Abstract Nonsense

Crushing one theorem at a time

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.

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August 30, 2012 - Posted by | Algebraic Topology, Topology | , , , , , , , ,

4 Comments »

  1. […] by topological Euclidean balls. Note, since connectedness is equivalent to path connectedness, the independence of base point for fundamental groups comes to connected manifolds. Our proof heavily follows […]

    Pingback by Topological Manifolds (Pt. III) « Abstract Nonsense | August 30, 2012 | Reply

  2. Hello. Sorry for my English. There is a little Error at the end, when you define “simply conected”, you should say that the space X is path conected but you said it is simply conected.

    Comment by Guest | August 30, 2012 | Reply

  3. […] the induced structure), and wanting to keep things topologically simple, we decide that we want simply connected Riemann surfaces. We look for a while, and we can’t find (up to equivalence!) any new ones. […]

    Pingback by Riemann Surfaces (Pt. I) « Abstract Nonsense | October 2, 2012 | Reply


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