3 Questions concerning the fundamental group of $S^1\vee S^1\vee S^1$

Question

I have the following questions:

1) What is the fundamental group of $S^1\vee S^1\vee S^1$ and why?

2) Is $S^1\vee S^1\vee S^1$ homeomorphic to the bouquet of 3 circles

3) Are $S^1\vee S^1\vee S^1$ and the bouquet of 3 circles homotopic to one another?

Thanks for the help.

Answer 1

1. The fundamental group of $S^1\lor S^1\lor S^1$ is a free group on three generators: each generator is the homotopy class of a loop about one of the three circles. More generally, if $X$ and $Y$ are path-connected spaces with fundamental group $\pi_1(X)$ and $\pi_1(Y)$ then the fundamental group $\pi_1(X\lor Y)$ is isomorphic to the free product of $\pi_1(X)$ and $\pi_1(Y)$. This follows from the Seifert-van Kampen Theorem.
2. $S^1\lor S^1\lor S^1$ is homeomorphic to the bouquet of $3$ circles -- in fact, it is usually taken as the definition of that bouquet.