Homotopically Hausdorff Spaces (Part 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.

History

This property appeared in two sets of unpublished notes before it appeared in a published paper with the now-standard name.

W.A. Bogley, A.J. Sieradski, Universal path spaces, Unpublished notes. 1998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A. Zastrow, Generalized $\pi_1$ -determined covering spaces, Unpublished notes. 2002.
- homotopically Hausdorff is equivalent to what the author calls “weak $\pi_1$ -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: $X$ 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

$[\beta]\left(\prod_{i=1}^{n}[\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}]\right)[\beta]^{-1}=\prod_{i=1}^{n}[\beta\cdot\alpha_i\cdot\gamma_i\cdot\alpha_{i}^{-}\cdot\beta^{-}]$ ,

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:

$X$ is homotopically Hausdorff,
$\widetilde{X}$ is Hausdorff ( $T_2$ ),
$\widetilde{X}$ is $T_1$ ,
$\widetilde{X}$ 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 $T_0$ . $\square$