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Stream: theory: algebraic topology

Topic: singular homology and fundamental groupoids

Leopold Schlicht (Oct 14 2021 at 17:18):

What would be an example of two spaces whose fundamental groupoids coincide (up to isomorphism) but with non-isomorphic singular $H_1$? What about the other direction?

Joe Moeller (Oct 14 2021 at 17:35):

First homology is the abelianization of the fundamental group by the Hurewicz map. So I don't think this is possible. (I hope I recalled everything correctly)

Leopold Schlicht (Oct 14 2021 at 19:19):

Nice result, thanks for telling me! So the first homology can be recovered from the fundamental group. But I think is only really makes sense if our space is path-connected (otherwise, what is "the" fundamental group?). So maybe there's still a counterexample for non-path-connected spaces?

Also, what about the other direction: what is an example of two spaces with the same first homology groups but with non-isomorphic fundamental groups?

What would be an example of two spaces whose fundamental groupoids coincide but which have non-isomorphic $H_2$?

Joe Moeller (Oct 14 2021 at 19:24):

For "fundamental group of a non-connected space" you can talk about the fundamental groupoid. I've never heard anybody say this, but I don't see why you couldn't talk about the first homology groupoid too.

Joe Moeller (Oct 14 2021 at 19:25):

You might be interested in looking at Eilenberg-Mac Lane spaces to cook up counter/examples.

John Baez (Oct 14 2021 at 19:28):

Leopold Schlicht said:

What would be an example of two spaces whose fundamental groupoids coincide but which have non-isomorphic $H_2$?

When you say "coincide", I hope you mean "are equivalent": that's the right notion here.

We should be able to find an example where both spaces are connected and simply connected. Then their fundamental groupoids will both be equivalent to the terminal groupoid. So, we just need to find two connected and simply connected spaces with different $H_2$.

And the easiest example is: the point and the 2-sphere.

John Baez (Oct 14 2021 at 19:35):

Also, what about the other direction: what is an example of two spaces with the same first homology groups but with non-isomorphic fundamental groups?

Remember, for connected spaces the homology group is the abelianization of the fundamental group, and the fundamental group can be anything. So in a way you're just asking "what is an example of two groups that aren't isomorphic, whose abelianizations are isomorphic?"

And a nice example is free group on 2 generators, $F_2$, and the free abelian group on 2 generators, $\mathbb{Z}^2$. They both have $\mathbb{Z}^2$ as their abelianization.

But what space has fundamental group $F_2$? And what space has fundamental group $\mathbb{Z}^2$? People who want to do homotopy theory should know these, or learn them. (I can give away the answer if it's too hard for someone.)

John Baez (Oct 14 2021 at 19:42):

Leopold Schlicht said:

What would be an example of two spaces whose fundamental groupoids coincide (up to isomorphism) but with non-isomorphic singular $H_1$?

I don't think there are any. First of all, I think you can write any space as a disjoint union of path-connected components, and its fundamental groupoid is the coproduct of the fundamental groupoids of its path-connected components, and its $H_1$ is the coproduct of those of its path-connected components. (This is certainly true for the "locally nice" spaces that algebraic topologists mainly care about, like simplicial complexes or CW complexes.)

Second of all, the $H_1$ of any path-connected space is the abelianization of its fundamental group.

Using these facts you can show that the fundamental groupoid determines $H_1$ for any space (or at least any "locally nice" one - and I don't really care about the others when doing homotopy theory).

Leopold Schlicht (Oct 16 2021 at 16:45):

Joe Moeller said:

You might be interested in looking at Eilenberg-Mac Lane spaces to cook up counter/examples.

Counterexamples for which question? I asked several questions.

Leopold Schlicht (Oct 16 2021 at 17:03):

@John Baez Thanks! That answers almost all of my question. But I'm still interested in whether there are counterexamples which aren't "locally nice".

Reid Barton (Oct 16 2021 at 18:09):

There aren't: singular homology and the fundamental groupoid are both unchanged under replacing your space by a weakly equivalent CW complex.

John Baez (Oct 18 2021 at 04:15):

My only confusion was whether the fundamental groupoid of a topological space is the coproduct of the fundamental groupoids of its path components, and similarly for the homology groups of this space. But I think it's true, even when the space itself isn't the coproduct of its path components. (Think of $\mathbb{Q}$ with its usual topology, for example.)

John Baez (Oct 18 2021 at 04:19):

Given this, the fundamental groupoid $\Pi_1(X)$ of a space $X$ always determines $\mathrm{H}_1(X)$. $\Pi_1(X)$ is a coproduct of groupoids that are equivalent to groups, one for each path component. You can abelianize each one of these groups and take the coproduct of the resulting abelian groups, and that's $\mathrm{H}_1(X)$.

John Baez (Oct 18 2021 at 04:21):

But I recommend avoiding spaces that aren't "locally nice" unless you want to dabble in exotic delights. For example, for a path-connected space $X$ that's not locally nice, the fundamental group $\pi_1(X)$ has a natural topology which may not be the discrete topology, and you can then go ahead and define $\pi_1(\pi_1(X))$. This is NOT what you should be thinking about if you're learning algebraic topology! :skull_and_crossbones:

John Baez (Oct 18 2021 at 04:24):

Of course, telling students not to study something makes them curious about it, so lovers of spaces that aren't locally nice can sate their unhealthy appetites here:

• Jeremy Brazas, The fundamental group as a topological group.

Reid Barton (Oct 18 2021 at 08:59):

John Baez said:

My only confusion was whether the fundamental groupoid of a topological space is the coproduct of the fundamental groupoids of its path components, and similarly for the homology groups of this space. But I think it's true, even when the space itself isn't the coproduct of its path components. (Think of $\mathbb{Q}$ with its usual topology, for example.)

That's right, and it's because for any simplex $\Delta^n$ and space $X$, the set (not space, as you point out!) of continuous maps from $\Delta^n$ into $X$ is the disjoint union of the sets of maps from $\Delta^n$ into the path-connected components of $X$. For the fundamental groupoid you only need the cases $n=0$, $1$, $2$.

More broadly, the "reason" that the fundamental groupoid and singular homology of $X$ only depend on the weak homotopy type of $X$ is that they both depend only on the sets of maps of simplices into $X$ and their face and degeneracy relationships, that is, they depend only on the simplicial set $\mathrm{Sing}\,X$ (and in a homotopy-invariant way--this isn't meant to be a complete argument).