Fundamental group of a kite shaped grid

Question

What is the fundamental group of a "kite-shaped" two dimensional figure, with lines connecting opposite pairs of corner-points? (So, a diamond with a cross in the middle.)

Doesn't this just deformation retract to a wedge of 4 circles?

Answer 1

Yes, it deformation retracts to a wedge of four circles. Additionally, $\pi_1(\bigvee^4 S^1)=\ast^4\pi_1(S^1)$, where $\ast$ is the free product of groups.