Munkres Exercise 70.1

This is question number 1 from section 70 (The Seifert-van Kampen Theorem) in Munkres.

Assume the hypotheses of the Seifert-van Kampen Theorem.

Suppose that the homomorphism $i_*$ induced by the inclusion $i : U \cap V \rightarrow X$ is trivial.

(a) Show that $j_1$ and $j_2$ induce an epimorphism $$h: (\pi_1(U, x_0)\space / \space N_1)\space * \space (\pi_1(V, x_0)\space / \space N_2)\rightarrow \pi_1(X,x_0)$$ where $N_1$ is the least normal subgroup of $\pi_1(U, x_0)$ containing image $i_1$ and $N_2$ is the least normal subgroup of $\pi_1(V, x_0)$ containing image $i_2$.

(b) Show that $h$ is an isomorphism

Note that $i_1$, $i_2$, $j_1$, and $j_2$ are homomorphisms induced by the inclusion map such that: $$i_1 : \pi_1(U \cap V, x_0) \rightarrow \pi_1(U, x_0)$$ $$i_2 : \pi_1(U \cap V, x_0) \rightarrow \pi_1(V, x_0)$$ $$j_1 : \pi_1(U , x_0) \rightarrow \pi_1(X, x_0)$$ $$j_2 : \pi_1(V, x_0) \rightarrow \pi_1(X, x_0)$$

Here is my attempt at part (a): Since $N$ is generated by elements of the form $i_i(w) \space * \space i_j(w)^{-1}$ where $w \in \pi_1(U \cap V)$, it contains the element $i_i(w) \space * \space identity$. Hence, it contains the elements of that form so it follows that it contains $N_1$. For similar reasoning, it will also contain $N_2$.

Since by Van Kampen, we have an isomorphism between $(\pi_1(U) * \pi_1(V)) \space / \space N$ and $\pi_1(X)$, and $N$ contains $N_1$ and $N_2$, it follows that $(\pi_1(U, x_0)\space / \space N_1)\space * \space (\pi_1(V, x_0)\space / \space N_2) \supseteq (\pi_1(U) * \pi_1(V)) \space / \space N$. Thus, the map $h$ is surjective.

• This looks like an interesting question, surprised no one answered it yet.
– Zee
Dec 4, 2017 at 7:11