# Abstract Nonsense

## The Fundamental Groupoid and Group (Pt. I)

Point of Post: In this post we describe the fundamental group and groupoid functors.

$\text{ }$

Motivation

$\text{ }$

We have discussed in previous posts how the notion of homotopy as being an equivalence that captures, and focuses in on, the various notions of connectivity that a space carries. Now, the first thing one does when one defines a new notion of equivalence is to try to classify all objects of interest up to this equivalence. For us, this means that we would (in a perfect world) be able to classify all spaces up to homotopy equivalence. Of course, as in most categories(subjects of study), this is an untenable goal. It is not practically possible to, say, explicitly classify all finite groups (even though we have made stupendous strides in this direction).

$\text{ }$

Now, while it is impossible to actually classify all spaces up to homotopy equivalence, it is possible to work towards this. One of the standard ways to work towards the classification of objects up to equivalence is to start looking for equivalence invariants of the object. This will give one an approach to test whether or not two objects are different–if they are the same, they must have the same invariant. Thus, we are on the lookout for homotopy invariants.

$\text{ }$

In this post we define two such homotopy invariants–the zeroth and first homotopy functors. These each measure connectivity in a specific way. The zeroth functor measures the usual notion of connectedness–in other words, it measures how connected is (in the traditional sense) by (in essence) counting the number of connected components of a space–a common topological, homotopy invariant. The first homotopy functor, alternatively called the fundamental group, measures the amount of “one-dimensional holes” a space has–in the exact same spirit as the first homology functor. And, just like the case of the first homology functor, the fundamental group functor does this by using groups. In a sense, we discretize the continuous notion of a topological spaces into algebra.

$\text{ }$

Now, when we tried to measure one-dimensional holes using $H^1$ we did it using the following observation: if a space has no one-dimensional holes, then every unfilled in triangle is the boundary if a filled in triangle. We make an entirely different observation to define the fundamental group. We begin by taking a space that has a one-dimensional hole, something simple, like the annulus $X=\{z\in\mathbb{C}:1<|z|<2\}$. How can we identify this hole without making explicit mention of it–i.e. how can we detect it using only things inside our space? A very clever way to do this is to consider loops in our space anchored at some point $x_0$.  If we consider a loop at $x_0$ which doesn’t encompass the hole, then it should be (visually) clear that we can contract this loop down to a point. Conversely, if the loop does encompass the hole, we can see that we can’t shrink it down (staying within the space) to a point–the hole prevents this.

$\text{ }$

Now, while the description of this method of finding one-dimensional holes seems a little wishy-washy, thanks to our notion of homotopy, we can make it entirely rigorous. Indeed, let $c_{x_0}$ denote the constant path $[0,1]\to\{x_0\}\subseteq X$. What we are really saying is that the existence of loops anchored at $x_0$ which aren’t homotopic to $c_{x_0}$ detect the obstruction–the one-dimensional hole. So, if we let $\pi_1(X,x_0)$ denote the homotopy classes of all loops $[0,1]\to X$ anchored at $x_0$, then the non-triviality of $\pi_1(X,x_0)$ indicates the existence of a topological obstruction–once again, in this case, a one-dimensional hole.

$\text{ }$

Now, where does the algebra come into play? So far we really only seem to care about the “function” (functor) which takes a space $X$ with a distinguished point $x_0$ and associates to it the number $\#\pi_1(X,x_0)$ of homotopy classes of loops based at $x_0$. The algebra comes into play by noticing that there is a natural way to multiply homotopy classes in $\pi_1(X,x_0)$. Indeed, intuitively given a loop $\alpha:[0,1]\to X$ and a loop $\beta:[0,1]$ their product, $\alpha\cdot\beta$, should be the loop $[0,1]\to X$ gotten by first doing $\alpha$, and then doing $\beta$. Noticing that with this product the (homotopy class of the) constant loop $c_{x_0}$ becomes an identity, and that the reversal loop (the loop done in reverse) becomes an inverse, we may conclude that, in fact, $(\pi_1(X,x_0),\cdot)$ becomes a group!

$\text{ }$

This group is a much more sensitive measure of the “one-holieness” of the space. For example, this group structure can pick up on the number of holes much easier than the mere cardinality of the group can. To get an intuition for how this is true, consider the two spaces given by the circle $\mathbb{S}^1$ and the torus $\mathbb{T}^2$. It’s obvious that they have a different amount of holes–the sphere has one, whereas the torus has two (one taken up by its center, and one taken up by the interior of the doughnut). That said, it is well-known amongst math people that $\#\pi_1(S^1)=\#\pi_1(\mathbb{T}^2)=\aleph_0$ (where here, we haven’t indicated a base-point because it is unnecessary, as we shall soon see). This does not yet $\pi_1(S^1)\cong\mathbb{Z}$ whereas $\pi_1(\mathbb{T}^2)\cong\mathbb{Z}^2$. So, the algebra (in this case, the rank of the group) was able to detect the existence of multiple holes whereas the mere cardinality was not.

$\text{ }$

Path Homotopies and the Fundamental Groupoid

$\text{ }$

For us a path from $x$ to $y$ on a space $X$ will be a continuous map $\alpha:[0,1]\to X$ with $\alpha(0)=x$ and $\alpha(1)=y$.  We will often denote a path from $x$ to $y$ as $\alpha:x\to y$ when the space $X$ is clear from context. Now, given two paths $\alpha:x\to y$ and $\beta:y\to z$ we can define a path $\alpha\cdot\beta:x\to z$ defined as follows

$\text{ }$

$(\alpha\cdot \beta)(s)=\begin{cases}\alpha(2s) & \mbox{if}\quad s\in[0,\tfrac{1}{2}]\\ \beta(2s-1) & \mbox{if}\quad s\in[\tfrac{1}{2},1]\end{cases}$

$\text{ }$

Since $\alpha(2(\tfrac{1}{2}))=\alpha(1)=y=\beta(0)=\beta(2(\tfrac{1}{2})-1)$ we see that $\alpha\cdot\beta$ is continuous by the gluing lemma. We note that this is precisely the formal way of deifning “do $\alpha$ first, and then do $\beta$” with necessary speed adjustments (so that we can do it all in the time contained in $[0,1]$.

$\text{ }$

For each pair of points $x,y\in X$ let $\mathcal{P}_X(x,y)=\mathcal{P}(x,y)$ denote the set of all paths $x\to y$. We see then that $\cdot$ is a pairing

$\text{ }$

$\mathcal{P}(x,y)\times\mathcal{P}(y,z)\to\mathcal{P}(x,z)$

$\text{ }$

I leave it to anyone interested to verify that if $\alpha,\beta\in\mathcal{P}(x,y)$ and $\gamma,\delta\in\mathcal{P}(y,z)$ are such that $\alpha\simeq\beta$ and $\gamma\simeq\delta$ then $\alpha\cdot\gamma\simeq\beta\cdot\delta$. Thus, if we let $\mathcal{P}[x,y]$ denote the homotopy classes $\mathcal{P}(x,y)/\simeq$ of paths $x\to y$ then $\cdot$ descends to a well-defined multiplication

$\text{ }$

$\mathcal{P}[x,y]\times\mathcal{P}[y,z]\to\mathcal{P}[x,z]:([\alpha],[\beta])\mapsto [\alpha\cdot\beta]$

$\text{ }$

$\text{ }$

References:

$\text{ }$

[1] Spanier, Edwin Henry. Algebraic Topology. New York: McGraw-Hill, 1966. Print.

[2] Hatcher, Allen. Algebraic Topology. Cambridge: Cambridge UP, 2002. Print.

[3] Bredon, Glen E. Topology and Geometry. New York: Springer-Verlag, 1993. Print.