Cutting a torus enough times disconnects it I am interested in showing that if you cut a torus too many times it becomes disconnected.
Let $\mathbb T^n$ be the standard $n$-dimensional flat torus. Let $M_1, \ldots, M_k$ be $k$ disjoint smooth $(n-1)$-dimensional connected closed submanifolds. I call the manifold $$\mathbb T^n \setminus (M_1 \cup \ldots \cup M_k)$$ a $k$-cut torus.
I want to show that there exists some $k$ (maybe depending on $n$) such that any $k$-cut torus is disconnected.
The difficulty: I have no background in algebraic topology (besides the notions of homotopy and fundamental group), specifically I haven't learned about cohomology, homology, and the like. So if the answer uses these tools, please use them in a gentle way and/or help me out by giving references I can learn from. Thanks!
 A: $k=2$ is enough to disconnect. I'm going to try to explain this using only the language of fundamental groups and homotopies. It's not a short answer, but what I am trying to do is to take the proof that I know using all kinds of algebraic topology (including cup product in cohomology and some differential topology) and turn it into elementary language. This is not always possible for every algebraic topology problem, but it is possible for this problem.
Suppose that $M_1 \cup M_2$ do not disconnect, so each $M_i$ ($i=1,2$) individually does not disconnect. Since $M_i$ has codimension 1, it is orientable and has, locally, two sides; more precisely, $M_i$ has a regular neighborhood diffeomorphic to $M_i \times (-1,+1)$ in which $M_i$ itself is embedded as $M_i \times 0$. Pick a transverse orientation for $M_i$, meaning that locally one side is the "back" and the other is the "front". So at each point where a path crosses $M_i$ transversally we can assign a crossing number $\pm 1$: assign $-1$ if the path crosses front to back and $+1$ if it crosses back to front.
Pick a base point $O \in \mathbb{T}^n - (M_1 \cup M_2)$. Consider the fundamental group $\pi_1(\mathbb{T}^n,O) \approx \mathbb{Z}^n$. For each of $i=1,2$ define a homomorphism
$$f_i : \pi_1(\mathbb{T}^n,O) \to \mathbb{Z}
$$
as follows: given a closed path $\gamma$ in $\mathbb{T}^n$ based at $O$, homotope $\gamma$ relative to its endpoints to be transverse to $M_i$, assign $\pm 1$ crossing numbers to each crossing point, and add them up to get $f_i(\gamma)$. By using homotopies one can show that this is well-defined and that it is a homomorphism. Also, since $M_i$ is nonseparating, there exists a loop $\gamma$ such that $f_i(\gamma)=+1$: after departing $O$, the path $\gamma$ approaches $M_i$ from its back side, then crosses to the front side; then returns to $O$. So $f_1,f_2$ are both surjective homomorphisms. Let $L_i = \text{kernel}(f_i)$, and so $L_i$ is a rank $n-1$ subgroup of $\pi_1(\mathbb{T}^n,O) \approx \mathbb{Z}^n$.
Now we break into two cases depending on whether $f_1 = \pm f_2$. In each case we will find a contradiction
Case 1: $f_1 = \pm f_2$. By changing the transverse orientation on $M_2$ we may assume $f_1 = f_2$. It follows that the homomorphism $f_1 + f_2 = 2 f_1$ is not surjective because its image of $2 \mathbb{Z}$. On the other hand, $f_1 + f_2$ is represented by taking any loop $\gamma$ based at $O$, making it transverse simultaneously to $M_1 \cup M_2$, and adding up local intersection numbers. From the hypothesis that $M_1 \cup M_2$ is nonseparating, we can construct a loop $\gamma$ that, after departing $0$, approaches one of $M_1$ or $M_2$ from its back side, then crosses to the front side, then returns to $O$, having only just the one crossing; it follows that $(f_1+f_2)(\gamma)=1$ and so $f_1 + f_2$ is surjective, contradiction.
Case 2: $f_1 \ne \pm f_2$. It follows that $L_1 \ne L_2$, because two surjective homomorphisms $\mathbb{Z}^n \mapsto \mathbb{Z}$ with equal kernels must be equal up to sign. It follows that $L_1 \cap L_2$ is a rank $n-2$ subgroup of $\pi_1(\mathbb{T}^n,O) \approx \mathbb{Z}^n$. 
Now the idea is to apply Van-Kampen's Theorem to get a certain calculation of $\pi_1(\mathbb{T}^n,O)$ which will contradict that $\pi_1(\mathbb{T}^n,O) \approx \mathbb{Z}^n$. I'll define two path connected open sets $V,W$ containing $O$, with path connected intersection, and with inclusion induced homomorphisms $j_V :\pi_1(V,O) \to \pi_1(\mathbb{T}^n,O)$ and $j_W :\pi_1(W,O) \to \pi_1(\mathbb{T}^n,O)$. We can then conclude from Van-Kampen's Theorem that $\pi_1(\mathbb{T}^n,O)$ is generated by $\text{image}(j_V)$ and $\text{image}(j_W)$. The desired contradiction will arise from  the specifics of the calculations of those two images.
First, let $V = \mathbb{T}^n - (M_1 \cup M_2)$, which is path connected and contains $O$. Since every loop in $V$ is disjoint from $M_1$ and from $M_2$, it follows that $\text{image}(j_V) \subset L_1 \cap L_2$.
Next, let $c$ be a circle embedded in $\mathbb{T}^n$ as follows: starting from $O$, travel to a point near the back of $M_1$, then cross $M_1$ to its front, then travel to a point near the front of $M_2$, then cross $M_2$ to its back, and finally travel back to $O$. Let $W$ be a regular neighborhood of $M_1 \cup c \cup M_2$, and so $W$ is path connected and it deformation retracts to $M_1 \cup c \cup M_2$. The loop $c$ decomposes at its intersection points with $M_1,M_2$ into two subarcs, one denoted $c_O$ containing $O$. So, by application of Van Kampen's theorem, $\text{image}(v_W)$ is generated by two families of loops: all loops based at $O$ contained in $M_1 \cup c_O \cup M_2$, and the single loop $c$. The loops based at $O$ and contained in $M_1 \cup c_O \cup M_2$ can all be homotoped to be disjoint from $M_1,M_2$, by homotoping their portions in $M_1$ to the backside of $M_1$ where $c_O$ approaches from, and by homotoping their portions in $M_2$ to the backwide of $M_2$ where $c_0$ also approahces from. It follows that the image under $j_v$ of $\pi_1(M_1 \cup c_O \cup M_2,O)$ is contained in $L_1 \cap L_2$. Since the image under $j_W$ of $c$ itself is just one single additional element, it follows that $\text(image(j_V))$ is contained in the subgroup generated by $L_1 \cap L_2$ plus one other element, a subgroup which has rank $\le n-1$. 
We've shown that $\text{image}(j_V)$ is contained in the rank $n-2$ subgroup $L_1 \cap L_2$, and that $\text{image}(j_W)$ is contained in a rank $n-1$ subgroup that contains $L_1 \cap L_2$, and so both images are contained in the same rank $n-1$ subroup of $\pi_1(\mathbb{T}^n,O) \approx \mathbb{Z}^n$. This contradicts that those images together generate $\mathbb{Z}^n$.
