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I am trying to solve this by Van-Kampen's theorem. What I do is just move a point at left side and a point at right side. Then I get two open sets whose intersection deformation retracts to a circle. Each open set deformation retracts to a figure 8.

So what is the next? Just free product with 4 basis module the normal subgroup generated by the fundamental group of a circle? I can not see what is this normal subgroup. Is it just Z?

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space

This is how I pictured the described space. It's easy to give a CW decomposition where the red arc above is a cell. It's contractible, so this CW complex is homotopically equivalent to what you get after contracting that red arc to a point (see Hatcher's Algebraic Topology, Proposition 0.17, pages 15-16); and that's three circles wedged at a point, whose fundamental group is simply $\mathbb{Z} * \mathbb{Z} * \mathbb{Z}$.

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  • $\begingroup$ Dear Pedro, a petty little nitpick: you should take the red arc (= the cell) and its extremities in order to get a subcomplex. Then you can contract it. That said, your solution is excellent and your picture very illuminating: +1. $\endgroup$ – Georges Elencwajg Oct 16 '15 at 9:56
  • $\begingroup$ Dear qwwqqq: I strongly encourage you to accept and upvote this elegant answer. $\endgroup$ – Georges Elencwajg Oct 16 '15 at 9:59
  • $\begingroup$ @GeorgesElencwajg Agree, this is truly elegant! $\endgroup$ – user198206 Oct 16 '15 at 22:57

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