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Math Help - Homotopy Loop problem

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    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.
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  2. #2
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    Quote Originally Posted by EinStone View Post
    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) P_1,\,P_2,\,P_2,\ldots, write a_k for (the homotopy class of) a clockwise loop around the point P_k, and write 1 for the homotopy class of a null-homotopic loop. The loop L_2 in the example linked to above is given by 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 P_k then the corresponding loop a_k becomes equal to 1. So for example if you remove the hole at P_1 then the path L_2 becomes 1a_21a_2^{-1} = a_2a_2^{-1} = 1.

    Now proceed by induction. Suppose that 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 L_{n+1} = L_na_{n+1}L_n^{-1}a_{n+1}^{-1}. If you remove one of the points P_k (for 1\leqslant k\leqslant n) then L_n becomes null-homotopic and so L_{n+1} becomes a_{n+1}a_{n+1}^{-1} = 1. And if you remove P_{n+1} then a_{n+1} becomes 1 and so L_{n+1} becomes L_nL_n^{-1} = 1. That completes the inductive proof that 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 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.
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