# What is the fundamental group?

29 May

I’m at Yale for my fifth reunion, and it’s my birthday!  Happy birthday to me!  I’m a little overwhelmed by seeing all the old friends/catching up/showing off pictures of the baby so I’m hiding in my suite and pumping milk and writing a quick post about the fundamental group.

We’ve talked before about what a group is– a set of elements with some operation that takes two elements to another one (like addition with the group of integers takes 5+3 to 8 or multiplication takes 1*9 to 9) which satisfies some group axioms.  Given a geometric or topological object, we can associate a group with it my defining these elements and an operation, and making sure that they satisfy the axioms.

First we fix a basepoint of our space, which means that you pick a point and say that’s the one, that’s the special one I want.  Then our group elements with be isotopy classes of loops (this means you can wiggle loops to be the same, as in the red ones below) that go through the basepoint.

Red curves are homotopic to each other; blue curve is not

The group operation is concatenation– first you do one loop starting and ending at the basepoint, then you do the next loop starting and ending at the basepoint.  You can homotope away the middle connection to more clearly see the resulting loop.

Here’s the example:

Red and green make… more red. I didn’t want to make a brown curve

I don’t want to FOMO my reunion (I already ran away to take a nap) so we’ll make this super fast and just look at the fundamental group of the circle and of the torus.

How can I make a loop around a circle?  Well, there’s one obvious way- make one full circle and end up where you starting.  You can homotope to something that backtracks for a bit and then comes forward again, and you could go around two different directions (counterclockwise vs. clockwise).  So let’s call these +1 and -1.

Red goes around once counterclockwise, even though it backtracks a bit, and orange goes around once clockwise.  Imagine the colors down on the black circle.

If you put the red and orange curves together, concatenating like we did above, you’d fully backtrack over yourself, which means you could homotope to just a point.  Let’s call that 0 (can you guess where we’re going here?)

This blue curve goes around the circle three times.

You can go around any integer number of times, but no fractions because you won’t end up back at the basepoint.  So this is a rough schematic of why the fundamental group of the circle is the integers.

I want to go back to my reunion, so I’ll just tell you that the fundamental group of the torus is $\mathbb{Z}^2$, a.k.a. ordered pairs of integers, as I hinted in a previous post using the following picture.

Left: follow the numbers to see the knot. Right: look at the bottom-most green line.

Sorry for the short and delayed post.  It’s my birthday, YOLO.  (I’m not that sorry about this post being delayed but I am sorry if it is unclear/too short please comment/let me know if you need more explanation).