Understanding the Fundamental Group

Question

Let us have a space $X$. Define the fundamental group $\pi_{1}(X,x_0)$ for some point $x_0 \in X$. If I understood it well, this group contains the path-homotopy classes that are consisting of paths starting and ending at $x_0$. Can you explain how do we have more than $1$ path-homotopy classes? Also what happens when we have a path-connected space?

Answer 1

The picture I have in mind is that of a room with a column at his center. Say that you are in front of the column, with a rope in your hand; you fix one end of the rope to the ground, walk around while holding the other end, and then go back to where you started.

Now stay there, and try to rewind the rope while staying where you are (you pull it back at you). There are two things that may happen:

1. You manage to rewind the rope;

2. At some point, the rope gets stuck around the column. In the easiest configuration, this happened because after fixing the first end of the rope to the ground, you walked around the column and then came back.

The room is of course path-connected (you can walk around freely), but you want to express mathematically this kind of problematic, which in turn hints at the fact that there is a column. The formalization of the problem - and its answer - is the fundamental group.

To elaborate still a bit, you may notice that there are actually many configurations. You may distinguish them in this way:

1. You rewind the rope successfullly.

2. You don't, but if you walk around the column clockwise once (+1) with the rope in your end and come back, you are able to rewind it;

3. Same, but with walking around counterclockwise (-1);

4. Twice clockwise and it's done (+2);

5. Twice counterclockwise and it's done (-2);

6. ...

Also, notice that walking around 1 time clockwise and one time counterclockwise is the same as doing nothing at all. We therefore say that the fundamental group is given by the integers, because there is one possible configuration for every integer, and the way the configurations relate one another is the same as the way the integer relate one other (the sum).

This is probably far too complicated, and different from what you've asked... but I think it's good to have something concrete in mind. :-)