Lecture notes for Tuesday, Feb. 7

We won’t be concerned with the technicalities of point set topology in this tutorial. You can follow everything by a) thinking “subset of \(\mathbb{R}^n\)” whenever we use the term “space” and b) ignoring some occasional comments about topological issues aimed at people who have studied point set topology. You should know what continuous functions are, what compact spaces are and a few basic results about them. (I mean you should know about these concepts at least for when the spaces involved are subsets of \(\mathbb{R}^n\).)

A path in a space \(X\) is a parametrized curve, that is a continuous function from the closed interval \([0,1]\) into \(X\). (When we draw pictures we only indicate the images of such a function and not any information about the parametrization beyond the direction of travel which is indicated by drawing an arrow on the curve; this is fine since we really won’t need to make much distinction between a path and a reparametrization of it.) We want to make a groupoid out of such paths, so let’s start by trying to define a (partial) multiplication of paths.

A natural thing to try is to glue paths together, this can be done when the starting point of one path is the endpoint of another. Given \(\alpha : [0,1] \to X\) and \(\beta : [0,1] \to X\) with \(\alpha(1) = \beta(0)\), we can define the composite path \(\beta \alpha\) by “first doing \(\alpha\) and then \(\beta\)”. Explicitly we define \(\beta \alpha (t) = \alpha(2t)\) for \(0 \le t \le 1/2\) and \(\beta \alpha (t) = \beta(2(t-1/2))\) for \(1/2 \le t \le 1\). Notice that to be able to fit the parameter domain of \(\beta \alpha\) into a unit interval we have to reparametrize \(\alpha\) and \(\beta\) to go twice as fast.

Now, this composition isn’t all that well behaved algebraically. Strictly speaking, it’s not associative and doesn’t have identities. Take associativity, for instance. The paths \(\gamma(\beta \alpha)\) and \((\gamma \beta) \alpha\) look the same when you draw them but have different parametrizations: the first one spends half the time doing \(\gamma\) (at double speed) but the second one only a quarter of the time (at quadruple speed). Similarly, there are no identities for this composition. On the level of pictures (i.e., considering only the images of the parametrized curves), a constant path \({\mathrm{id}}_{x}\) is an identity, but \(\alpha\ {\mathrm{id}}_{\alpha(0)}\) is (except in one case!) never equal to \(\alpha\).

Exercise. What is that one case where \(\alpha\ {\mathrm{id}}_{\alpha(0)} = \alpha\)? Prove that it really is the only exception.

You probably already know the punchline here: these problems are fixed by considering instead of paths, homotopy classes of paths. But there is another way to fix this which is interesting: using paths of different “lengths”.

The failure of associativity and identities was all caused by having to speed up curves to define the composition, so let’s try not doing that. We’ll let each curve proceed at its own pace. So instead of only using functions with domain \([0,1]\), we’ll allow any interval of the form \([0,t]\). A continuous function \([0,t] \to X\) is sometimes called a Moore path in \(X\). We can still define composition of paths by basically the same formulas, the key difference is that there is no speeding up involved now:

Given two Moore paths \(\alpha : [0,t_0] \to X\) and \(\beta : [0,t_1] \to X\) with \(\alpha(1) = \beta(0)\), we define the composite path \(\beta \alpha : [0, t_0+t_1]\) as follows: \(\beta \alpha (t) = \alpha(t)\) for \(0 \le t \le t_0\) and \(\beta \alpha (t) = \beta(t-t_0)\) for \(t_0 \le t \le t_0+t_1\).

Composition of Moore paths is associative and has identities. The identities are given by “instantaneous” paths \([0,0] = \{0\} \to X\), \(0 \mapsto x\). Recall that the structure of a partially defined multiplication which is associative and has identities is called a category. So we have the path category \(PX\) of a space \(X\). It’s objects are the points of \(X\), and the morphisms from \(x_0\) to \(x_1\) are the Moore paths going from \(x_0\) to \(x_1\).

This is an algebraic invariant of sorts associated to a space, but not a very practical one; to show two spaces are different you wouldn’t want to compute their path categories and show they’re not equivalent. And these path categories are ridiculously large even for simple spaces like \([0,1]\). We don’t really want to distinguish all of these paths. Also, notice that this trick is really one-dimensional. You can’t do something analogous for the higher homotopy groups by say, thinking of maps out of general boxes rather than cubes. The problem is that while glueing together intervals of different lengths always gives you an interval, glueing together rectangles of different sizes need not give another rectangle. (This just says a naive version of the Moore trick won’t work in higher dimensions, but in fact there is a fundamental reason –beyond the scope of this course– that there is nothing similar in dimension at least 3: strict 3-groupoids do not model all homotopy 3-types.)

The path category is almost never¹ a groupoid, since we don’t have inverses for paths: since the composite of two paths has a domain of length the sum of the lengths of the domains of the paths composed, the composition can’t be instantaneous unless both paths are. So Moore’s wonderful trick doesn’t give us a groupoid. This means we need to modify our strategy a little bit, and what we’ll do is instead of using paths (either unit-time or Moore) for morphisms, we’ll introduce an equivalence relation on paths and use the equivalence classes as morphisms.

Let’s think about what an equivalence relation \(\sim\) on paths (of either kind) needs to satisfy for the equivalence classes to be the morphisms of a groupoid. The first three requirements are just to maintain the status quo: to keep the good properties we already managed to get for paths themselves (as opposed to equivalence classes of paths).

1. If \(\alpha \sim \beta\), then the starting points of \(\alpha\) and \(\beta\) coincide and their ending points also coincide. We need this condition because we want the objects of the fundamental groupoid to be the points of \(X\) (not some kind of equivalence classes of points of \(X\)).
2. The equivalence relation should respect path composition, i.e., if \(\alpha \sim \alpha'\) and \(\beta \sim \beta'\), then \(\beta \alpha \sim \beta' \alpha'\). We need this so that the operation of composition is well-defined on equivalence classes.
3. If we’re not using Moore paths, we need to require also that identities and associativity for the equivalences classes, i.e., that \(\alpha\ {\mathrm{id}}_{\alpha(0)} \sim \alpha \sim {\mathrm{id}}_{\alpha(0)}\ \alpha\) and \((\gamma \beta) \alpha \sim \gamma (\beta \alpha)\).
4. Every path \(\alpha\) should have an “inverse up to \(\sim\)”, a path \(\alpha^{\leftarrow}\) such that \(\alpha \alpha^{\leftarrow} \sim {\mathrm{id}}_{\alpha(1)}\) and \(\alpha^{\leftarrow} \alpha \sim {\mathrm{id}}_{\alpha(0)}\). This of course implies the class of \(\alpha^{\leftarrow}\) is an inverse for the class of \(\alpha\).

For concreteness, instead of requiring in condition 4 that there is some path \(\alpha^{\leftarrow}\), let’s ask that condition 4 be satisfied for the following specific choice of \(\alpha^{\leftarrow}\): the path \(\alpha\) run in reverse, that is \(\alpha^{\leftarrow}(t) = \alpha(t_0-t)\) (if the domain of \(\alpha\) is \([0,t_0]\)).

Now, there are many possible choices of \(\sim\) satisfying these requirements and any such choice leeds to a groupoid of \(\sim\)-classes. To get the fundamental groupoid we take \(\sim\) to be homotopy relative to the endpoints (this is explained below). I remember in my undergraduate topology course learning about the fundamental group² and wondering what happened for weaker relations. Let’s look at a few examples.

1. The smallest possible equivalence relation satisfying the requirements. This is not a very interesting relation and I don’t think anybody has studied it. Roughly, two paths are equivalent according to this relation if one can be obtained from the other by several steps of “pruning or sprouting branches”. Here a branch is a piece of curve of the form \(\alpha^{\leftarrow} \alpha\), sprouting a branch means passing from \(\gamma \beta\) to \(\gamma \alpha^{\leftarrow} \alpha \beta\), and pruning a branch is going in the opposite direction. Notice that under this equivalence relation reparametrizing a path (almost) always gives you a non-equivalent path!
2. The smallest equivalence relation satisfying the requirements above plus the requirement that paths are equivalent to reparametrized versions of themselves. This is a bit more reasonable, but still not something people would care to study, as far as I know. But it feels pretty close to thin homotopy which has been considered and is the next example.
3. Thin homotopy The idea here is to regard to paths in \(X\) as equivalent if one can be deformed to the other with out sweeping out any area during the process, or, expressing this idea differently, if they can be deformed to one another inside a one-dimensional subspace of \(X\). There are at least two version of this which have been studied:
   1. For smooth manifolds \(X\) you can make the “without sweeping out any area” idea precise by saying that two smooth paths are equivalent if there is a smooth homotopy (see next section for the definition of homotopy) \(H\) between them such that it’s derivative has rank less than 2 everywhere. Using this equivalence relation you get the path groupoid of a manifold, which is useful in the study of (not necessarily flat) bundles on the manifold.
   2. For general topological spaces you can make the idea of being able to deform one path to another inside a one-dimensional subspace of \(X\) by declaring two paths to be equivalent if there is a homotopy \(H\) between them that factors through a finite \(T\) tree, that is, \(H = G \circ F\) where \(F : [0,1] \times [0,1] \to T\) and \(G : T \to X\), such that the paths in \(T\) given by \(F(t,0)\) and \(F(t,1)\) are piecewise-linear. You can read about this notion in the paper A homotopy double groupoid of a Hausdorff space by Ronald Brown, Keith A. Hardie, Klaus Heiner Kamps, Timothy Porter.
4. Homotopy, which is the topic of the next section.

A homotopy between two paths³ \(\alpha : [0,t_0] \to X\) and \(\beta : [0,t_0] \to X\) is a function \(H : [0,t_0] \times [0,1] \to X\) such that for all \(0 \le t \le t_0\) we have \(H(t,0) = \alpha(t)\) and \(H(t,1) = \beta(t)\). Given a homotopy \(H\), for any \(0 \le s \le 1\) we can consider the path \(\gamma_s : [0,t_0] \to X\) given by \(\gamma_s(t) = H(t,s)\); in these terms we have that \(\gamma_0 = \alpha\) and \(\gamma_1 = \beta\). We can think of a homotopy as giving a path between the paths \(\alpha\) and \(\beta\), that is, we can think of it as giving a function \(\Gamma : [0,1] \to \{\text{paths in } X\}\), where \(\Gamma(s) = \gamma_s\). It is possible, with some mild assumptions on \(X\), to make the set of paths in \(X\) into a topological space in such a way that this means of producing a \(\Gamma\) from an \(H\) gives a bijection between homotopies as defined above and “paths between paths”, but that’s precisely the kind of point-set topological technicality we will avoid dealing with here. So instead, we will define homotopies as map from a rectangle and use the path-of-paths imagery for intuition.

This (unrestricted) notion of homotopy is not very useful for us because every path is homotopic to (i.e., there exist a homotopy from the path to) a constant path: \(H(t,s) = \alpha((1-s)t)\). We don’t really want to consider two paths as equivalent if there is a path connecting them in the space of all paths, but only there is a path connecting in the space of paths having the same endpoints as the initial path. So we say a homotopy \(H\) fixes the endpoints or is relative to the endpoints if \(H(0,s)\) and \(H(t_0,s)\) are both constant functions of \(s\). Notice that if there is a homotopy fixing the endpoints between two paths, these must the same starting point and end point. Now, when talking about homotopies between paths we will almost always want the homotopies to be relative to the endpoints, so we’ll make the convention that “homotopy” means “homotopy relative to the endpoints” unless we explicitly say otherwise. (A few we will mention the endpoints anyway for emphasis.)

So at this point we can make the definition of fundamental groupoid using unit-time paths: the fundamental groupoid \(\pi_{\le 1}X\) of a space \(X\) has as objects the points of \(X\) and as morphisms between \(x_0\) and \(x_1\) the equivalence classes of (unit-time) paths from \(x_0\) to \(x_1\), where two paths are considered equivalent if there is a homotopy (fixing the endpoints) between them.

We can also get the same groupoid by using Moore paths, but we need one more step: so far we have only defined homotopies for Moore paths with the same domain. For paths with different “lengths”, we need a special definition of homotopy: we will say a homotopy between two Moore paths \(\alpha\) and \(\beta\) is a homotopy in the previous sense between the paths obtained by padding \(\alpha\) and \(\beta\) to a common length (kind of like putting fractions on a common denominator before adding them). There are a couple of ways we could implement this:

1. Only pad the shorter of the two paths. Say \(\alpha : [0,t_0] \to X\) and \(\beta : [0,t_1] \to X\) where \(t_0 \le t_1\), we define a homotopy from \(\alpha\) to \(\beta\) to be a homotopy from \(\alpha'\) to \(\beta\) where \(\alpha' : [0,t_1] \to X\) agrees with \(\alpha\) on \([0,t_0]\) and is constant (equal to \(\alpha(t_0)\)) on \([t_0,t_1]\).
2. Allow padding on both, that is, with notation as above, define a homotopy to be a path between \(\alpha''\) and \(\beta''\) where these both have domain \([0,t]\) for some \(t \ge t_0, t_1\) and are defined analogously to \(\alpha'\) above.

Exercise. Show that both definitions of homotopy would give rise to the same equivalence relation on paths of “existence of a homotopy fixing the endpoints”.

This allows us to define the fundamental groupoid in terms of Moore paths, and it’s straightforward to check we really do get an isomorphic fundamental groupoid. The proof boils down to checking that any Moore path \(\alpha : [0,t_0] \to X\) with \(t_0>0\) is homotopic to the unit-time path \(\alpha^u : [0,1] \to X\), \(t \mapsto \alpha(t\ t_0)\), and this passage from \(\alpha\) to \(\alpha^u\) gives a bijection between homotopy classes of Moore paths and homotopy classes of unit-time paths (always considering homotopies relative to endpoints).

Some simple spaces have simple fundamental groupoids. Take \(\mathbb{R}^n\), for example. Any two paths \(\alpha : [0,1] \to X\) and \(\beta : [0,1] \to X\) are homtopic by means of a so-called straight-line homotopy: \(H(t,s) = (1-s)\alpha(t) + s\beta(t)\). (This homotopy fixes the endpoints if \(\alpha\) and \(\beta\) share endpoints.) So we get that the fundamental groupoid of \(\mathbb{R}^n\) is just the indiscrete⁴ groupoid on the set \(\mathbb{R}^n\). This straight-line homotopy works a little more generally for any convex subset of \(\mathbb{R}^n\).

Exercise. A space \(X\) is called contractible if there is a function \(C : X \times [0,1] \to X\) and a point \(x_0\) in \(X\), such that for all \(x\) in \(X\) we have \(C(x,0) = x\) and \(C(x,1) = x_0\). You can think of this as giving for every \(x\) a path \(t \mapsto C(x,t)\) between \(x\) and \(x_0\), in such a way that the path depends continuously on \(X\). From the point of view of homotopy theory, a contractible space is equivalent to a single point. Prove that the fundamental groupoid of a contractible space is indiscrete.