What is a semicovering map?

I’ve heard twice in the past year from folks who study non-Archimedian geometry and have found connections to “semicoverings,” which are a generalization of covering maps used in wild topology. The questions I received had me revisiting the basics and motivated this post.

There is a massive history and literature of attempts to generalize covering space theory. The goal of such generalizations is usually to expand the incredibly useful symbiotic relationship between topology (covering spaces) and algebra (the fundamental group) that covering space theory provides. One can distinguish spaces using fundamental groups and, on the flip-side, one can also prove algebraic results by realizing algebraic objects as invariants of a space, e.g. a group realized as the fundamental group of some space that topologically “encodes” the structure of the group. Generalizations of covering space theory often result in an even richer symbiotic relationship between more complicated spaces (maybe failing to be locally path-connected or semilocally simply connected) or space-like objects and more intricate/enriched algebraic objects (like pro-groups, topological groups, etc.) In my experience, the tricky part of this business is starting with an intended application and then finding just the right generalized notion of “covering map” that does exactly what you want it to do. This can require a lot of fussing around with the hierarchy of properties that covering maps enjoy and seeking out the appropriate combination.

Background

“Semicovering maps” and their classification are something that I worked on at the end of grad school in 2011. I had spent a lot of time studying topological versions of homotopy groups for my thesis, including the “tau topology,” and I had my eye on using it to fill in some long-stand gaps in topological group theory. If only I had the right generalization of covering maps to do the job! It all feels “obvious” in hindsight but back then it was not so easy to do and it took a lot of wrong definitions to sort out one that ended up working. Even then, the original definition of semicoverings in [1] was not the simplest possible one. Honestly, I can’t remember if I made any attempt to simplify it back in 2011. I just wanted a working classification and toward that end I found a combination of properties that made it work. Not too much later, several folks realized that the definition could be simplified.

In this post, I’ll introduce semicoverings using the refined definition. Perhaps in a future post, I’ll discuss their classification and applications. Spoiler: semicoverings over a space $X$ are classified by open subgroups of a natural topologized version of the fundamental group $\pi_1(X,x_0)$ . Since “nice” spaces have discrete fundamental groups, this contains the usual classification of covering spaces as a special case.

Much of the original work in [1] is done in categorical language using topologically enriched groupoids. I love groupoids but some of my colleagues don’t and this groupoid-heavy approach was partly the product of a referee’s preferences. I was but a baby-child of a mathematician and wanted this published so I didn’t push back…what can I say?. Anyway, the fundamental group version of the classification appears at the very end of the paper anyway.

Getting to the point…What is a semicovering?

We’ll say that a map $p:E\to X$ has unique lifting of all paths rel. basepoint if whenever $\alpha:[0,1]\to X$ is a path and $e\in p^{-1}(\alpha(e))$ , there exists a unique path $\beta:[0,1]\to E$ such that $\beta(0)=e$ and $p\circ\beta=\alpha$ .

Definition: A map $p:E\to X$ is a semicovering map if $p$ is a local homeomorphism and has unique lifting of all paths rel. basepoint. We refer to $E$ as a semicovering space of $X$ .

Notice that $p$ is NOT defined to be locally trivial like a covering map is. So although a semicovering must have discrete fibers (because it is a local homeomorphism) it will not typically be a fiber bundle.

Two-of-three: in general, the composition of a two coverings maps is not always a covering map. However, compositions of local homeomorphisms are local homeomorphisms. Also, maps with unique lifting of all paths rel. basepoint are closed under composition. Therefore, semicoverings are closed under composition. In fact, semicoverings satisfy the two-of-three property that if $f\circ g=h$ where two of the maps $f,g,h$ are semicoverings, then the third one must also be a semicovering (provided the spaces involved are path-connected).

Every covering map is a semicovering map: this is true essentially by choice of our definition. Every covering map is a local homeomorphism and the usual theory shows that covering maps have unique lifting of paths rel. basepoint. It’s not really obvious at this point but if you have a space $X$ with the usual conditions (path-connected, locally path-connected, and semilocally simply connected), then any semicovering $p:E\to X$ where $E$ is connected, will be a true covering map. So, the situations where semicoverings are intended to be useful in non-trivial ways are related to locally complicated spaces like the earring space, other locally path-connected spaces, and even many non-locally path-connected spaces.

Example: The two-of-three property immediately tells you how to find examples of semicoverings that are not covering maps: take two covering maps whose composition is not a covering map. The composition will be a semicovering but not a covering map. There are more extreme examples in [1] that can’t be realized this way, but this is a good start.

A composition of two covering maps, which is a semicovering map but not a covering map. The lower map “unwinds” the outermost circle of the earring. The upper map is a 2-fold covering where as you look both to the left and right you see more and more circles based at fiber points becoming arcs connecting the two fiber points.

What is the difference between a covering and a semicovering over the earring space? Let $\mathbb{E}$ be the earring space with wild point $b_0$ and $n$ -th circle $C_n$ . If you have a true covering map $p:E\to \mathbb{E}$ , you can find some neighborhood $U$ of $b_0$ which is evenly covered by $p$ . Now all but finitely many of the circles of $\mathbb{E}$ , will be contained in $U$ . In particular we have a smaller copy of $\mathbb{E}_{\geq N}=\bigcup_{n\geq N}C_n$ in $U$ . Since $U$ is evenly covered by $p$ , we have $p^{-1}(U)$ decomposing as a disjoint collection of open neighborhoods $\coprod_{k\in K}V_k$ where $p$ maps $V_k$ homeomorphically onto $U$ . In particular, we have a copy of $\mathbb{E}_{\geq N}$ in each $V_k$ that gets mapped homeomorphically onto $\mathbb{E}_{\geq N}$ . In this way covering spaces of $\mathbb{E}$ are “uniformly wild” by which I mean there is a copy of $\mathbb{E}_{\geq N}$ hanging around each point in the wild fiber. Said another way, there must be some uniform upper bound $N$ on the number of circles $C_n$ can lift to an arc in $E$ .

Now, a semicovering map $p:E\to \mathbb{E}$ is still a local homeomorphism but there may not be a neighborhood of $b_0$ that is evenly covered by $p$ . Being a local homeomorphism means that at each wild fiber point $e\in p^{-1}(b_0)$ there will be exist natural number $m_e$ and a copy of $\mathbb{E}_{\geq m_e}$ attached at $e$ . However, like in the illustrated example, $F=\{m_e\in\mathbb{N}\mid e\in p^{-1}(b_0)\}$ may not be bounded above (see the image below). In fact $p$ will be a true covering if and only if $F$ is bounded above.

A non-trivial semicovering over the earring space, which is not a traditional covering map. Each step that moves you one edge away from the given basepoint, unwraps the next circle of the earring.

This informs how we should think about semicovering spaces in general. We should still think of a semicovering space $E$ over $X$ as looking like $X$ locally but as we “unwind” the path-homotopy classes of $X$ to obtain $E$ , we are allowed to unwind smaller and smaller paths as we move further away from a fixed basepoint $e\in E$ .

Continuous Lifting

What’s the most interesting property of semicoverings? I think it’s that even without local triviality, paths lift not only uniquely but also continuously. This is what makes the connection to topologized fundamental groups possible.

Let $P(X,x_0)$ be the space of paths $\alpha:[0,1]\to X$ with $\alpha(0)=x_0$ equipped with the compact-open topology. Every based map $f:(X,x)\to (Y,y)$ induces a continuous function $P(f):P(X,x)\to P(Y,y)$ , $P(f)(\alpha)=f\circ\alpha$ .

Using path spaces provides a nice way to understand lifting: A map $p:E\to X$ has unique lifting of paths rel. basepoint if and only if for every $e\in E$ , the map $P(p):P(E,e)\to P(X,p(e))$ is a bijection.

Lemma (Continuous Lifting): If $p:E\to X$ is a semicovering, then for every $e\in E$ , the induced map $P(p):P(E,e)\to P(X,p(e))$ is a homeomorphism.

Proof sketch. Set $x=p(e)$ . A subbasic set in $P(E,e)$ is of the form $\langle K,U\rangle=\{\alpha\in P(E,e)\mid \alpha(K)\subseteq U\}$ where $K\subseteq [0,1]$ is compact and $U$ is open in $E$ . Let $K_{n}^{j}=\left[\frac{j-1}{n},\frac{j}{n}\right]$ . With some basic general topology arguments, you can show that there is a basis for the topology of $P(E,e)$ consisting of open sets of the form $\mathcal{U}=\bigcap_{j=1}^{n}\langle K_{n}^{j},U_j\rangle$ for open sets $U_1,U_2,\dots, U_n\subseteq E$ . One should think of $\mathcal{U}$ as a finite set of instructions determining how the paths it contains can proceed through $E$ .

It’s tempting to think that $P(p)$ will map $\mathcal{U}$ onto the open set $\bigcap_{j=1}^{n}\langle K_{n}^{j},p(U_j)\rangle$ in $P(X,x)$ . But this doesn’t exactly work out because of how the intersections might overlap. Therefore, we need to define

$\mathcal{V}=\bigcap_{j=1}^{n}\langle K_{n}^{j},p(U_j)\rangle\cap \bigcap_{j=1}^{n-1}\langle \{\frac{j}{n}\},p(U_{j}\cap U_{j+1})\rangle.$

which is still a basic open set in $P(X,x)$ . With this set defined, showing that $P(p)(\mathcal{U})=\mathcal{V}$ requires a direct set-inclusion argument.

What comes for free? Lifting of path-homotopies!

Corollary: If $p:E\to X$ is a semicovering map and $H:(D^2,(1,0))\to (X,x)$ is a map from the closed unit disk, then for every $e\in p^{-1}(x)$ , there is a unique map $\widetilde{H}:(D^2,(1,0))\to (E,e)$ such that $p\circ\widetilde{H}=H$ .

Proof. Recall that a map $f:(D^2,(1,0))\to (Y,y)$ may be identified uniquely with a loop $F:[0,1]\to P(Y,y)$ using exponential properties of spaces. Specifically $F(t)$ , $0<t<1$ is the path given by restricting $f$ to the line from $(1,0)$ to $(\cos(2\pi t),\sin(2\pi t))$ . Therefore, may view $H$ is a loop $h:[0,1]\to P(X,x)$ . Recall that $P(p):P(E,e)\to P(X,x)$ is a homeomorphism. Therefore, $P(p)^{-1}\circ h:[0,1]\to P(E,e)$ is a loop. Going backward in the adjunction, have a map $\widetilde{H}:(D^2,(1,0))\to (E,e)$ that satisfies $p\circ \widetilde{H}=H$ . $\square$

With unique lifting of paths and path-homotopies established, we can conclude that nearly all of the lifting properties that covering maps enjoy follow as well. A little less obvious is that every semicovering is a Hurewicz fibration (see Theorem 7.5 in [2]). Thus, semicoverings fit snuggly between the two:

Covering map $\Rightarrow$ Semicovering map $\Rightarrow$ Hurewicz fibration with discrete fibers.

I don’t know of a Hurewicz fibration with discrete fibers that is not a semicovering map but I expect that one exists. Think you can find one?

References:

[1] J. Brazas, Semicoverings: a generalization of covering space theory. Homology Homotopy Appl. 14, (2012) 33–63. Open Access.

[2] J. Brazas, A. Mitra, On maps with continuous path lifting. Preprint. 2020. https://arxiv.org/abs/2006.03667

[3] H. Fischer, A. Zastrow, A core-free semicovering of the Hawaiian Earring. Topology Appl. 160, (2013) 1957–1967. Open Access.

[4] M. Kowkabi, B. Mashayekhy, H. Torabi, When is a local homeomorphism a semicovering map? Acta Mathematica Vietnamica, 42, (2017) 653-663. https://arxiv.org/abs/1602.07260

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