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
.

The harmonic archipelago . Notice the Hawaiian earring
is a subspace and every loop based at the one wild point
may be deformed over finitely many hills to lie within an arbitrary neighborhood of
.
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:
and so
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.

At height , the homotopy
pictured is first
, constant in the black region, and then the reverse of
.
We construct a null-homotopy
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
(1.
2.)
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.
(2.
3.)
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.
(3.
2.)
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 .
We have
where the last equality is allowed according to Remark 2. But by Lemma 4.
Therefore . This completes the proof!
Takeaway
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.
Pingback: Homotopically Hausdorff Spaces (Part I) | Wild Topology
Pingback: Homotopically Reduced Paths: Part I | Wild Topology