Algebraic Topology, Homology - Cohomology This problem made me crazy, so please take a look at it:

"$X=S^1 \cup_\phi D^2$" where $\phi$ is a degree $g$ map. (That means $X$ is obtained by glueing a disk to $S^1$ with a degree $g$ map.)
  $H_n(X, Z_g)=$?
  $H^n(X, Z_3)=$?
  $H^n(X, Z_g)=$?

Thanks
 A: The cellular chain complex looks like 
$$0 \to  \mathbb{Z} \stackrel{d_2}{\to} \mathbb{Z} \stackrel{d_1}{\to} \mathbb{Z} \to 0$$
where $d_2 = g$ and $d_1=0$. These maps come from

Cellular boundary formula: $d_n(e_{\alpha}^n) = \sum_{\beta} d_{\alpha\beta}e_{\beta}^{n-1}$ where $d_{\alpha\beta}$ is the degree of the map $S_{\alpha}^{n-1} \to X^{n-1} \to S^{n-1}_{\beta}$ which is the composition of the attaching map and the quotient map collapsing $X^{n-1}-e^{n-1}_{\beta}$ to a point. [I lifted this theorem from Hatcher.]

(In our case $X^1\cong S^1$, so $X^1-S^1 = \emptyset$, hence $d_2$ is essentially multiplication by the degree of $\phi$, i.e. $g$. The space is connected, so $H_0(X) \cong \mathbb{Z}$, hence $d_1$ is forced to be 0).
From here, you can go about this two ways. You can find the homology with integer coefficients and then apply universal coefficients. The other way is to tensor with $\mathbb{Z}_g$ and take homology (for $H_*(X,\mathbb{Z}_g)$), apply $\text{Hom}(-,\mathbb{Z}_g)$ and take cohomology (for $H^*(X,\mathbb{Z}_g)$), etc. The computations should be pretty straight-forward, nothing tricky beyond this point.
