Recovering a topological space from its fundamental groupoid

Question

Given only the fundamental groupoid of a topological space X, we can recover the underlying set of X since objects of the groupoid (as a category) are precisely the elements of X. I would like to know if there is a way to recover the topology of X using the information contained by its fundamental groupoid.

Answer 1

This is an interesting question, and the quick answer is "no", but the long answer is "kind of". Let me say a bit more.

First, there's a quibble here that I don't fully agree with, but I feel obligated to mention. Usually when we talk about categories, our notion of "sameness" is equivalence, rather than isomorphism. This means that we aren't allowed to ask about the "set of objects" of a category, since this set is not well defined up to equivalence (for instance, up to equivalence, the category of finite dimensional $\mathbb{R}$ vector spaces could have one object for every natural number, or it could have one for every set of size continuum). You can talk about strict categories which allow you explicit access to the underlying set, and for these the correct notion of "sameness" is isomorphism. You should think about (categories vs strict categories) and (equivalence vs isomorphism) as being rather analogous to ("homotopy types" vs "topological spaces") and ("homotopy equivalence" vs "homeomorphism").

There's many ways to make this analogy precise, but for now it's enough to know that it's too much to expect the fundamental groupoid to recover the space up to homeomorphism. It's better to ask if it can recover the space up to homotopy equivalence.

Another, more serious, issue is that the fundamental groupoid only sees "one dimensional information" about your space. This is because we remember paths in the space up to homotopy, but importantly we don't remember what precisely the homotopies were. In the language of the last paragraph, if $X$ is path connected, then the fundamental groupoid $\Pi_1 X$ is equivalent to the fundamental group $\pi_1 (X,x_0)$ for any point $x_0 \in X$ (where we view this group as a one object groupoid). In particular, for path connected spaces, the fundamental groupoid can't distinguish between spaces with the same fundamental group. But there are plenty of non homotopy equivalent spaces which happen to have the same fundamental group!

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So that's why the answer is "no". First, since we talk about categories up to equivalence, it doesn't make sense to get anything more than the homotopy-type of $X$ from its fundamental groupoid. Second, since the fundamental groupoid is equivalent to the fundamental group, we can't even expect to recover the homotopy type from just this information.

Of course, you might remember that there are higher homotopy groups, and that these remember the "higher dimensional information" about $X$ which the fundamental group forgets. Is there a way to make the fundamental groupoid remember this bonus information?

The answer here, as promised, is "kind of"!

We can build a bicategory $\Pi_2 X$ whose objects are points of $X$, arrows are paths in $X$, and 2-cells homotopies (up to higher homotopy). This construction, roughly, remembers all the "$\leq 2$-dimensional information" of $X$. We can also build a tricategory $\Pi_3 X$ where our 2-cells are all homotopies, and now we have 3-cells between these, which are higher homotopies (up to even higher homotopy). This tricategory remembers all the "$\leq 3$-dimensional information" of $X$. But, of course, why stop there? One can imagine building an $\infty$-category $\Pi_\infty X$ where we remember all the homotopies in all dimensions! This is the famous Fundamental $\infty$-Groupoid of $X$, and this makes your dream a reality: From the fundamental $\infty$-gropuoid $\Pi_\infty X$ you can recover $X$ up to homotopy!

There's only one catch...

Groupoids are algebraic objects that you can really compute with, and present (by generators and relations, say), which is what makes them so useful as invariants of a space. Unfortunately we have no such algebraic description of $\infty$-groupoid. At time of writing, it's not too much of an exaggeration to say that we basically define "$\infty$-groupoid" to be "topological space up to homotopy" (this is made precise with the language of model categories). This makes computations with the fundamental $\infty$-groupoid feel much more like doing topology than doing algebra, and while the concept is still useful, it's perhaps not as useful as it might one day be.

Finding a good algebraic definition of the $\infty$-groupoid, and proving that we can recover a space (up to homotopy) from (the algebraic description of) its fundamental $\infty$-groupoid is called the Homotopy Hypothesis, which is one of the most famous open problems in algebraic topology today!

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I hope this helps ^_^

Answer 2

There is no way. For example, take $\mathbb{R}$ with the Euclidean topology and $\mathbb{R}_{indisc}$ with the indiscrete topology. The underlying set is the same and both spaces are contractible, so their fundamental groupoids are the same.