# Abstract Nonsense

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

Point of Post: This is a continuation of this post.

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The Fundamental Group of Path Connected Spaces

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It turns out that even though, a priori, to talk about the fundamental group of a space we have to distinguish a base point, it turns out that for nice spaces this really is irrelevant. In particular, we shall show that there is (in some sense) a unique isomorphism $\pi_1(X,x_0)\xrightarrow{\approx}\pi_1(X,x_1)$ whenever the points $x_0$ and $x_1$ share the same path component (maximally path connected subset of $X$) of $X$.

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In particular, let’s assume that $x_0$ and $x_1$ are two points of $X$ such that there exists a path $\gamma_0:x_0\to x_1$. Define a map from $\pi_1(X,x_0)$ to $\pi_1(X,x_1)$ by $[\alpha]\mapsto [\gamma_0][\alpha][\gamma_0]^{-1}$. In essence, this takes a path $\alpha\in\mathcal{P}(x_0,x_0)$ and creates a path in $\mathcal{P}(x_1,x_1)$ by first travelling back from $x_1$ to $x_0$ by $\gamma_0$, performing $\alpha$, and then making our way back to $x_1$ via $\gamma_0^{\leftarrow}$.

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Now, it should be evident that this correspondence $\pi_1(X,x_0)\to\pi_1(X,x_1)$ is a group map, but if not it follows from the following general fact:

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Theorem: Let $\mathcal{G}$ be any groupoid and $x,y$ objects in $\mathcal{G}$. Then, for any $f\in\text{Hom}_\mathcal{G}(x,y)$ the map $i_f:\text{End}_\mathcal{G}(x)\to\text{End}_\mathcal{G}(y)$ given by $i_f(g)=fgf^{-1}$ is a group isomorphism.

Proof: To see that $i_f$ is a group map we merely verify that

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$i_f(gh)=fghf^{-1}=(fgf^{-1})(fhf^{-1})=i_f(g)i_f(h)$

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The fact that $i_f$ is a bijectiion follows immediately since it is trivial to see that the map $i_{f^{-1}}:\text{End}_\mathcal{G}(y)\to\text{End}_\mathcal{G}(x)$ defined by $i_{f^{-1}}(g)=f^{-1}gf$ is an inverse for $i_f$. $\blacksquare$.

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If we then, following the previous proof, denote $i_{\gamma_0}$ to be the map $\pi_1(X,x_0)\to\pi_1(X,x_1)$ defined by conjugating by $[\gamma_0]$ we get that:

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Theorem: Let $X$ be a topological space and $x_0,x_1\in X$. If $\gamma_0:x_0\to x_1$ then $i_{\gamma_0}:\pi_1(X,x_0)\xrightarrow{\approx}\pi_1(X,x_1)$

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Remark: Up until this point the consideration of the fundamental groupoid seemed contrived and unappealing. It seemed an unnecessarily complicated construction which served no real purpose. Sure, it’s definitely made the proof that the fundamental group is actually a group simpler, but really this was because all of the work (which would have looked exactly the same whether done in the context of the fundamental group or groupoid!) was done in the groupoid section. That said, the above theorem is where the total structure of the fundamental groupoid becomes so powerful. Imagine if we had started with just defining the fundamental group $\pi_1(X,x_0)$ as homotopy classes of loops at $x_0$. Up until this point, everything would have worked out exactly the same. But, now imagine trying to define this isomorphism $\pi_1(X,x_0)\to\pi_1(X,x_1)$. We would have two choices. We could make everything intuitive, explain how it’s just conjugation by some path from $x_0\to x_1$, but to do this we’d really have to open up the whole can of worms that is involved with the fundamental groupoid: how to multiply arbitrary paths, why everything is an isomorphism, why notions like associativity, etc. still apply to this general non-loop situation. Or, we could instead just define the map without mentioning that it’s conjugation–this would be simple, in the sense that the proof would be concise, but it would lose the big algebraic picture. This is precisely why, even though it seems abstract and useless, the fundamental groupoid is a necessary discussion topic prior to the fundamental group.

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Because of this theorem, if we start with a path connected space $X$ we know that $\pi_1(X,x_0)\cong\pi_1(X,x_1)$ for any two points $x_0,x_1\in X$. Thus, we may unabashedly define the fundamental group $\pi_1(X)$ of $X$, to be the fundamental group of $X$ based at any point of $X$. It is extremely important to note that this isomorphism isn’t natural–we have to choose a path $x_0\to x_1$ to produce an isomorphism $\pi_1(X,x_0)\xrightarrow{\approx}\pi_1(X,x_1)$. That said, it’s not so bad, because at least the class of isomorphisms we have described is just a conjugacy class in the group of isomorphisms $\pi_1(X,x_0)\to\pi_1(X,x_1)$.

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As a last note, we define the notion of simple connectedness. In particular, we call a space $X$ simply connected if $X$ is path connected and $\pi_1(X)$ is trivial. Simple connectedness intuitively means that $X$ is a path connected space that has no one-dimensional holes.

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References:

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[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.

August 30, 2012 -