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

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*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.

*$S^1\lor S^1\lor S^1$ is homeomorphic to the bouquet of $3$ circles -- in fact, it is usually taken as the definition of that bouquet.

