When is a local homeomorphism a semicovering map?

In a previous post “What is a Semicovering Map?,” I gave an introduction to semicovering maps. A semicovering is a slight generalization of covering map that becomes particularly relevant when you’re dealing with locally complicated spaces. In particular, a semicovering p:E\to X is a local homeomrphism with unique lifting of all paths (for every path \alpha:[0,1]\to X and point e\in E with p(e)=\alpha(0), there is a unique lift \beta:[0,1]\to E with \beta(0)=e). For context, I recommend taking a look at the introductory post (or the original papers [1] and [3]) first to gain intuition for the subtle difference between covering maps and semicovering maps.

A natural question to ask is: how much can we weaken the definition of “semicovering map?” More precisely, is there a weaker condition that allows us to promote a local homeomorphism to a semicovering? I really enjoy the logical investigations that these kinds of questions lead to.

This question comes from the paper [4], which gives an answer similar to the one I’ll give in this post. My interest in this question was piqued by questions of some non-Archimedian geometers, who identified a geometric analogue of semicoverings. I shared some of my suspicions with these geometers and this post is just me writing down the details. Certainly, the authors of [4] deserve the credit for originally answering this question.

Definition: We say a map p:E\to X has the endpoint-lifting property whenever \alpha:[0,1]\to X is a path and \beta:[0,1)\to E is a map satisfying p\circ \beta=\alpha|_{[0,1)}, \beta extends (not necessarily uniquely) to a path \beta:[0,1]\to E such that p\circ\beta=\alpha.

The endpoint lifting property

The endpoint-lifting property should be thought of as a completeness-type property of E relative to p: if you have a map [0,1)\to E and it agrees with the start of a true path in X, then you can complete it to a true path [0,1]\to E. There are some conditions on E (without referring to p) that imply this, but these include very strong compactness conditions and do not improve known results.

Example: Let T=\{0\}\times[-1,1]\cup\{(x,\sin(1/x)\mid 0<x\leq 1\} be the closed topologist sine curve and let p:T\to [0,1] be the projection onto the x-axis. This map does not have the endpoint-lifting property because we can define continuous map \beta:[0,1)\to T by \beta(t)=(1-t,\sin(\frac{1}{1-t})). Now \alpha:[0,1]\to [0,1], \alpha(t)=1-t is a path and agrees with p\circ\beta on [0,1). However, we cannot assign a value to \beta(0) so that \beta is continuous at 0. You can have both spaces path conecnted if you take the projection of the Warsaw circle onto the ordinary circle.

Throughout this post, we’ll assume the space X is non-empty and path connected. With this assumption, a map p:E\to X with the endpoint lifting property (or stronger property) will always be surjective.

Theorem: Suppose convergent sequences in E have unique limits and p:E\to X is a local homeomorphism. Then the following are equivalent:

  1. p is a semicovering map,
  2. p is a Hurewicz fibration,
  3. p has the endpoint-lifting property.

Note: the condition that convergent sequences in E have unique limits is formally weaker than being Hausdorff. Sometimes a space with this property is called a “US-space.”

Proof of Theorem. The proof of 1. \Leftrightarrow 2. is Theorem 7.5 in [2]. I’d like to focus on the equivalence of 1. and 3 because this part is, in my view, a little more fun. By definition, a semicovering has “unique lifts of all paths rel. basepoint.” Clearly, this implies the endpoint-lifting property. The direction 1. \Rightarrow 3. follows. The rest of this post will be to prove 3. \Rightarrow 1.

Suppose p is a local homeomorphism with the endpoint-lifting property. We must show that p has unique lifting of all paths. Let \alpha:[0,1]\to X be a path starting at \alpha(0)=x and p(e)=x. We must find a unique path \beta:[0,1]\to E such that \beta(0)=e and p\circ \beta=\alpha. We proceed similar to how you might prove that closed intervals are compact, that is, by defining a convenient set and analyzing the supremum. Let $$A=\{t\in (0,1]\mid \exists \beta_t:[0,t]\to E\text{ s.t. }\beta_t(0)=e\text{ and }p\circ\beta_t=\alpha|_{[0,t]}\}$$

Existence of path lifts: To show that lifts exist, we must show that 1\in A. Since p is a local homeomorphism, we can find an open neighborhood U of e that p maps homeomorphically onto the neighborhood p(U) of x. Find s>0 such that \alpha([0,s])\subseteq p(U). The formula \beta_s(t)=p|_{U}^{-1}(\alpha(t)) gives a lift \beta_s latex of \alpha|_{[0,s]}. Therefore s\in A and we have \sup(A)>0. Moreover, the definition of A also ensures that A is an interval of the form [0,t) or latex [0,t] for 0<t\leq 1. Now the assumption that p has the endpoint-lifting property tells us that if if we knew \alpha|_{[0,t)} could be lifted, then we can also lift \alpha|_{[0,t]}. Hence, A must actually be a closed interval of the form [0,v] where v=\max(A).

But if v<1, we can only lift \alpha|_{[0,v]} to a path \beta:[0,v]\to E and no more. But p is still a local homeomorphism. So we can find an open neighborhood U' that p maps homeomorphically onto the open set p(U'). Since \alpha(v)\in p(U'), we can find v<w such that \alpha([v,w])\subseteq p(U'). Like before we define \beta(t)=p|_{U'}^{-1}(\alpha(t)) when t\in [v,w]. This extends \beta to a map \beta:[0,w]\to E that is a lift of \alpha|_{[0,w]}.

However w>max(A); a contradiction. Since v>0 and v<1 is false, we must have v=\max(A)=1. This proves existence.

Uniqueness of path-lifts: To prove uniqueness, suppose \beta_1,\beta_2:[0,1]\to E both satisfy \beta_i(0)=e and p\circ\beta_i=\alpha. Let B=\{t\in (0,1]\mid \beta_{1}|_{[0,t]}=\beta_{2}|_{[0,t]}\}.

Using the neighborhood U of e (as above) that is mapped homeomorphically to p(U), we can see that we must have \beta_{1}|_{[0,t]}=\beta_{2}|_{[0,t]} for at least some small t>0. Hence, B\neq \emptyset. Just like A, the set B must be an interval containing 0. What if B=[0,s) for some 0<s\leq 1? Then \beta_{1}|_{[0,s)}=\beta_{2}|_{[0,s)} but \beta_1(s)\neq \beta_2(s). Picking a convergent increasing sequence s_1<s_2<s_3<\cdots \to s in [0,s], we see that \{\beta_1(s_n)\} and \{\beta_2(s_n)\} are equal but converge to distinct limits points \beta_1(s) and \beta_2(s).

However, we have assumed that convergent sequences in E have unique limits. Therefore, B must be a closed interval.

Now we must show v=\max(B)=1. Suppose that v<1. Then \beta_{1}|_{[0,v]}=\beta_{2}|_{[0,v]}. Take a neighborhood U' that maps homeomorphically on the neighborhood p(U') of \alpha(v). Because p|_{U'}:U'\to p(U') is injective, \beta_1([v,w]) and \beta_2([v,w]) lie in U' for some v<w, and p(\beta_1(t))=p(\beta_2(t)) for all v\leq t\leq w, we must have \beta_{1}|_{[v,w]}=\beta_{2}|_{[v,w]}. Thus w\in B; a contradiction that v is the maximum. We conclude that v=1, i.e. \beta_1=\beta_2. \square

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

This entry was posted in Algebraic Topology, Covering Space Theory, Fundamental group, Generalized covering space theory, semicovering. Bookmark the permalink.

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