Fundamental Groups in Punctured Space

When learning concepts in math, I often find it helpful to look at "toy" examples. They help me more intuitively see facts about the objects I'm looking at. This is particularly true for Fundamental Groups, which are what my last post was about. I found a simple example to be the plane with some points removed from it — as if I'd come along with a dart and poked holes in @@\mathbb{R}^2@@.

It's actually a lot more useful than it might initially seem. Suppose we have some region that's simply connected save for some "sufficiently nice" holes. We can pick one of the holes and deform the set such that the hole becomes a disk. This is guaranteed to be possible if the border of the hole can be parameterized due to Jordan-Schoenflies. Then, we might "contract" the hole down to a point if it's open, or contract it arbitrarily small if it's closed. We could finally repeat this process on all the other holes, ending up with some subset of a "punctured plane."

At least, that's the intuition. It's far from complete — there's still work to do in the case of a closed hole, for instance. Perhaps to make our efforts more concrete, we can focus on @@\mathbb{R}^2@@ with some disk (either open or closed, and possibly of zero radius) removed. It's clear the fundamental group of this space is @@\mathbb{Z}@@. Why? Since a loop never goes through the origin, we can write all its points in terms of polar coordinates @@\gamma(s) = (r,\theta)@@, then collapse them all to a fixed @@R@@ as %% h(s,t) = (tR + (1-t)r, \theta). %% Thus, the every loop is homotopic to one on a fixed circle, whose fundamental group is known to be @@\mathbb{Z}@@. Indeed, that's where we get the concept of winding numbers. However, it may not be obvious that this is true (it certainly wasn't to me). I might try to prove it formally in a later post.

So that's the case of one hole, but what if there are multiple holes? I don't know how to do this formally, but here's a possible approach. We might start by partitioning the plane into circular sectors, "based" at some point @@B@@, each containing one hole. If all the holes are isolated points, this is always possible since there are only finitely many of them. If not, you might have to argue based off the fact the holes can be made arbitrarily small.

We might then take an arbitrary loop and "contract" it along the sector division lines, as shown above. The end result of this homotopy will be a loop @@\gamma@@ that only crosses sectors at their "base point." This clearly gives rise to the same "winding" structure as earlier. If that's not obvious, do the following. In each sector pick a loop @@\ell_i@@ homeomorphic to a circle that stays entirely in the sector and that goes through @@B@@. Then, similar to what we did above, project each of @@\gamma@@'s "segments" onto their relevant sector's loop @@\ell_i@@.

Looking at this construction, we may guess at the fundamental group for a punctured plane with @@n@@ holes. We might start with @@\mathbb{Z}^n@@, but then I'd point out that the order we wrap in would matter. For instance, in the image above, wrapping around the bottom hole, then the top, then the bottom is clearly different from wrapping around the bottom, then the bottom again, then the top. Thus, we might conjecture that the fundamental group of this punctured space is the free group on @@n@@ elements.

How correct this reasoning is? How easy it is to formalize? How well it extends to closed holes? I don't know. Fundamental groups are quite interesting, though, and they raise a lot of questions. For instance, are all fundamental groups free? What about on non-planar spaces, like tori or mobius strips? What if you have an infinite number of "holes," as in @@\mathbb{R} \setminus \mathbb{Q}@@? I don't have the machinery to answer these questions. I know nothing about Topology, but I look forward to studying it if I can.