In this last post about reduced paths, I’m going to work through the details of one of the most useful results in wild topology. Writing this post helped me work out my own way of proving this result and hopefully will help bring together some ideas from the literature in a unique way in a way that is helpful to folks trying to learn about some of the techniques of the field.
Unique Reduced Path Theorem: If is a one-dimensional Hausdorff space, then every path is path-homotopic to a reduced path that is unique up to reparameterization. Moreover, the homotopy between and has image in .
Although stated a little differently, this was basically proven in . The contemporary version appears in . This theorem changed the game for me. Instead of using inverse limits all of the time, this allowed me to understand and prove things using order theory and unique reduced representatives.
First, it might be helpful to give more explanation of the statement itself.
- What does one-dimensional mean? Actually, you can use any of the three standard notions of dimension (Lebesgue covering, small inductive, or large inductive) and this theorem would still be true. Generally, Lebesgue covering dimension is the typical choice.
- Recall from Part I that a path being reduced means that has no null-homotopic subloops, i.e. there is no such that is a null-homotopic loop.
- What does “unique up to reparameterization” mean? A path is a reparameterization of a path if there is an increasing homeomorphism such that . I use the notation when this occurs. Reparameterization is an equivalence relation on the set of paths in a space. So “unique up to reparameterization” means that if and are homotopic reduced paths in a 1-dim. space, then .
- The last statement of the theorem implies that deforming to its reduced representative only requires deleting portions of . As a special case, if is a null-homotopic loop, then it contracts in its own image.
One-dimensional Peano continua
We need to dive into some terminology and “well-known” results from General Topology here. I’ve used these terms and results before but here’s a reminder:
An arc (respectively, simple closed curve) in a space is a subspace of that is homeomorphic to (respectively, ). If is an embedding onto the arc , we call and the endpoints of the arc. Notice that an “arc” is technically not the same thing as an embedding . Rather an arc is the image of such an embedding with the subspace topology.
A space is uniquely arcwise connected, if given any distinct , there is a unique arc with endpoints and .
A Peano continuum is a connected, locally path-connected compact metric space. The Hahn-Mazurkiewicz Theorem says that a space is a Peano continuum if and only if is Hausdorff and there exists a continuous surjection .
A dendrite is a Peano continuum that has no simple closed curves. I go into some of the theory of dendrites in this old post on shape injectivity. In particular, I mention a well-known structure theorem, which says that a dendrite is homeomorphic to an inverse limit of trees where is with a single edge attached and each bonding map collapses that edge to the vertex at which it is attached. In Part II of the Shape Injectivity Post, we used this structure theorem to prove that dendrites are contractible.
To put these old posts to work here’s a theorem that I mentioned at the end of Part II.
Inverse Limit Representation Theorem [3, Theorem 1]: Every one-dimensional path-connected compact metric space can be written as an inverse limit of finite graphs .
In fact, this is improved a bit in  where it is shown that it is possible to improve a given inverse system to ensure that all bonding maps map surjectively and simplicially onto some finite subdivision of .
Dendrite Factorization Lemma
The next lemma is at the heart of why we can do so much in wild one-dimensional spaces like the earring space, Menger Cube, and higher-dimensional constructions that start with one-dimensional spaces, e.g. the Harmonic Archipelago. The proof for a general space is the same as that for loops where so I went ahead and wrote out the general proof.
Lemma: If is a one-dimensional Hausdorff space, is a Peano continuum, and is a null-homotopic map based at , then factors through a dendrite, that is, there is a dendrite a map and a map such that .
Proof. Assuming that is non-constant, notice that is one-dimensional and Hausdorff (as a subspace of a one-dimensional Hausdorff space that admits a non-constant path). Since is a Hausdorff continuous image of a Peano continuum, it is a Peano continuum by the Hahn-Mazurkiewicz Theorem. Hence, we may assume that and that is a Peano continuum.
Using the Inverse Limit Representation Theorem, write with bonding maps and finite graphs . If are the projections, we take to be the basepoints in the graphs. Let be the universal covering map where is a tree. As we did in Part II of the Shape Injectivity Post, once we choose basepoints , the maps induce unique based maps that give the following inverse system of covering maps.
Since is null-homotopic in and each latex is a Hurewicz fibration, the map is null-homotopic in the graph and has a unique lift to a based map . These based lifts agree with the bonding maps and give the inverse system of covering maps you see below. The universal property of inverse limits gives a unique map based at such that. . In fact, Hurewicz fibrations are closed under inverse limits so is also a Hurewicz fibration!
Let . Since the inverse limit of trees is clearly Hausdorff, is a Peano continuum. Moreover, I gave a detailed proof in Part I of the Shape Injectivity Post that an inverse limit of trees contains no simple closed curves. Since is a Peano continuum with no simple closed curves, it must be a dendrite! Taking to be the restriction of to , completes the proof.
We could replace with any Peano continuum in the next Corollary, but I’ll try to keep it focused.
Corollary: Every null-homotopic loop in a one-dimensional Hausdorff space contracts in its own image.
Proof. Suppose is a null-homotopic map. By the previous Lemma, we have for a map and map where is a dendrite. Set . Since is contractible, there is a null-homotopy latex with , . Now is a null-homotopy of .
The null-homotopy in the last proof is a “free” null-homotopy but since is well-pointed, you could just as easily construct a basepoint-preserving homotopy.
Getting back on track, we’d like to apply the general results in Part II, which says that homotopy classes of paths will have reduced representatives if our space has well-defined transfinite -products. So let’s make sure that happens.
Proposition: Every one-dimensional Hausdorff space has well-defined transfinite -products.
Proof. Let be a one-dimensional Hausdorff space. Suppose is a closed set containing and are paths such that and such that for every connected component of , we have . We must show that . For each component of , the loop is null-homotopic and therefore (by the last Corollary) contracts by a null-homotopy in . In particular, there exists an endpoint-relative homotopy , where
We just need to put these all together! Define by
Notice that is the constant homotopy at .
Checking continuity of
would be considered “routine” for those who make these constructions a lot so sometimes these things are skipped in the literature. But a blog is a good place to lay out the details for those who are still getting used to the proof techniques. So let’s do it!
Fix and an open neighborhood of in . Since each is continuous, is continuous at if for some component of . So we may assume . Since and are continuous at , there exists such that . We now have . If , then by the definition of . What remains is to consider components having as an endpoint.
If is not an endpoint of a component of , then we may find such that . If is a right endpoint of a component latex , then we may choose small enough so that (by the continuity of ). Similarly, if is a left endpoint of a component , then we may choose small enough so that . In any case, we can find an open interval of such that .
The main point of Part II, was to show that every path in a Hausdorff space with well-defined transfinite -products is path-homotopic to a reduced path. This path-homotopy was defined by “deleting” null-homotopic subloops on a maximal cancellation and therefore had image in the image of . Combining this old stuff with the above proposition, it must be that every path in a one-dimensional Hausdorff space is path-homotopic to a reduced path by a homotopy that takes place in the image of that path itself. This proves the existence portion of the Unique Reduced Path Theorem as well as the last statement about the size of the homotopy required.
Uniqueness of reduced paths in one-dimensional spaces
Dendrites have their infinitely many little fingers all over this content. We’ll need them again to finish the proof of uniqueness.
Lemma: If is a one-dimensional Hausdorff space and are reduced and path-homotopic to each other, then , i.e. there exists an increasing homeomorphism such that .
Proof. By replacing with the image of a homotopy from to , we may assume that is a Peano continuum. Now is a null-homotopic loop and so by Dendrite factorization, there exists a dendrite , a loop , and a map such that . Write so that and . Let and . Recall that dendrites are uniquely arcwise connected and so there is a unique arc in with endpoints and . Now is a path in from to . We check that is injective and therefore a parameterization of . If such that , then would be a loop. Since dendrites are contractible, is a null-homotopic loop. Then must also be a null-homotopic loop. However, this violates the assumption that is reduced. Since is also reduced, applying the same argument to shows that also parameterizes . Since both are homeomorphisms with the same orientation, we consider the increasing homeomorphism . Now , which completes the proof. .
What’s the Takeaway?
Imagine you’ve got a based loop where is the earring space or, more amazingly, the Menger cube. The homotopy class in the fundamental group is represented by a “tightest” loop that has absolutely no homotopical redundancy. Every point is crucial to that homotopy class and the order in which those points are traced out in is completely unique.
This also tells you about the operation in the fundamental groupoid too. Suppose you’ve got two composable path-homotopy classes . Write and for reduced paths and . Then the product is represented by the concatenation . However, may not be reduced. But, it’s still homotopic to some reduced path and that reduced representative is obtained by deleting null-homotopic subloops on a maximal cancellation. But wait! There’s only one possible way for this to happen because the entirety of and are both reduced. A maximal cancellation of can only contain one interval, which must contain the concatenation point . Hence, there exists and such that . There’s more! If a path is reduced, then all of its subpaths are reduced too. Since and are homotopic reduced paths, which means they are actually reparameterizations of each other.
are reduced paths in a one-dimensional Hausdorff space satisfying
. Then either
is reduced or there exists unique
is a reduced path representing
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