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 is a local homeomrphism with* unique lifting of all paths* (for every path and point with , there is a unique lift with ). 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 has the *endpoint-lifting property* whenever is a path and is a map satisfying , extends (not necessarily uniquely) to a path such that .

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

**Example:** Let be the closed topologist sine curve and let be the projection onto the x-axis. This map does not have the endpoint-lifting property because we can define continuous map by . Now , is a path and agrees with on . However, we cannot assign a value to so that is continuous at . 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 is non-empty and path connected. With this assumption, a map with the endpoint lifting property (or stronger property) will always be surjective.

**Theorem: **Suppose convergent sequences in have unique limits and is a local homeomorphism. Then the following are equivalent:

- is a semicovering map,
- is a Hurewicz fibration,
- has the endpoint-lifting property.

**Note:** the condition that convergent sequences in 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. 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. 3. follows. The rest of this post will be to prove 3. 1.

Suppose is a local homeomorphism with the endpoint-lifting property. We must show that has unique lifting of all paths. Let be a path starting at and . We must find a unique path such that and . 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 . Since is a local homeomorphism, we can find an open neighborhood of that maps homeomorphically onto the neighborhood of . Find such that . The formula gives a lift latex of . Therefore and we have . Moreover, the definition of also ensures that is an interval of the form or latex for . Now the assumption that has the endpoint-lifting property tells us that if if we knew could be lifted, then we can also lift . Hence, must actually be a closed interval of the form where .

But if , we can only lift to a path and no more. But is still a local homeomorphism. So we can find an open neighborhood that maps homeomorphically onto the open set . Since , we can find such that . Like before we define when . This extends to a map that is a lift of .

However ; a contradiction. Since and is false, we must have . This proves existence.

*Uniqueness of path-lifts: *To prove uniqueness, suppose both satisfy and . Let

Using the neighborhood of (as above) that is mapped homeomorphically to , we can see that we must have for at least some small . Hence, . Just like , the set must be an interval containing . What if for some ? Then but . Picking a convergent increasing sequence in , we see that and are equal but converge to distinct limits points and .

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

Now we must show . Suppose that . Then . Take a neighborhood that maps homeomorphically on the neighborhood of . Because is injective, and lie in for some , and for all , we must have . Thus ; a contradiction that is the maximum. We conclude that , i.e. .

## 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