How to relate algebraic properties of the fundamental group to the topological properties of a space

Question

After just finally getting a somewhat intuitive grasp on just what the fundamental group of a topological is, I'm curious as to how one would relate the algebraic properties of the group to the topological properties of the space. For instance, in an annulus, one has the identity loop, and a loop which wraps around the "hole" in the space, and all multiples and inverses thereof, obtaining the infinite cyclic group. So what it looks like is that the fact that we have this single cyclic group being generated by one loop is telling us that there is precisely one classic "hole" that can be "looped around" any number of times. Similarly, how can one infer other, more exotic types of "holes" in the space from the fundamental group?

An example that, to me, is a little spooky is that the real projective plane has fundamental group isomorphic to $\mathbb Z_2$. So, essentially, there is a sort of strange hole around which one can wrap a loop, but if you wrap around it twice, the loop contracts back to nothing. And I have no idea what it might mean intuitively for a space to have a fundamental group that is non-abelian.

I'm also interested in the higher homotopy groups $\pi_n$ of higher dimensional spaces, and what other kinds of strange holes might exist. I might add that I was imagining that something like $\pi_3(S^2)$ couldn't even exist, since it seems it shouldn't be possible to embed a 3-sphere into a 2-sphere, but now I've seen a cursory explanation of Hopf fibrations, and apparently topology is even more bizarre than it seems. I don't really feel ready to think about that one quite yet.

I don't actually have much rigorous knowledge of topology, aside from things like the basic axioms of a topologial space, definitions of continuity, compactness, etc., so I would greatly appreciate a special emphasis on the intuitive concepts that the fundamental group and the other homotopy groups demonstrate.

Answer 1

"Holes" can only get you so far, and at some point I recommend you discard this intuition and just look at what's actually happening.

Here is a simple-to-visualize example of a space with a nonabelian fundamental group: take the plane $\mathbb{R}^2$ and remove two points, say $0$ and $1$. Pick a basepoint that's not $0$ or $1$. Then there's a loop $a$ where you go around $0$ and a loop $b$ where you go around $1$. These are two elements of the fundamental group, and I claim that the fundamental group is free on these two elements. In particular, $ab \neq ba$, so the loop "go around $0$, then go around $1$" is not homotopic to the loop "go around $1$, then go around $0$." These are some very concrete statements I'm making about, say, rubber bands on a board with pegs in it.

(But note that if you take $\mathbb{R}^3$ and remove two points, the resulting space has trivial fundamental group, although these two points can be detected using higher homotopy or homology. One really has to be careful with "holes.")

The higher homotopy groups are a bit more mysterious, but you can get some traction by thinking of them as describing "loops of loops." For example, $\pi_2$ describes all the ways that a loop in a space can start out small, then get big and wander around your space for awhile, then get small again. In particular, you can visualize the generator of $\pi_2(S^2) \cong \mathbb{Z}$ as being a loop at some point starting out very small, then wrapping around one side of $S^2$, then shrinking along the other side of $S^2$ back to where it was before. The other elements of $\pi_2(S^2)$ correspond to the loop wrapping around a bunch of times before settling down.

In general I recommend becoming familiar with more examples and doing more calculations, then backtracking to see if you can get any intuition about them.