Abstract Nonsense

Crushing one theorem at a time

The Fundamental Groupoid and Group (Pt. III)


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

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What we claim is that, in fact, \Pi_1 is a functor \mathbf{Top}\to\mathbf{Grpd} (where, of course, \mathbf{Grpd} is the category whose objects are groupoids and whose arrows are functors). Of course, while we know how \Pi_1 acts on objects of \mathbf{Top} we have yet to mention how it acts on arrows. Given two topological spaces X and Y and a continuous map f:X\to Y we define \Pi_1(f):\Pi_1(X)\to\Pi_1(Y) to be the functor which takes an object x\in X to the object f(x)\in Y, and which takes an arrow [\alpha]:x\to y to the arrow [f\circ\alpha]:f(x)\to f(y). This is well-defined because homotopy respects compositions. Clearly \Pi_1 preserves identities,  for

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\Pi_1(\text{id}_X)([\alpha])=[\text{id}_X\circ\alpha]=[\alpha]

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for all \alpha an arrow in \Pi_1(X), and \Pi_1(\text{id}_X)(x)=\text{id}_X(x)=x, so that \Pi_1(\text{id}_X) is the identity functor \Pi_1(X)\to\Pi_1(X). Moreover, it’s clear that \Pi_1 preserves composition since if f:X\to Y and g:Y\to Z are continuous maps, we see that for any object x in \Pi_1(X) we have that

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\Pi_1(g\circ f)(x)=(g\circ f)(x)=g(f(x))=\Pi_1(g)(\Pi_1(f)(x))=(\Pi_1(g)\circ\Pi_1(f))(x)

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and

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\begin{aligned}\Pi_1(g\circ f)([\alpha]) &=[(g\circ f)\circ\alpha]\\ &=[g\circ(f\circ\alpha)]\\ &=\Pi_1(g)[f\circ\alpha]\\ &=\Pi_1(g)(\Pi_1(f)([\alpha]))\\ &=(\Pi_1(g)\circ\Pi_1(f))([\alpha])\end{aligned}

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Thus, we may put, in theorem form, this all as:

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Theorem: \Pi_1 is a functor \mathbf{Top}\to\mathbf{Grpd}.

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Pointed Topological Spaces and The Fundamental Group

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The issue with the fundamental groupoid is simple–it’s just too much information. The method of finding one-dimensional holes described in the motivation singled out a single point and attempted to find loops anchored at that point which are incontractible (i.e. not null-homotopic). What we have described though is an object which captures (up to homotopy) every single possible path in the space–this is just too much information. So, to get back to our original intent we focus in on one point.

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To this end, if X is a space and x_0\in X let \pi_1(X,x_0) denote \mathcal{P}[x_0,x_0]–the arrows x_0\to x_0 in \Pi_1(X). By definition \pi_1(X,x_0) is the homotopy classes of loops anchored at x_0. Now, while the above may seem like general nonsense, it makes the fact that \pi_1(X,x_0) has extra structure incredibly nice. Indeed:

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Theorem: Let \mathcal{G} be a groupoid and x an object of \mathcal{G}. Then, \text{End}_\mathcal{G}(x)=\text{Hom}_\mathcal{G}(x,x) is a group under the composition of \mathcal{G}.

Proof: It is simple category theory that a monoid is just a category with a single object, and in particular that \text{End}_\mathcal{G}(x) is a monoid. The fact that every arrow in \mathcal{G} is an isomorphism then implies that \text{End}_\mathcal{G}(x) is, in fact, a group. \blacksquare

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Corollary: Let X be a topological space and x_0\in X, then \pi_1(X,x_0) is a group.

Proof: By definition \pi_1(X,x_0) is \text{End}_{\Pi_1(X)}(x_0) for the groupoid \Pi_1(X). \blacksquare

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We call the group \pi_1(X,x_0) either the first homotopy group of X with base point x_0 or the fundamental group of X with base point x_0.

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Now, the issue with passing from the fundamental groupoid \Pi_1(X) to the fundamental groupoid \pi_1(X,x_0) is that now we must feed in more specific information–not only a space, but a space and a specific point. To this end, we define a pointed space to be an ordered pair (X,x_0) with X a topological space and x_0 a distinguished point x_0\in X called the base point. A pointed map between pointed spaces (X,x_0)\to (Y,y_0) is a continuous map f:X\to Y such that f(x_0)=y_0–so a continuous map which respects base points. It is easy to see that the collection of pointed spaces and pointed map forms a category, denoted \mathbf{Top}_\bullet, the category of pointed topological spaces.

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Remark: For those who are interested in this kind of thing, \mathbf{Top}_\bullet is the comma category (\bullet\downarrow\mathbf{Top}) where \bullet is the discrete category with one object.

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What we claim is that, like the fundamental groupoid, the fundamental group is a functor, this time to \mathbf{Grp}. But, as should be expected, it is a functor not from \mathbf{Top} but from \mathbf{Top}_\bullet. Indeed, define \pi_1:\mathbf{Top}_\bullet\to\mathbf{Grp} on objects as (X,x_0)\mapsto \pi_1(X,x_0). On arrows, f:(X,x_0)\to (Y,y_0) define \pi_1(f)([\alpha])=[f\circ\alpha]–we shall often denote \pi_1(f) as f_\ast and call it the induced map on fundamental groups. Note that the pointedness really only played an assignment role in this functor (domain and codomain assignment) and that for all intents and purposes this is just the functor \Pi_1. Thus, the exact same argument that shows that \Pi_1 is a functor shows that

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Theorem: \pi_1 is a functor \mathbf{Top}_\bullet\to\mathbf{Grp}.

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This shows us that \pi_1 is, in some sense, a topological invariant. Really \pi_1 (as defined here) is an invariant of pointed spaces, that said if f:X\to Y is a homeomorphism, then \pi_1(X,x_0) better be isomorphic to \pi_1(Y,f(x_0)) for all x_0. Indeed:

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Theorem: Let X and Y be topological spaces. If f:X\to Y is a homeomorphism, then \pi_1(X,x_0)\cong\pi_1(Y,f(x_0)) for all x_0\in X.

Proof: Since f is a homeomorphism it’s obvious that f:(X,x_0)\to(Y,f(x_0)) is an isomorphism of pointed spaces. The result then follows from the functorality of \pi_1. \blacksquare

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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 | , , , , , , , ,

1 Comment »

  1. […] The Fundamental Groupoid and Group (Pt. IV) Point of Post: This is a continuation of this post. […]

    Pingback by The Fundamental Groupoid and Group (Pt. IV) « Abstract Nonsense | August 30, 2012 | Reply


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