Celebrating the Career and Contributions of Katsuya Eda: Master of the Hawaiian Earring

Last week, I had the pleasure of speaking at the Arches Topology Conference in Moab, Utah. The conference was in honor of the career and contributions of Katsuya Eda, who retired not too long ago.

K. Eda considers himself a logician and set theorist, however, most of his research has been related to topology, what we now consider to be wild algebraic topology. He developed many of the technologies and proved many of the foundational theorems being used today in wild algebraic topology.

Katsuya Eda giving a talk at the 2018 Arches Topology Conference

Katsuya Eda at the 2018 Arches Topology Conference

Although a few phenomena of wild algebraic topology had appeared in a disjointed fashion in literature in the 1950’s and 60’s (e.g. H.B. Griffiths Twin Cone, Curtis and Fort’s asphericity of 1-dimensional spaces and Barratt and Milnor’s Anomalous Singular Homology) and shape theory was fashionable in the 70’s, the first systematic study of wild algebraic topology began with Eda’s series of papers on the Hawaiian earring and one-dimensional spaces in the 1990’s. In many ways, Katsuya Eda is the pioneer of what is now the developing field of wild algebraic topology.

While Eda was not the first to describe the fundamental group of the Hawaiian earring, he certainly was the first to offer new ways of thinking about it so that it could be used in serious ways.

Among Eda’s many contributions are:

  1. The homotopy classification of one-dimensional Peano continua: one-dimensional Peano continua X and Y are homotopy equivalent if \pi_1(X,x_0)\cong\pi_1(Y,y_0). In particular, the subspace of wild points is a homotopy invariant.
    • This includes the Hawaiian earring, Sierpinski Carpet, Menger Curve, etc. All are determined up to homotopy by their fundamental group. Absolutely remarkable!
    • The proof is highly technical and is completed in a series of papers.
  2. Automatic Continuity: In the process of 1. Eda proved that any self-homomorphism \pi_1(\mathbb{H},b_0)\to\pi_1(\mathbb{H},b_0) of the Hawaiian earring group is induced (up to conjugation) by a continuous function \mathbb{H}\to\mathbb{H}. His No-Head-Tail Lemma needed to prove this is an example of truly great mathematics. He later extended this result to show that all homomorphisms between fundamental groups of one-dimensional Peano continua are induced (up to conjugation) by a continuous function.
  3. Singular Homology: Computing the first singular homology group H_1(\mathbb{H}) of the Hawaiian earring \mathbb{H}. Eda did this by showing that H_1(\mathbb{H}) splits as a direct sum of the first Cech homology group \mathbb{Z}^{\mathbb{N}} and an algebraically compact group abstractly isomorphic to \mathbb{Z}^{\mathbb{N}}/\oplus_{\mathbb{N}}\mathbb{Z}. Eda later showed that H_1 of any wild 1-dimensional Peano continuum has the same singular homology. This strange phenomenon is very much a result of the decomposition theory of infinite abelian groups.
  4. Introduction of non-commutatively slender or n-slender groups, which is an analogue of “slenderness” for non-abelian groups. His seminal paper on Free \sigma-products and noncommutatively slender groups  introduces the notion of transfinite word, i.e. a word with letters indexed by a countable linear order so that no single letter appears more than finitely many times. Although the term “transfinite word” was later later by Jim Cannon and Greg Conner, Eda used this idea to allow the application of linear order theory to study the Hawaiian earring group and to use it as a non-commutative analogue of the Baer-Specker group.
  5. Higher homotopy groups: Proving \pi_n(\mathbb{H}_n)\cong \mathbb{Z}^{\mathbb{N}} where \mathbb{H}_n is the n-dimensional Hawaiian earring.
  6. Much more: A huge list of other results relating to coverings/overlays of topological groups, the relationship between singular and Cech (co)homology, combinatorial aspects of the Hawaiian earring group and other infinitely generated non-abelian groups, and the introduction of many new examples and phenomena.

Katsuya Eda has a unique way of thinking about mathematics and I personally have been inspired by his remarkable work and ingenuity.

Jeremy Brazas and Katsuya Eda at the 2018 Arches Topology Conference

Posted in Algebraic Topology, Conferences, Fundamental group, Singular homology | Leave a comment

Homotopically Hausdorff Spaces II

In my first post on homotopically Hausdorff spaces, 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. Preprint. 2017. 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 X, the following are equivalent:

  1. X is homotopically Hausdorff,
  2. Every map every map f:\mathbb{HA}\to X from the harmonic archipelago induces the trivial homomorphism f_{\#}:\pi_1(\mathbb{HA},b_0)\to\pi_1(X,f(b_0)) on \pi_1.
  3. For every map g:\mathbb{H}\to X such that g_{\#}([\ell_n])=1\in \pi_1(X,f(b_0)) for every n\geq 1, then f_{\#}([\ell_1\cdot\ell_2\cdots])=1.

The harmonic archipelago \mathbb{HA}. Notice the Hawaiian earring \mathbb{H} is a subspace and every loop based at the one wild point b_0 may be deformed over finitely many hills to lie within an arbitrary neighborhood of b_0.

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 X and basepoint x_0, we considered the subgroup

\pi(\alpha,U)=\{[\alpha\cdot\gamma\cdot\alpha^{-}]|\gamma([0,1])\subseteq U\}\leq\pi_1(X,x_0)

where \alpha is a path starting at x_0 and U is an open neighborhood of \alpha(1). We say X is homotopically Hausdorff if for every path \alpha:[0,1]\to X with \alpha(0)=x_0, we have \bigcap_{U\in\mathcal{T}_{\alpha(1)}}\pi(\alpha,U)=1 where \mathcal{T}_x denotes the set of all open sets in X containing x.

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 C_n is the circle of radius \frac{1}{n} centered at \left(\frac{1}{n},0\right), then \mathbb{H}=\bigcup_{n\geq 1}C_n is the usual Hawaiian earring space with basepoint b_0=(1,0). Let \ell_n:[0,1]\to\mathbb{H} be the loop which traverses C_n once in the counterclockwise direction. The homotopy classes [\ell_n], n\geq 1 freely generate the subgroup F=\langle [\ell_n]|n\geq 1\rangle.

Definition: A sequence \{\alpha_n\}_{n\geq 1} of paths \alpha_n:[0,1]\to X is null at a point x\in X if for every open neighborhood U of x, there is an N such that \alpha_n([0,1])\subseteq U for all n\geq 1, equivalently if \{\alpha_n\}_{n\geq 1} converges to the constant path at x.

Given a sequence of paths \{\alpha_n\}_{n\geq 1} satisfying \alpha_n(1)=\alpha_{n+1}(0) and which is null at x\in X, we may define the infinite concatenation to be the path \alpha=\prod_{n=1}^{\infty}\alpha_n to be the path defined to be \alpha_n on the interval \left[1-\frac{1}{n},1-\frac{1}{n+1}\right] and \alpha(1)=x.

Sometimes, we may expand the notation as


Infinite Concatenation

Example: An important infinite concatenation for this post will be the loop


that winds once around each hoop C_n of \mathbb{H}.

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:

\prod_{n=1}^{\infty}\alpha_n\simeq \left(\prod_{n=1}^{m}\alpha_n\right)\cdot\left(\prod_{n=m+1}^{\infty}\alpha_n\right)

and so

\left[\prod_{n=1}^{\infty}\alpha_n\right]= \left[\prod_{n=1}^{m}\alpha_n\right]\left[\prod_{n=m+1}^{\infty}\alpha_n\right]

in the fundamental groupoid for any m

Proposition 3: A sequence of loops \{\alpha\}_{n\geq 1} based at x is null at x if and only if there is a map f:(\mathbb{H},b_0)\to (X,x) such that f\circ\ell_n=\alpha_n.

Proof. The key here is to observe that a function f:(\mathbb{H},b_0)\to (X,x) is continuous if and only if f|_{C_n} is continuous for each n and if for every neighborhood U of x, f maps all but finitely many of the circles C_n into U. The latter condition is clearly equivalent to the sequence of loops f\circ\ell_n being null at x. \square

Lemma 4: Let \{\alpha\}_{n\geq 1} be a null sequence of paths in X such that \alpha_n(0)=x for all n. Then the infinite concatenation


is a null-homotopic loop.

Proof. Recall that for any path \alpha with \alpha(0)=x, we can contract \alpha\cdot\alpha^{-} to the constant path at x by a homotopy contracting the loop back along its own image.

At height t, the homotopy h(s,t) pictured is first \alpha|_{[0,t]}, constant in the black region, and then the reverse of \alpha|_{[0,t]}.

We construct a null-homotopy H:[0,1]\times[0,1]\to X of \prod_{n=1}^{\infty}(\alpha_{n}\cdot\alpha_{n}^{-}) by creating an infinite concatenation of the individual contractions h_n of \alpha_n\cdot\alpha_{n}^{-}. It will looks something like this:

Infinite concatenation of null-homotopies.

where H is defined as h_n on \left[1-\frac{1}{n},1-\frac{1}{n+1}\right]\times [0,1]. You can use the pasting lemma to verify continuity at every point except those on the right vertical wall. To verify continuity of H on the right edge recall that \{\alpha\}_{n\geq 1} is null at x. This means that given any open neighborhood U of x, there is an N such that \alpha_n has image in U for all n\geq N. But h_n has image in \alpha_n([0,1]) for each n. Therefore, H\left(\left[1-\frac{1}{N},1\right]\times [0,1]\right)\subseteq U. We conclude that there is an open set V containing \{1\}\times [0,1] which is mapped into U by h.

This verifies the continuity of H. \square

Functorality of the Obstruction

Lemma 5: Let f:X\to Y be a map, \alpha:[0,1]\to X be a path, and U be an open neighborhood of f(\alpha(1)). Then f_{\#}(\pi(\alpha,f^{-1}(U))\leq\pi(f\circ \alpha,U).

Proof. If \gamma is a loop in f^{-1}(U), then [\alpha\cdot\gamma\cdot\alpha^{-}] is a generic element of \pi(\alpha,f^{-1}(U)). Since f\circ \gamma has image in U, it follows that

f_{\#}([\alpha\cdot\gamma\cdot\alpha^{-}])=[(f\circ\alpha)\cdot(f\circ\gamma)\cdot(f\circ \alpha)^{-}]\in\pi(f\circ \alpha,U). \square

Corollary 6: Let f:X\to Y be a map, \alpha:[0,1]\to X be a path from x_0 to x, and set f(x_0)=y_0 and f(x)=y. Then


as subgroups of \pi_1(Y,y_0).

Proof. Suppose g\in\pi(\alpha,V) for all V\in\mathcal{T}_x and pick any U\in\mathcal{T}_y. Then g\in\pi(\alpha,f^{-1}(U)) and by Lemma 5, we have f_{\#}(g)\in\pi(f\circ\alpha,U). Thus f_{\#}(g)\in\bigcap_{U\in\mathcal{T}_y}\pi(f\circ\alpha,U). \square

Interpretation: Corollary 6 can be thought of as saying that the “obstruction” subgroups \bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U) 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. \Rightarrow 2.)

In the harmonic archipelago \mathbb{HA} every loop based at b_0 may be continuously deformed within an arbitrary neighborhood U of the basepoint b_0. Thus if \alpha:[0,1]\to\mathbb{HA} is the constant path at the wild point b_0, then \pi(\alpha,U)=\pi_1(\mathbb{HA},b_0) for every open neighborhood U of b_0. Hence \bigcap_{U\in\mathcal{T}_{b_0}}\pi(\alpha,U)=\pi_1(\mathbb{HA},b_0).

Now suppose f:\mathbb{HA}\to X is a map such that the induced homomorphism f_{\#}:\pi_1(\mathbb{HA},b_0)\to\pi_1(X,f(b_0)) is not the trivial homomorphism, then by Corollary 6, we have

1\neq f_{\#}(\pi_1(\mathbb{HA},b_0))=f_{\#}\left(\bigcap_{V\in\mathcal{T}_{b_0}}\pi(\alpha,V)\right)\leq\bigcap_{U\in\mathcal{T}_{f(b_0)}}\pi(f\circ\alpha,U).

Now f\circ \alpha is a constant path at f(b_0) such that \bigcap_{U\in\mathcal{T}_{f(b_0)}}\pi(f\circ\alpha,U)\neq 1, which means X cannot be homotopically Hausdorff.

Note: this direction of Theorem 1 doesn’t actually require first countability.

(2. \Rightarrow 3.)

The main fact that we need is that if j:\mathbb{H}\to\mathbb{HA} is the inclusion map, then j_{\#}([\ell_{\infty}])\neq 1 in \pi_1(\mathbb{HA},b_0). 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 j\circ\ell_{\infty} can only intersect finitely many of the hills of \mathbb{HA}. So if E_n is the interior of the n-th hill, then j\circ\ell_{\infty} is null-homotopic in \mathbb{H}\cup\bigcup_{1\leq n\leq N}E_n for some N but this is impossible since \ell_{\infty} winds around the circle retracts C_n, n>N in a non-trivial way.

We prove the contrapositive. Suppose 3. does not hold. Then there exists a map g:\mathbb{H}\to X such that g_{\#}([\ell_n])=1 for all n\geq 1 and g_{\#}([\ell_{\infty}])\neq 1.

For each n\geq 1, the loop g\circ(\ell_n\cdot\ell_{n+1}^{-}):S^1\to X is null-homotopic loop in X and therefore extends to a map on the unit disk. Since each of the holes in \mathbb{H} can be extended to “large” disks, g extends to a map f:\mathbb{HA}\to X such that f\circ j=g. So we have [j\circ\ell_{\infty}]\neq 1 and f_{\#}([j\circ\ell_{\infty}])=g_{\#}([\ell_{\infty}])\neq 1. Therefore f_{\#}:\pi_1(\mathbb{HA},b_0)\to\pi_1(X,f(b_0)) is not the trivial homomorphism.

Note: This part of the proof does not require first countability either.

(3. \Rightarrow 2.)

For this direction of Theorem 1, we do need the assumption that X is homotopically Hausdorff. We prove the contrapositive.

Suppose that X is first countable and that X fails to be homotopically Hausdorff.  Then there exists a path \alpha:[0,1]\to X from x_0 to x and a loop \gamma based at x such that


Notice that if we conjugate by [\alpha]^{-1}, then we see that 1\neq [\gamma]\in\pi_1(X,x).

Let U_1\supset U_2\supset U_3\supset... be a countable neighborhood base at x. Then

1\neq[\alpha\cdot\gamma\cdot\alpha^{-}]\in\bigcap_{n\geq 1}\pi(\alpha,U_n)\leq\pi_1(X,x_0).

Hence, for each n\geq 1, there is a loop \gamma_n:[0,1]\to U_n based at x such that [\alpha\cdot\gamma\cdot\alpha^{-}]=[\alpha\cdot\gamma_n\cdot\alpha^{-}]. In particular, [\gamma]=[\gamma_n]  in \pi_1(X,x) for each n\geq 1.

By construction, the sequence of loops \{\gamma_n\}_{n\geq 1} is null at x. Therefore, the sequence \{\gamma_n\cdot\gamma_{n+1}^{-}\}_{n\geq 1} of loops is also null at x. Using Proposition 3, we put this sequence together to construct a continuous function g:(\mathbb{H},b_0)\to (X,x) defined by g\circ\ell_n=\gamma_n\cdot\gamma_{n+1}^{-}.

Notice that g_{\#}([\ell_n])=[\gamma_n][\gamma_{n+1}]^{-1}=[\gamma][\gamma]^{-1}=1 for each n\geq 1.

Now, we use a “telescoping product” to prove that g_{\#}([\ell_{\infty}])\neq 1.

We have


where the last equality is allowed according to Remark 2. But \left[\prod_{n=2}^{\infty}(\gamma_n\cdot\gamma_{n}^{-})\right]=1 by Lemma 4.

Therefore g_{\#}([\ell_{\infty}])=[\gamma](1)=[\gamma]\neq 1. This completes the proof! \square


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.

  1. Because the abelianization of \pi_1(\mathbb{HA},b_0) is isomorphic to \prod_{n=1}^{\infty}\mathbb{Z}/\bigoplus_{n=1}^{\infty}\mathbb{Z} (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. \Leftrightarrow 2. is that if \pi_1(X,x_0) is abelian and cotorsion free, then X is homotopically Hausdorff.
  2. 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.


Posted in Algebraic Topology, Fundamental group, Group homomorphisms, harmonic archipelago, Hawaiian earring | Tagged , , | Leave a comment

Homotopically Hausdorff Spaces I

In previous posts, I wrote about the harmonic archipelago \mathbb{HA} (see also here and here):


Harmonic Archipelago

as well as the Griffiths Twin Cone \mathbb{G}.


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 \mathbb{HA} and \mathbb{G} 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.

  1. 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.”
  2. A. Zastrow, Generalized \pi_1-determined covering spaces, Unpublished notes. 2002.
    • homotopically Hausdorff is equivalent to what the author calls “weak \pi_1-continuity”
  3. 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: X will be a path-connected Hausdorff space and x_0\in X will be a basepoint.

Definition: Given a path \alpha:[0,1]\to X with \alpha(0)=x_0 and an open neighborhood U of \alpha(1), let

\displaystyle \pi(\alpha,U)=\{[\alpha\cdot\gamma\cdot\alpha^{-}]\in \pi_1(X,x_0)|\gamma([0,1])\subseteq U\}.

We can describe \pi(\alpha,U) as the subgroup of \pi_1(X,x_0) consisting of “lolipop” loops that go out on the fixed path \alpha, move around in U, and then go back along the reverse of \alpha.


An element of \pi(\alpha,U)

Definition: Given x\in X and a neighborhood U of x, let

\displaystyle \pi(x,U)=\langle \pi(\alpha,U)|\alpha(1)\in U\rangle\leq \pi_1(X,x_0)

to be the subgroup generated by all \pi(\alpha,U) where \alpha ranges over all paths from x_0 to x. This means a generic element of \pi(x,U) is of the form \displaystyle\prod_{i=1}^{n}[\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}] where all the loops \gamma_{i} have image in U.


An element of \pi(x,U)

Observation: For any loop \beta based at x_0, [\beta]\pi(\alpha,U)[\beta]^{-1}]=\pi(\beta\cdot\alpha,U).

Observation: \pi(\alpha,U)\leq\pi(x,U) whenever \alpha(1)=x.

Notational Remark: The notation for \pi(\alpha,U) and \pi(x,U) is influenced by E.H. Spanier’s excellent Algebraic Topology textbook.

Proposition: For any x\in X, the subgroup \pi(x,U)\trianglelefteq\pi_1(X,x_0) is a normal subgroup of \pi_1(X,x_0).

Proof. If [\beta]\in\pi_1(X,x_0) and \pi(x,U) is of the form \displaystyle \prod_{i=1}^{n}[\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}] is a generic element of \pi(x,U) where each loop \gamma_i has image in U, then


which is an element of \pi(x,U). \square

Defining the homotopically Hausdorff property

Definition: If x\in X, let \mathcal{T}_x be the set of open neighborhoods in X containing x. We say a space X is homotopically Hausdorff at x\in X if for every path \alpha from x_0 to x, we have

\displaystyle \bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)=1

where 1 denotes the trivial subgroup. We say X is homotopically Hausdorff if X is homotopically Hausdorff at all of its points.

Intuition: Recall that \gamma is a null-homotopic loop based at x if and only if \alpha\cdot\gamma\cdot\alpha^{-} is a null-homotopic loop based at \alpha(0). So a space X fails to be homotopically Hausdorff if there is a point x\in X and a non-null homotopic loop \gamma based at x which may be homotoped within an arbitrary neighborhood of x.

So what happens if X is not homotopically Hausdorff? It means that there is a point x\in X, a path \alpha from x_0 to x, and a non-null-homotopic loop \gamma based at x such that the class [\alpha\cdot\gamma\cdot\alpha^{-}] can be represented by \alpha\cdot\gamma\cdot\alpha^{-} where \gamma may be chosen to have image in an arbitrary neighborhood of x.

The conjugating loops \alpha are simply the way of describing this property as ranging over all points x\in X while still using a fixed basepoint x_0. 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 X is semilocally simply connected at x\in X if there exists an open neighborhood latex V of x such that the inclusion i:V\to X induces the trivial homomorphism i_{\#}:\pi_1(V,x)\to\pi_1(X,x), i.e. if every loop in V based at x is null-homotopic in X by a (possibly large) homotopy in X. We say X is semilocally simply connected if it is semilocally simply connected at all of its points.

Observation: A space X is semilocally simply connected at x if and only if there is an open neighborhood V of x such that \pi(x,V)=1. In this case, for every \alpha with \alpha(1)=x, we have

\displaystyle\bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U)\leq \bigcap_{U\in\mathcal{T}_x}\pi(x,U)\leq\pi(x,V)=1

Hence, semilocally simply connected \Rightarrow homotopically Hausdorff.

In fact, there are many important intermediate properties to explore…perhaps later.

Corollary: If X admits a simply connected covering space, then X is homotopically Hausdorff.

Proof. It’s a nice exercise to show that every such space X 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 \widetilde{X} be the set of homotopy (rel. endpoint) classes of paths in X starting at x_0. 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

B([\alpha],U)=\{[\alpha\cdot\gamma]|\gamma([0,1])\subseteq U\}

where U is an open neighborhood of \alpha(1).


An element in B([\alpha],U). Such an element can only differ from [\alpha] at its terminal end but there it may be a complicated extension within U.

It’s a nice exercise in covering space theory to show that these sets form a basis for a topology on \widetilde{X}. 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:

  1. X is homotopically Hausdorff,
  2. \widetilde{X} is Hausdorff (T_2),
  3. \widetilde{X} is T_1,
  4. \widetilde{X} is T_0.

Proof. 1. \Rightarrow 2. Suppose \widetilde{X} is not Hausdorff. Then there are paths \alpha,\beta:[0,1]\to X starting at x_0 such that [\alpha] and [\beta] are distinct homotopy classes which cannot be separated by disjoint basic open sets in \widetilde{X}. Since X is assumed to be Hausdorff, if \alpha(1)\neq \beta(1), then we could separate [\alpha] and [\beta] in \widetilde{X} simply by taking B([\alpha],U), B([\beta],V) where U\cap V=\emptyset. Thus we must have x=\alpha(1)=\beta(1). Notice [\alpha\cdot\beta^{-}]\neq 1 in \pi_1(X,x_0). Let U be an arbitrary open neighborhood of x. By assumption, B([\alpha],U)\cap B([\beta],U)\neq \emptyset so we have [\alpha\cdot\delta]=[\beta\cdot\epsilon] for paths \delta,\epsilon in U. Since [\beta]=[\alpha\cdot\delta\cdot\epsilon^{-}], we have

[\alpha\cdot\beta^{-}]=[\beta\cdot\alpha^{-}]^{-1}= ([\alpha\cdot\delta\cdot\epsilon^{-}\cdot\alpha^{-}])^{-1}=[\alpha\cdot\epsilon\cdot\delta^{-}\cdot\alpha^{-}].

This means [\alpha\cdot\beta^{-}]\in \pi(\alpha,U). Since U was arbitrary, it follows that 1\neq [\alpha\cdot\beta^{-}]\in \bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U). Thus X is not homotopically Hausdorff.

2. \Rightarrow 3. \Rightarrow 4. are basic facts of separation axioms.

4. \Rightarrow 1. Suppose X is not homotopically Hausdorff. Then there is a path \alpha from x_0 to \alpha(1)=x and an element 1\neq [\beta]\in\bigcap_{U\in\mathcal{T}_x}\pi(\alpha,U). Since [\beta]\neq 1, we have [\alpha]\neq[\beta\cdot\alpha]. Let B([\alpha],U) be a basic neighborhood of [\alpha] in \widetilde{X}. By choice of [\beta], we may write [\beta]=[\alpha\cdot\gamma\cdot\alpha^{-}] where \gamma is a loop in U. Thus [\beta\cdot\alpha]=[\alpha\cdot\gamma]\in B([\alpha],U). Since [\beta\cdot\alpha] lies in every neighborhood of [\alpha], the two are topologically indistinguishable in \widetilde{X}, i.e. \widetilde{X} is not T_0. \square

Posted in Covering Space Theory, Fundamental group, Griffiths twin cone, harmonic archipelago, Homotopy theory, Uncategorized | Tagged , , , | 1 Comment