For a pair \((X, A)\) of path-connected spaces, show that \(\pi_1(X, A, x_0)\) can be identified in a natural way with the set of cosets \(H\alpha\) of the subgroup \(H \subset \pi_1(X, x_0)\) represented by loops in \(A\) at \(x_0\).
Solution
Use Hatcher’s interval model for \(\pi_1(X,A,x_0)\): an element is represented by a path \[ u:I\to X, \qquad u(0)\in A, \quad u(1)=x_0, \] with homotopies keeping the endpoint at \(x_0\) and allowing the initial point to move in \(A\).
Since \(A\) is path-connected, choose a path \(v\) in \(A\) from \(x_0\) to \(u(0)\). Then \(vu\) is a loop at \(x_0\). Associate to the relative class of \(u\) the left coset \[ H[vu], \qquad H=i_*\pi_1(A,x_0)\subseteq\pi_1(X,x_0). \] If \(v'\) is another such path, then \(v'v^{-1}\) is a loop in \(A\) at \(x_0\), so \[ [v'u]=[v'v^{-1}][vu] \] and hence \(H[v'u]=H[vu]\). A relative homotopy of \(u\) changes \(vu\) by the same sort of left multiplication, so the coset depends only on the relative class.
Conversely, a loop \(\alpha\) at \(x_0\), viewed as a relative path with initial point \(x_0\in A\), gives an element of \(\pi_1(X,A,x_0)\). Two loops \(\alpha,\alpha'\) give the same relative class exactly when \[ [\alpha'][\alpha]^{-1}\in H, \] that is, exactly when \[ H[\alpha']=H[\alpha]. \] Thus the two constructions are inverse natural bijections of pointed sets: \[ \boxed{\pi_1(X,A,x_0)\cong H\backslash\pi_1(X,x_0).} \]