\rightarrow X$ be a loop at $a$. Prove that there exists a unique element $g(c)$ of F satisfying the following property: for every integer $n$ and every sequence $(s_1, . . . , s_n)$ of elements of S such that $c([(k-1)/n, k/n])\subset s_kM$ for every integer $k\in \{1, . . . , n\}$, one has $g(c) =x_{s^{-1}_ns_1}x_{s^{-1}_1s_2}. . . x_{s^{-1}_{n-1}s_n}$.

b) Prove that there exists a unique group homomorphism $\gamma$ of $\pi_1(X, a)$ into F such that $\gamma ([c]) =g(c)$ for every loop $c$ in X at $a$.

c) Prove that the kernel of $\varphi$ is equal to the image of the homomorphism $\gamma$. d) Prove that the kernel of $\gamma$ is generated by the union of the images of the groups $\pi_1(M\cup s\cdot M, a)$, for $s\in S$.