I'm working on this problem: Find all connected $3$-sheeted covering space of the wedge sum of a circle and the projective plane.
Here is my work:
Let me explain: So we start with the preimage of the common point in the wedge sum, which is 3 points. Now in case 1, I suppose the preimage of the projective plane is 3 copies of it. Then each copy is linked to each point. To make this connected, I need a "circle" connecting each 2 points, hence the two lines I drew. I only have 1 line left to add (because this is a 3-sheeted covering), but I need at least 2 more lines to make this a covering, hence this is impossible.
In case 2, I suppose the preimage of the projective plane is a copy of it and its 2-sheeted covering space, the sphere $S^2$. Then $S^2$ must connect 2 points and the copy of projective plane has nowhere to go but to link with the remaining point. I need the covering to be connected, hence I drew 2 "circles" to connect the isolated points to others. I have one line left to add, and the lowest point in my work need to connect with a circle, so I put it there.
To sum up, there is only one 3-sheeted cover?
My question: is this correct? I don't know the optimal way to consider all cases so any other methods will be great.