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I have the following problem of the book "Massey, Algebraic Topology an Introduction" (Chapter 4, section 4, problem 1):

Let $X$ arcwise connected and $U,V$ open arcwise connected subsets of X. Suppose that $U\cup V=X$, $U\cap V$ non-empty and arcwise connected.

Consider $\varphi_1:\pi(U\cap V)\to\pi(U)$, $\varphi_2:\pi(U\cap V)\to\pi(V)$, $\psi_1:\pi(U)\to\pi(X)$, $\psi_2:\pi(V)\to\pi(X)$, all morphism induced by the inclusions.

Prove that if $\varphi_2$ is isomorphism onto, then so is $\psi_1$.

I tried to use the first version of Seifert-Van Kampen Theorem but I have no result. Maybe is a wrong way to do this problem.

Any solution or hint would be appreciated.

Thanks in advance.

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  • $\begingroup$ I think this indeed follows from SVK theorem. Basically the amalgamated product turns out to be isomorphic to the fundamental group of X, you can see this from the presentation of the amalgamated product. $\endgroup$ – Nick L Nov 6 '18 at 20:05
  • $\begingroup$ @NickL with amalgamated you mean free? $\endgroup$ – jnaf Nov 6 '18 at 20:07
  • $\begingroup$ Not quite, amalgamated free product is different from free product $\endgroup$ – Nick L Nov 6 '18 at 20:08

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