Finding map from Klein bottle to $RP^2$ that induces an epimorphism of fundamental groups. I want to find a map from the Klein bottle to $RP^2$ that induces an epimorphism of fundamental groups. Since the fundamental group of $RP^2$ is $Z_2$, intuitively it feels like a good start would be to look for any non-trivial map from $K$ to $RP^2$, but frankly even that is difficult for me. After reading quite a bit about the Klein bottle i still have problems even visualizing it, so any help would be appreciated.

Let $X$ and $Y$ be two connected manifolds of dimension $n \geq 2$. 
There is a map $\varphi : X\# Y \to X$ given by mapping $Y$ to a disc $D$. By the Seifert van Kampen theorem, $\pi_1(X\# Y) \cong \pi_1(X^{\circ})*_{\pi_1(S^{n-1})}\pi_1(Y^{\circ})$ where $X^{\circ}$ denotes $X$ with an embedded open disc removed; likewise for $Y^{\circ}$. The map $\varphi$ induces a map 
$$\varphi_* : \pi_1(X^{\circ})*_{\pi_1(S^{n-1})}\pi_1(Y^{\circ}) \to \pi_1(X^{\circ})*_{\pi_1(S^{n-1})}\pi_1(D) \cong \pi_1(X)$$
which is the identity on the subgroup $\pi_1(X^{\circ})$ and necessarily trivial on $\pi_1(Y^{\circ})$ (because $\pi_1(D)$ is trivial). In particular, $\varphi_*$ is an epimorphism.
As $K = \mathbb{RP}^2\#\mathbb{RP}^2$, we obtain a map $\varphi : K \to \mathbb{RP}^2$ inducing an epimorphism $\varphi_* : \pi_1(K) \to \pi_1(\mathbb{RP}^2)$.
