Use van Kampen’s theorem to compute \(A *_C\) as a quotient of \(A * \mathbb{Z}\), as stated in the text.
Solution
Goal. Use van Kampen to express the amalgamated product \(A *_C\) as a quotient of \(A * \ZZ\).
1. Setup: \(A\) is a space with \(\pi_1(A) = A\) (abusing notation), and \(C\) is a subspace with \(\pi_1(C) = C\), attached along a map.
1.1. The amalgamated product \(A *_C\) is \(\pi_1(A) *_{\pi_1(C)} \pi_1(\cdot)\).
this is the fundamental group of the space obtained by gluing \(A\) to another space along \(C\).
2. Attach a \(2\)-cell (or a cylinder) to \(A\) along a loop representing a generator of \(C\).
2.1. Attaching a \(2\)-cell along a loop \(\gamma\) kills \(\gamma\) in \(\pi_1\).
the attaching map makes \(\gamma\) null-homotopic.
2.2. The resulting space has fundamental group \(\pi_1(A) / \langle\!\langle \gamma \rangle\!\rangle\).
van Kampen: attaching a \(2\)-cell along \(\gamma\) quotients \(\pi_1(A)\) by the normal closure of \(\gamma\).
3. \(A *_C\) is the quotient of \(A * \ZZ\) by the relation identifying the generator of \(\ZZ\) with the image of \(C\).
3.1. \(A * \ZZ\) is \(\pi_1\) of \(A\) wedged with a circle.
van Kampen for a wedge: \(\pi_1(A \vee S^1) = \pi_1(A) * \pi_1(S^1) = A * \ZZ\).
3.2. The amalgamated product \(A *_C\) is obtained by identifying the generator of \(\ZZ\) with the image of the generator of \(C\) in \(A\).
this is the definition of the amalgamated free product: \(A *_C = (A * \ZZ)/\langle\!\langle i_*(c) \cdot t^{-1} \rangle\!\rangle\) where \(t\) is the generator of \(\ZZ\) and \(i_*: C \to A\) is the inclusion.
4. Q.E.D.
3.2 expresses \(A *_C\) as the quotient of \(A * \ZZ\) by the stated relation.