In lectures it was said that the fundamental group of the following shape, which we called $X_3$ (why?):
is $\pi_1(X_3, P) \cong \mathbb{Z} * \mathbb{Z} * \mathbb{Z} * \mathbb{Z}$, i.e. the free group on four generators.
Why is this the case?
2026-03-26 19:03:05.1774551785
Why is the fundamental group of a triangle in a circle the free group on four generators?
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Your space is homotopically equivalent to a wedge sum of $4$ circles. The result now follows from van Kampen.