I know this probably has a really straightforward answer, specially if it is as standard as it is entuitive to visualize. Still, because i'm not experienced at all on working with this objects and im just learning by myself, i would like to see a formal proof of the following facts, please : (# means the CONNECTED SUM) (i) $(\mathbb{R}\mathbb{P}^{2})\#(\mathbb{R} \mathbb{P}^{2}) = K^{2}$ (where $\mathbb{R}\mathbb{P}^{2}$ is the projective plane and $K^{2}$ is the Klein Bottle) (ii) $(\mathbb{R}\mathbb{P}^{2})\#(T^{2}) = (\mathbb{R}\mathbb{P}^{2})\#(K^{2})$ (where $T^{2}$ is a torus)
Im sorry if the question is annoying, but i dont know how to start, i can just see how it works if i draw or visualize, but i would like to see a rigourous proof. Again, sorry if its really obvious, but im just learning this by myself.
Thank you a lot
See this book page 35-45 for a rigorous proof. A non-rigorous proof can be done by drawing fundamental polygons (for example $\mathbb{R}\mathbb{P}^{2}$ is a square identifying opposite sides with different orientation and identify them). Then the conected sums is an identification of edges in different diagrams having the same orientation. It should not be difficult for you to visualize.