Archive for April 7th, 2009

Homotopy Classes and the Fundamental Group

If you imagine a hole on a plane—say, a circular one with radius \epsilon > 0—as being an infinitely tall column rising above the plane, you can determine how many holes exist on that surface in the following natural way:

Suppose that you take a string, tie it into a loop, and lay it on the surface. If you can pull the loop in a free manner to any location on the surface without breaking it, then it must not loop around any of these columns. Suppose that every conceivable loop on this surface has the same property. Then that surface must logically not have any of these columns—i.e., it must not have any holes.

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