Let $S$ be the closed orientable surface surface given in the picture below, where we attach three disks $D^2$ to $S$: the first along $a$ via a degree $2$ map, the second along $b$ via a degree $3$ map, and at $c$ via a degree $4$ map. Compute $\pi_1(S)$ and $H_*(S)$.
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I don't see why the disks should change $\pi_1(S)$ from the ordinary surface $M_g$ so this can be computed the ordinary way. As for $H_*(S)$, clearly $H_0(S)=\mathbb{Z}$ since the space is path connected. I would think $H_n(S)=0$ for $n>2$. It is now a matter of computing $H_1(S)$ and $H_2(S)$. I think I should be able to 'ignore' the disk attached at $c$ because $S$ deformation retracts to $S$ without the disk attached at $c$ (correct?). But figuring out how to deal with the rest of the space is stumping me.
Do I just break the space into the $3$ pieces using Mayer-Vietoris and compute the homology of the $3$ pieces and then piece together? I would think any of the 'legs' can be deformed down to just a wedge of two circles and the disk is attached to one of these circles with the given degrees. Is this correct?