# Fundamental groups of the projective plane

I'm practicing for my topology final. Munkres' Topology, Theorem 74.4 states that:

Let $X$ denote the $m$-fold projective plane, that is the space obtained by taking the connected sum of the projective plane with itself $m$ times: $$X = \underbrace{P^2 \# \dots \# P^2}_{m \text{ times}}.$$

Then $\pi_1(X, x_0)$ is isomorphic to the quotient of the free group of $m$ generators $\alpha_1, \ldots, \alpha_m$ by the least normal subgroup containing the element $$(\alpha_1)^2(\alpha_2)^2\ldots(\alpha_m)^2$$

I'm just trying to figure out what this statement is actually saying. What is the fundamental group of the $m$-fold projective plane? I saw somewhere that it's $\mathbb{Z}_2 * \mathbb{Z} * \ldots * \mathbb{Z}$ but how do you formally prove this?

• The fundamental group is precisely what's described - there's a smallest normal subgroup of $F_m$, the free group on $m$ generators, that contains that element. I would be surprised if this is isomorphic to $\Bbb Z_2 * \Bbb Z * \dots * \Bbb Z$ in general, though it is for $m \leq 2$. – user98602 Jan 14 '15 at 4:15