Need help with the following exercise;
"Let M be a compact orientable surface of genus g. Prove that M with a point removed has the same homotopy type as 2g circles with a point in common."
I have really no idea how to start with this exercise. I have the intuition for the case of a torus but not for this general case. Any help is appreciated
Here we assume that you know the basic construction of a genus $g$ surface via the fundamental polygon. If not, you should talk to your instructor about it before you work on this problem. Now assume this, you can look at Hatcher's book page 5 and figure out from the fundamental domain's glueing maps. I suppose knowing CW structure on the surface will be helpful, but not necessary.