I'm looking for help with the following question, specifically the Fundamental Group part. I won't complain if the Homology Groups are calculated as well, but I think I should be okay on this part:
I think van Kampen is the way to go with $U$ being the space minus a point, and $V$ being a neighborhood about the point, but I'm having difficulty grinding through it. When calculating things like the Torus or Klein bottle, one is easily able to identify what the respective $U$ deformation retracts to. What does our $U$ deformation retract to in this case?
Edit: Actually, we could kill questions like this in general if there is a standard way to recognize what the $U$ deformation retracts to in the general case. I remember reading somewhere, maybe on this site, that you can realize the fundamental group of all spaces built in this fashion using this setup of applying van Kampen.
Edit 2: This isn't HW, I'm studying for a qual.
Edit 3: I clarified that I am mostly interested in knowing the solution for the fundamental group part. This editing is getting out of control, sorry about that.
$U$ retracts to the boundary of a hexagon $H$ modulo your equivalence relation $\sim$. Under $\sim$, there are two equivalence classes of vertices of $H$: The ones being pointed at and the ones being pointed away from. These two equivalence classes are connected by 6 edges, but only 2 equivalence classes of edges: The single arrowed ones and the double arrowed ones. Hence $U$ retracts to a space consisting of two points, connected by two paths, i.e. $S^1$.