Find a presentation for the fundamental group of $P^2\#T$

Question

I have to find a presentation for the fundamental group of $ P^2\# T $. Which I believe to be identified by the following labeling scheme: $(a_1b_1a_1b_1)(a_2b_2a_2^{-1}b_2^{-1})$. $P^2$ is the projective plane and T is the torus. I honestly have no idea how to tackle this. Kind regards :)

Answer 1

Think about van Kampen's theorem. The natural decomposition corresponds to open sets that look like punctured copies of $\mathbb{R}P^2$ and $T^2$ intersecting in an open annulus. The fundamental group is then the free product of $\pi_1(\mathbb{R}P^2\setminus \{pt\})$ and $\pi_1(T^2 \setminus \{pt\})$ with an additional relation that comes from analyzing the loop in the intersection. I'm happy to give a more complete answer if need be, but this hint should probably suffice.