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So im doing an exercise that asks me to find a topological space that has a fundamental group isomorphic to $\mathbb{Z}_n * \mathbb{Z}_m$. I would like to know if the space that I found is correct or not. So we know that if we do the labeling of polygon of $n$ sides with the labeling scheme $a...a , n$ times by the Seifert Van-Kampen Theorem its fundamental group is isomorphic to $\mathbb{Z}_n$. So know we have found spaces that gives fundamental groups of $\mathbb{Z}_n$ for any $n$ natural. Now what I did was the wedge the spaces wich give the fundamental groups required, and since my spaces are the quotient of polygons I can always find a neighborhood $W_i$ in each one so that $p$ is a deformation retract of $W_i$ and so I can use the Seifert Van Kampen Theorem with some appropriate open sets that cover up the space and again I can conclude that the fundamental group of this spaces is isomorphic to what I want. Is this correct ?

Someone
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  • You could also take the product: https://math.stackexchange.com/questions/291311/the-fundamental-group-of-a-product-is-the-product-of-the-fundamental-groups-of-t – Oiler Jan 01 '20 at 18:12
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    I dont think it works, i want the free product of the groups not the cartesian product, if i did that how would i get the free product? – Someone Jan 01 '20 at 18:18
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    Yes your argument is correct. In general if $(X_1, x_1)$ and $(X_2, x_2)$ are pointed spaces so that $x_i$ is a (strong) deformation retract of some open neighbourhood $U_i \subset X_i$ then $\pi_1(X_1 \vee X_2) \cong \pi_1(X_1) * \pi_1(X_2)$ by Seifert-van Kampen – William Jan 01 '20 at 18:45

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