In my post homotopically Hausdorff spaces (Part I), I wrote about the property which describes the existence of loops that can be deformed into arbitrarily small neighborhoods but which are not actually null-homotopic, i.e. can’t be deformed all the way back to that point. In this post, we’ll offer up different viewpoints on this property based on an approach taken from a recent paper:
Jeremy Brazas, Hanspeter Fischer, Test map characterizations of local properties of fundamental groups. To Appear in the Journal of Topology and Analysis. Click here for the arXiv paper.
In particular, we’ll discuss the following characterization of the homotopically Hausdorff property.
Theorem 1: For a first countable space , the following are equivalent:
- is homotopically Hausdorff,
- Every map every map from the harmonic archipelago induces the trivial homomorphism on .
- For every map such that for every , then .
Condition 2. in the theorem clarifies the notion that the harmonic archipelago really is the prototypical non-homotopically Hausdorff space since maps from it detect the same failure.
Condition 3. suggests that the homotopically Hausdorff property should be thought of as a closure property of the trivial subgroup.
Just as a reminder, here is the definition of the property we’re focusing on.
Definition: Given a path-connected space and basepoint , we considered the subgroup
where is a path starting at and is an open neighborhood of . We say is homotopically Hausdorff if for every path with , we have where denotes the set of all open sets in containing .
Infinite Concatenations of Paths and Homotopies
We’re going to use the Hawaiian earring as a kind of “test space” so let’s recall its construction. If is the circle of radius centered at , then is the usual Hawaiian earring space with basepoint . Let be the loop which traverses once in the counterclockwise direction. The homotopy classes , freely generate the subgroup .
Definition: A sequence of paths is null at a point if for every open neighborhood of , there is an such that for all , equivalently if converges to the constant path at .
Given a sequence of paths satisfying and which is null at , we may define the infinite concatenation to be the path to be the path defined to be on the interval and .
Sometimes, we may expand the notation as
Example: An important infinite concatenation for this post will be the loop
that winds once around each hoop of .
Warning: Notice here that we’re only considering infinite concatenations or “products” of loops – not homotopy classes of loops. Indeed, this operation is well defined for paths but the notion of “infinite product” of homotopy classes is not well defined in all fundamental groups.
Remark 2: What we are allowed to do with these infinite products is reparameterize them. This allows us to treat them like infinite sums and products in Calculus:
in the fundamental groupoid for any
Proposition 3: A sequence of loops based at is null at if and only if there is a map such that .
Proof. The key here is to observe that a function is continuous if and only if is continuous for each and if for every neighborhood of , maps all but finitely many of the circles into . The latter condition is clearly equivalent to the sequence of loops being null at .
Lemma 4: Let be a null sequence of paths in such that for all . Then the infinite concatenation
is a null-homotopic loop.
Proof. Recall that for any path with , we can contract to the constant path at by a homotopy contracting the loop back along its own image.
We construct a null-homotopy of by creating an infinite concatenation of the individual contractions of . It will looks something like this:
where is defined as on . You can use the pasting lemma to verify continuity at every point except those on the right vertical wall. To verify continuity of on the right edge recall that is null at . This means that given any open neighborhood of , there is an such that has image in for all . But has image in for each . Therefore, . We conclude that there is an open set containing which is mapped into by .
This verifies the continuity of .
Functorality of the Obstruction
Lemma 5: Let be a map, be a path, and be an open neighborhood of . Then .
Proof. If is a loop in , then is a generic element of . Since has image in , it follows that
Corollary 6: Let be a map, be a path from to , and set and . Then
as subgroups of .
Proof. Suppose for all and pick any . Then and by Lemma 5, we have . Thus .
Interpretation: Corollary 6 can be thought of as saying that the “obstruction” subgroups which detect the failure of the homotopically Hausdorff property are functorial since continuous maps induce homomorphisms that always map obstruction subgroups into obstruction subgroups.
Proof of Theorem 1
In the harmonic archipelago every loop based at may be continuously deformed within an arbitrary neighborhood of the basepoint . Thus if is the constant path at the wild point , then for every open neighborhood of . Hence .
Now suppose is a map such that the induced homomorphism is not the trivial homomorphism, then by Corollary 6, we have
Now is a constant path at such that , which means cannot be homotopically Hausdorff.
Note: this direction of Theorem 1 doesn’t actually require first countability.
The main fact that we need is that if is the inclusion map, then in . I give a simple explicit proof of this fact in this post. A quick reminder of how this is done: compactness of the unit disk means that a null-homotopy of can only intersect finitely many of the hills of . So if is the interior of the n-th hill, then is null-homotopic in for some but this is impossible since winds around the circle retracts , in a non-trivial way.
We prove the contrapositive. Suppose 3. does not hold. Then there exists a map such that for all and .
For each , the loop is null-homotopic loop in and therefore extends to a map on the unit disk. Since each of the holes in can be extended to “large” disks, extends to a map such that . So we have and . Therefore is not the trivial homomorphism.
Note: This part of the proof does not require first countability either.
For this direction of Theorem 1, we do need the assumption that is homotopically Hausdorff. We prove the contrapositive.
Suppose that is first countable and that fails to be homotopically Hausdorff. Then there exists a path from to and a loop based at such that
Notice that if we conjugate by , then we see that .
Let be a countable neighborhood base at . Then
Hence, for each , there is a loop based at such that . In particular, in for each .
By construction, the sequence of loops is null at . Therefore, the sequence of loops is also null at . Using Proposition 3, we put this sequence together to construct a continuous function defined by .
Notice that for each .
Now, we use a “telescoping product” to prove that .
where the last equality is allowed according to Remark 2. But by Lemma 4.
Therefore . This completes the proof!
There are a few things I hope you can take away from this post. Ultimately, we have taken this important obstruction and teased it apart into different viewpoints. To me, that makes good mathematics.
- Because the abelianization of is isomorphic to (a highly non-trivial fact), Condition 2. in Theorem 1 looks like a non-abelian generalization of the cotorsion free property defined for abelian groups. In fact, a direct Corollary of 1. 2. is that if is abelian and cotorsion free, then is homotopically Hausdorff.
- Condition 3 in Theorem 1 looks like a closure property – something like subgroup-closure under infinite products…we make this more precise and apply the idea widely in the paper I shamelessly plug at the top of the post.