In previous posts, I wrote about the harmonic archipelago (see also here and here):
as well as the Griffiths Twin Cone .
One special feature of these 2-dimensional spaces is that any loop either of these spaces can be deformed to lie within an arbitrarily small neighborhood of the basepoint. In fact, these are the prototypical spaces for this pathology. The existence of non-trivial loops that can be deformed into arbitrarily small neighborhoods can be thought of as an obstruction to applying covering space and shape theoretic techniques to understand the fundamental group. It turns out there is a named property that gets to the heart of this obstruction.
It’s actually an open question whether or not and have isomorphic fundamental groups! They are known to have isomorphic first singular homology groups. The difficulty of this question stems from the fact that they are not homotopically Hausdorff.
This property appeared in two sets of unpublished notes before it appeared in a published paper with the now-standard name.
- W.A. Bogley, A.J. Sieradski, Universal path spaces, Unpublished notes. 1998
- homotopically Hausdorff is equivalent to the author’s notation of the trivial subgroup being “totally closed.”
- A. Zastrow, Generalized -determined covering spaces, Unpublished notes. 2002.
- homotopically Hausdorff is equivalent to what the author calls “weak -continuity”
- J.W. Cannon, G.R. Conner, On the fundamental groups of one-dimensional spaces, Topology Appl. 153 (2006) pp. 2648-2672.
- the introduction of the term “homotopically Hausdorff”
Since it’s introduction, this fundamental property has appeared in a large number of publications.
Some Important Subgroups
Assumptions: will be a path-connected Hausdorff space and will be a basepoint.
Definition: Given a path with and an open neighborhood of , let
We can describe as the subgroup of consisting of “lolipop” loops that go out on the fixed path , move around in , and then go back along the reverse of .
Definition: Given and a neighborhood of , let
to be the subgroup generated by all where ranges over all paths from to . This means a generic element of is of the form where all the loops have image in .
Observation: For any loop based at , .
Observation: whenever .
Notational Remark: The notation for and is influenced by E.H. Spanier’s excellent Algebraic Topology textbook.
Proposition: For any , the subgroup is a normal subgroup of .
Proof. If and is of the form is a generic element of where each loop has image in , then
which is an element of .
Defining the homotopically Hausdorff property
Definition: If , let be the set of open neighborhoods in containing . We say a space is homotopically Hausdorff at if for every path from to , we have
where denotes the trivial subgroup. We say is homotopically Hausdorff if is homotopically Hausdorff at all of its points.
Intuition: Recall that is a null-homotopic loop based at if and only if is a null-homotopic loop based at . So a space fails to be homotopically Hausdorff if there is a point and a non-null homotopic loop based at which may be homotoped within an arbitrary neighborhood of .
So what happens if is not homotopically Hausdorff? It means that there is a point , a path from to , and a non-null-homotopic loop based at such that the class can be represented by where may be chosen to have image in an arbitrary neighborhood of .
The conjugating loops are simply the way of describing this property as ranging over all points while still using a fixed basepoint . We could have defined it without them, but there is also a subgroup-relative version of the homotopically Hausdorff property for which these conjugating paths are necessary.
Indeed, the harmonic archipelago and Griffiths twin cone spaces are not homotopically Hausdorff. It turns out that many spaces are homotopically Hausdorff though. Obvious ones include spaces that admit a simply connected covering space (including manifolds, CW-complexes, etc.). Note the following doesn’t actually require local path connectivity.
Definition: We say is semilocally simply connected at if there exists an open neighborhood latex of such that the inclusion induces the trivial homomorphism , i.e. if every loop in based at is null-homotopic in by a (possibly large) homotopy in . We say is semilocally simply connected if it is semilocally simply connected at all of its points.
Observation: A space is semilocally simply connected at if and only if there is an open neighborhood of such that . In this case, for every with , we have
Hence, semilocally simply connected homotopically Hausdorff.
In fact, there are many important intermediate properties to explore…perhaps later.
Corollary: If admits a simply connected covering space, then is homotopically Hausdorff.
Proof. It’s a nice exercise to show that every such space is semilocally simply connected. This does not require any local path connectivity assumptions.
The homotopically Hausdorff property is actually much weaker than the semilocally simply connected property but it is a necessary property to have in order to applying generalized covering space theories. Many, many other spaces without traditional universal covers are also homotopically Hausdorff, including all 1-dimensional spaces (e.g. Hawaiian earring, Menger Sponge, etc.), all planar spaces, and many (but not all) 2-dimensional spaces, among others.
Why is “Hausdorff” in the name?
The name suggests that there is some kind of separation-like axiom here. Indeed, if a space is homotopically Hausdorff, then we can separate homotopy classes in a certain topological sense.
The standard universal covering space construction: Let be the set of homotopy (rel. endpoint) classes of paths in starting at . We give this set the standard topology which is sometimes called “whisker topology”. A basic open set generating the standard topology is of the form
where is an open neighborhood of .
It’s a nice exercise in covering space theory to show that these sets form a basis for a topology on . Recent work of my own actually shows that this topology is the only topology of generalized covering space theories for locally path-connected spaces – any other notion of generalized covering space based on homotopy-lifting must be equivalent to it. Here is the reasoning for the name.
Theorem: The following are equivalent:
- is homotopically Hausdorff,
- is Hausdorff (),
- is ,
- is .
Proof. 1. 2. Suppose is not Hausdorff. Then there are paths starting at such that and are distinct homotopy classes which cannot be separated by disjoint basic open sets in . Since is assumed to be Hausdorff, if , then we could separate and in simply by taking , where . Thus we must have . Notice in . Let be an arbitrary open neighborhood of . By assumption, so we have for paths in . Since , we have
This means . Since was arbitrary, it follows that . Thus is not homotopically Hausdorff.
2. 3. 4. are basic facts of separation axioms.
4. 1. Suppose is not homotopically Hausdorff. Then there is a path from to and an element . Since , we have . Let be a basic neighborhood of in . By choice of , we may write where is a loop in . Thus . Since lies in every neighborhood of , the two are topologically indistinguishable in , i.e. is not .
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