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.