1. ## Homotopy Loop problem

Hi, the following is also known as the picture-nail problem.

Consider X = the euclidean plane with N holes. Explain how to construct a loop L in X that is not null-homotopic in X, but if you remove any hole in X, the loop becomes null-homotopic in the new space.

For N = 1 this is trivial. I am mainly concerned with the cases N = 2 and N = 3. The problem is not about proving anything, simple construction is enough.

I actually found an example for N = 2 (click here), I am only looking for an example of N = 3 now.

2. Originally Posted by EinStone
Hi, the following is also known as the picture-nail problem.

Consider X = the euclidean plane with N holes. Explain how to construct a loop L in X that is not null-homotopic in X, but if you remove any hole in X, the loop becomes null-homotopic in the new space.

For N = 1 this is trivial. I am mainly concerned with the cases N = 2 and N = 3. The problem is not about proving anything, simple construction is enough.

I actually found an example for N = 2 (click here), I am only looking for an example of N = 3 now.
Given points (or holes) $\displaystyle P_1,\,P_2,\,P_2,\ldots$, write $\displaystyle a_k$ for (the homotopy class of) a clockwise loop around the point $\displaystyle P_k$, and write 1 for the homotopy class of a null-homotopic loop. The loop $\displaystyle L_2$ in the example linked to above is given by $\displaystyle L_2 = a_1a_2a_1^{-1}a_2^{-1}$ (where the inverse denotes the corresponding anticlockwise loop, of course). If you remove one of the points $\displaystyle P_k$ then the corresponding loop $\displaystyle a_k$ becomes equal to 1. So for example if you remove the hole at $\displaystyle P_1$ then the path $\displaystyle L_2$ becomes $\displaystyle 1a_21a_2^{-1} = a_2a_2^{-1} = 1$.

Now proceed by induction. Suppose that $\displaystyle L_n$ is a path around n points such that if you remove the hole at any one of those n points then the path becomes null-homotopic. Define $\displaystyle L_{n+1} = L_na_{n+1}L_n^{-1}a_{n+1}^{-1}.$ If you remove one of the points $\displaystyle P_k$ (for $\displaystyle 1\leqslant k\leqslant n$) then $\displaystyle L_n$ becomes null-homotopic and so $\displaystyle L_{n+1}$ becomes $\displaystyle a_{n+1}a_{n+1}^{-1} = 1$. And if you remove $\displaystyle P_{n+1}$ then $\displaystyle a_{n+1}$ becomes 1 and so $\displaystyle L_{n+1}$ becomes $\displaystyle L_nL_n^{-1} = 1$. That completes the inductive proof that $\displaystyle L_{n+1}$ has the required property.

Once you have that formula, it's easy in principle to draw a geometric picture of the loop $\displaystyle L_3$ for example. But the complexity of the loops increases alarmingly as the number of holes increases. Even for n=3 it will look fairly messy.