I'm particularly interested in the case of $n=3$ but the most general formulation of the problem I'm trying to tackle is the following:
Let $\mathbf{T}^n$ be the n-dimensional torus and $W=L_1\sqcup\dots\sqcup L_k$ be a disjoint union of circles (possibly non-trivial torus cycles). I would like to compute the cohomology groups $H^p(\mathbf{T}^n\setminus W)$ using the Mayer-Vietoris sequence. I can seem to get some mild progress with deRham cohomology, although singular would be preferred, although I believe the Universal Coefficient Theorem should imply that it doesn't particularly matter much in this case.
My current approach is as follows: Surround each $L_i$ by tubular neighbourhoods $N_i$ and take $V$ to be the union of these tubular neighbourhoods. Set $U=\mathbf{T}^n\setminus W$. The set $U\cap V$ will be simply be the union of these tubular neighbourhoods with each circle $L_i$ removed, these retract to the boundary which gives us a bunch of disjoint $n-1$ dimensional tori, call these $D_i$. Meanwhile $V$ will simply be homeomorphic to a disjoint union of $S^1\times B_{n-1}$'s where $B_{n-1}$ is an ball of dimension $n-1$. I shall denote this by $L_i\times B_{n-1}$ to keep track of the $i$. Since the ball has vanishing cohomology except for in degree 0, the Kunneth theorem gives
$$H^p(V)=\bigoplus_{i}H^p(L_i\times B_{n-1})\cong \bigoplus_{i}H^{p}(L_i)$$ which is only non-zero for $p=0,1$.
Hence the Mayer-Vietoris sequence becomes
$$\dots\rightarrow H^p(\mathbf{T}^n)\rightarrow H^p(\mathbf{T}^n\setminus W)\oplus \bigoplus_{i}H^p(L_i)\rightarrow \bigoplus_{i}H^p(D_i)\xrightarrow{\partial}H^{p+1}(\mathbf{T}^n)\rightarrow\dots$$
I believe I can compute the boundary map $\partial:\bigoplus_{i}H^{n-1}(D_i)\rightarrow H^n(\mathbf{T}^n)$ in deRham cohomology, take a form $\omega$ representing an element of $H^{n-1}(U\cap V)$, we can write this as a difference $\omega=\eta_U-\eta_V$ However, since $V$ is a disjoint union, we can simply write $\eta_V$ as a sum $\sum_{i}\eta_{N_i}$. The boundary map $\partial$ sends $\omega$ to a globally defined form on $\mathbf{T}^n$, $d\eta_V=\sum_{i}d\eta_{N_i}$ which is supported on $U\cap V$, we can then apply Stokes' theorem and at the end of the day, should get that the boundary map $\partial$ is simply the sum, sending $$\bigoplus_{i}H^{p}(D_i)\ni(x_1,\dots x_k)\xrightarrow{\partial} \sum_{i}x_i\in H^{n}(\mathbf{T}^n).$$
Since this map is surjective it follows that $H^{n}(\mathbf{T}^n\setminus W)=0$.
We can also conclude from this for $n>2$ that $H^{n-1}(\mathbf{T}^n\setminus W)$ sits in an exact sequence $$0\rightarrow H^p(\mathbf{T}^n)/\text{im}(\partial^{n-2}) \rightarrow H^{n-1}(\mathbf{T}^n\setminus W)\rightarrow \bigoplus_{i}H^{n-1}(D_i)\xrightarrow{\partial} H^{n}(\mathbf{T}^n)\rightarrow 0$$ where $\partial^{n-2}$ is the boundary map $$H^{n-1}(U\cap V)\rightarrow H^{n-1}(\mathbf{T}^n).$$ So we can deduce that $H^{n-1}(\mathbf{T}^n\setminus W)$ is isomorphic to some subgroup of $\bigoplus_{i}H^{n-1}(D_i)$ such that each element $(x_1,\dots x_k)$ satisfies $\sum_{i}x_i=0$ with some extra quotient relation coming from the previous boundary map.
However, this is the best that I can get, I'd like to know if there are any means to compute the earlier boundary maps (since one cannot simply use Stokes' theorem to check the integral of the forms like the last boundary map) and whether anything more concrete can be deduced.