# Fundamental group of the sand clock

I computed the fundamental group of the following figure with the usual topology:

The figure corresponds to the set:

$X = \{(x,y,z) \in \mathbb{R}_3:x^2+y^2 = z^2,|z| \le 1\} \cup \{(-1,0,z):|z| \le 1\} \cup \{(1,0,z):|z| \le 1\}$

And I obtained that it is the trivial group using Seifert-Van Kampen theorem. However, other people claim that it is the free group with two generators.

My solution

I basically took as open sets one of the lines with a cone plus a mini cone. That way I have to open, path-connected-sets with the hypotheses of Seifert-Van Kampen.