I have some difficulties understanding a proof of the Wirtinger presentation using the Van Kampen theorem, found in John Stiwell's "Classical Topology and Combinatorial Group Theory".
I perfectly understand the proof except for its very end (which is crucial) : "The typical generator of $\pi_1(A \cap B)$, a circuit round a trench (Figure 161) has the form $a_ia_j^{-1}a_{i+1}^{-1}a_j$ in $\pi_1(A)$ and 1 in $\pi_1(B)$. Thus, the SVK theorem gives precisely the Wirtinger relations for $\pi_1(A \cup B)$".
(I understand we use a deform retract argument in the last sentence).
What does "has the form … in $\pi_1(A)$" mean?
Using SVK, we compute $\pi_1(A) \star_{\pi_1(A \cap B)} \{1\}$ (since $\pi_1(B) = \{1\}$), but I don't understand how the said relations apply to the amalgamated free product.
Thank you very much!
Respectfully,
AF
Best Answer
The group $\pi_1(\mathscr A)$ is a free group, with free basis $a_1,...,a_n$, and so $a_i a_j^{-1} a_{i+1}^{-1} a_j$ is an element of $\pi_1(\mathscr A)$.
The inclusion map $i : \mathscr A \cap \mathscr B \to \mathscr A$ induces a group homomorphism $i_* : \pi_1(\mathscr A \cap \mathscr B) \to \pi_1(\mathscr A)$, and this homomorphism plays an important role in the statement and application of Van Kampen's Theorem. In particular, part of the process of applying Van Kampen's Theorem is to take certain elements of the group $\pi_1(\mathscr A \cap \mathscr B)$, which is the domain of $i_*$, and determine their $i_*$-images in the group $\pi_1(\mathscr A)$, which is the range of $i_*$.
So to say that a certain element of $\pi_1(\mathscr A \cap \mathscr B)$ has the form $a_i a_j^{-1} a_{i+1}^{-1} a_j$ in $\pi_1(\mathscr A)$ means that the image of that element under the group homomorphism $i_*$ is equal to $a_i a_j^{-1} a_{i+1}^{-1} a_j \in \pi_1(\mathscr A)$.