How to compute the relative cohomology groups of a simple example? I'm starting to learn homology and cohomology, and I would like to try an example to understand more concretely what's going on. Let $T$ be the 2-torus, and $S^1$ the circle. Let $i: S^1 \to T$ be an embedding. I want to compute the relative cohomology groups $H^k(T,S^1)$ in two cases: the embedding $i$ is trivial (contractible), and $i$ maps the circle once around the handle of the torus. I'm working in $\mathbb{Z}$ coefficients.
I'm reading through Hatcher to try to solve this, and I see two ways to approach it. Using the long exact sequence of cohomology, and doing everything from scratch using cocycles and so on. I tried the long exact sequence but I couldn't see how to use the specific embeddings to get an answer. And I'm pretty much lost when it comes to doing it from scratch. All I know is that in homology, the trivial embedding of the circle is a boundary since it is the manifold-boundary of a submanifold (a disc), while the non-trivial embedding is a cycle that is not a boundary. So the homology class of the first is $0$ while that of the second is non zero. I don't understand the cohomology counterpart to these facts.
Could I get some guidance for this problem?
 A: Alright, with the help of Eric Wofsey's hints, I think I have an answer to my question.
First we have that $H^k(T,S^1) \cong \tilde{H}^k (T/S^1)$ for all $k$, where the quotient map is induced from the embedding of the circle. (I cannot find a reference for this fact.) Note that whenever a (nice enough?) space is path connected, the $0$th reduced homology is always $0$.
Now in the first case, let $S^1$ be embedded in $T$ such that it is contractible. Then $T/S^1$ is a torus wedge a $2$-sphere. Call this space $X_1$. Using the Mayer-Vietoris sequence, we get that $\tilde{H}^k (X_1) \cong \tilde{H}^k (T) \oplus \tilde{H}^k (S^2)$. The summands are widely known, so we get that
\begin{equation}
H^k(T,S^1) \cong
  \begin{cases}
  0 & \text{if}\ k=0\\
  \mathbb{Z}^2 & \text{if}\ k=1,2\\
  0 & \text{if}\ k \geq 3.
  \end{cases}
\end{equation}
Next, the second case, where $S^1$ is wrapped once around the handle of the torus, we get that $T/S^1$ is the pinched torus, and the wiki contains its cohomology groups. Changing the $0$th to be $0$, we get that
\begin{equation}
H^k(T,S^1) \cong
  \begin{cases}
  0 & \text{if}\ k=0\\
  \mathbb{Z} & \text{if}\ k=1,2\\
  0 & \text{if}\ k \geq 3.
  \end{cases}
\end{equation}
The next challenge would be to apply this to other non-trivial loops in the torus.
