Klein Bottle with two points removed and path $\alpha$
Hi guys, I'm trying to calculate the homotopy class of path $\alpha$ in the Klein Bottle with points Q and R removed (picture above). For now I've managed to show that this space is homotopy equivalent to $S^1 \vee S^1 \vee S^1$.
My question is if it's possible to calculate it easely just looking at the square rapresenting the space (knowing its Fundamental Group's generators) or if it's easier to draw the entire space and proceed in a more "intuitive" way.
(I've basic notions of Algebraic Topology, Van Kampen's theorem is the most advanced result I've seen)
Thanks in advance.