Say you’ve got some path-connected space and you want to know about it’s fundamental group . But isn’t locally path-connected so pretty much any standard tools in algebraic topology aren’t going to help you out. What’s an algebraic topologist to do? This post is about a simple but remarkably useful construction that will give you a locally path-connected spaces which has the same underlying set as but which does not change the fundamental group (or any homotopy or homology groups).
The construction is based on the following basic fact from general topology: If is locally path-connected and is an open set of , then the path-components of are open in .
Definition 1: Suppose is a topological space. Let be the space with the same underlying set as but whose topology is generated by the path-components of the open sets of .
This means a basic open sets in is the path-component of an open set in . Let be the collection of such basic open sets. The rest of this post will be devoted to exploring the basic properties of this construction.
Some preliminary facts
Proposition 2: is actually a basis for a topology on the underlying set of .
Proof. Certainly every point of is contained is some path-component of . Suppose and are open in and . Let be the path-component of in for and be the path-component of in the intersection . It suffices to show that . If , then there is a path from to in Since is the path-component of in , the path must have image in both and . Thus .
Proposition 3: The topology of is finer than the topology of . Equivalently, the identity function is continuous.
Proof. If is open in , then is the union of it’s path-components and is therefore a union of basic open sets in . Therefore the topology of is finer than the topology of .
Here is the most important property of .
Theorem 4: Suppose is locally path-connected and is a continuous function. Then the function is also continuous.
Proof. Suppose is the path-component of an open subset of (so that is a basic open set in ). Suppose such that . Since is continuous and is an open neighborhood of in , there is an open neighborhood of in such that . Now since is locally path-connected, we may find a path-connected open set in such that . It suffices to check that . If , then there is a path from to . Now is a path from to . Since is the path-component of in , we must have . This proves .
Another way to think about this is in terms of hom-sets of continuous functions. Here denote the category of topological spaces and continuous functions. Thus is the set of all continuous functions .
Corollary 5: If is locally path-connected, then the continuous identity induces a bijection given by composing a map with .
Proof. Injectivity of follows from the injectivity of the identity function and surjectivity of follows from Theorem 4.
Of course, an important case of Theorem 4 is when we take to be the unit interval. In this case, the above corollary can be interpreted as the fact that and have the same paths and homotopies of paths. For one, if is path-connected, then so is .
The original intent was to construct a locally path-connected version of a space in an “efficient” way. Let’s continue to check that we’ve actually done this.
Theorem 6: is locally path-connected. Moreover, if and only if is locally path-connected.
Proof. Suppose is a basic open neighborhood of a point . By construction of , is the path-component of an open neighborhood in . It is important to notice here that the subspace topologies with respect to the topologies of and may be different so it is not completely obvious that is path-connected as a subspace of . The above theorem will help us out though. Let . Then there is a path with image in and and . Since is also continuous and has image in the subset , we can conclude that and can be connect by a path in the subspace of . Thus is indeed path-connected as a subspace of confirming that is indeed locally path-connected.
For the second statement, it is now clear that if , then is locally path-connected. Conversely, if is locally path-connected, then according to Theorem 4, the continuity of the identity function implies the continuity of the identity function . We already knew the identity was continuous so the topologies of and must be identical.
A categorical interpretation
The construction of is a special type of functor called a coreflection function – the idea being that the category of locally path-connected spaces is a subcategory of such that for every object of there is a “most efficient” way to construct a corresponding object of .
Definition 7: Suppose is a category and is a subcategory. We say is a coreflective subcategory of if the inclusion functor has a right adjoint called a coreflection functor.
If we break this definition down, the fact that is right adjoint to the inclusion means that for every object of , there is an object of and a morphism in which induces a bijection
for every object of . This is precisely our situation: and is the continuous identity.
Theorem 8: is a functor right adjoint to the inclusion functor .
Proof. We’ve already confirmed that we have all the right ingredients. Let’s just put them together. First, we check that is a functor. We have left to see what it does to morphisms. If is a continuous function of any spaces, then we may compose it with the continuous identity to get a continuous function . Since is locally path-connected, Theorem 4 guarantees that is continuous (notice this is actually the same function, it’s just the spaces have different topologies). Thus is the identity on both underlying sets and functions. From here it is more or less obvious that preserves identities and composition.
Theorem 4 then shows that is in fact right adjoint to the inclusion since if and are locally path-connected, then (i.e. is a full subcategory of ). The natural bijection
also is described in Corollary 5.
This is why it is appropriate to call the locally path-connected coreflection of or the -coreflection of .
For a real number , let be the circle of radius centered at . Additionally, if , let .
Example: Let . Then is a non-locally path-connected, compact planar set that looks something like this:
This is something like a generalized wedge of circles; in fact is homeormophic to the reduced suspension of . What is ? Well the topology should only change near points where is not locally path-connected. Here that is the set . A basic neighborhood of a point in this set is a union of intervals, which are precisely the the path-components of .
In particular, the arc is open in illustrating the fact that the circles no longer converge to . In particular, is homeomorphic to the following planar set where the “limit” circle is no longer a topological limit.
Since the circles in are “discrete,” the resulting space is a wedge of circles but technically does not have the CW-topology (which would not be first countable). Instead, it has a metrizable topology. To be fair, I kind of doubt that can always be embedded in whenever the space can. Regardless, we know spaces and have the same homotopy groups but are not homotopy equivalent (an easy way to prove this is using a topologized version of the fundamental group). In fact, all higher homotopy groups are trivial and both fundamental groups are free on a countably infinite set of generators.
Similarly, if , then is not locally path-connected – it looks like a wedge of circles in which the circles are parameterized by the rationals. But the -coreflection is also a countable wedge of circles (with a metrizable topology) – in fact .
If we take , then we get the usual Hawaiian earring space. This is already locally path-connected so .
- For any totally path-disconnected space (i.e. a space in which every path-component is a point) the -coreflection must be discrete. So if is the rationals, then is a countable discrete space. More generally, cannot be the countable disjoint union of closed sets so, in general, if is a countable space, then must be discrete. Similarly, if is the cantor set, then is an uncountable discrete space.
- One could replace circles in the above example with a similar construction using n-spheres in and obtain examples with non-trivial higher homotopy and homology groups.
More algebraic topology
For based spaces and , let denote the set of based homotopy classes of based maps .
Theorem 9: If is locally path-connected, the identity function induces a bijection of homotopy classes .
Proof. Surjectivity follows directly from Theorem 4. Suppose are maps such that are homotopic. Then is locally path-connected and the homotopy is also continuous with respect to the topology of . Thus we obtain a based homotopy between . This shows the function on homotopy classes is injective. .
In the case that is the two-point space, we see that induces a bijection of path-components. When is the n-sphere, we get the following corollary.
Corollary 10: The identity function induces an isomorphism of homotopy groups for all and .
Replacing maps on spheres with maps on the standard n-simplex , we see there is a canonical bijection between singular n-chains in and . This means similar arguments give the same result for homology groups.
Corollary 11: The identity function induces isomorphisms and of singular homology and cohomology groups for all .
One of the limitations of algebraic topology is that most techniques do not apply to non-locally path-connected spaces. For instance, covering spaces of locally path-connected spaces are uniquely determined (up to isomorphism) by the corresponding action on the fiber, but this convenience only translates to very special types of non-locally path-connected spaces. As long as the goal is to understand the homotopy and (co)homology groups of the space, and not to characterize the homotopy type, the -reflection allows one to assume the space in question is locally path-connected.
Definition 12: A space is semi-locally simply connected if for every point , there is an open neighborhood of such that the inclusion induces the trivial homomorphism on fundamental groups.
It’s an important fact from covering space theory that every path-connected, locally path-connected and semi-locally simply connected admits a universal (simply connected) covering .
Proposition 13: is semi-locally simply connected if and only if is semi-locally simply connected.
Proof. First suppose is semi-locally simply connected. Suppose and is an open neighborhood of such that the inclusion induces the trivial homomorphism . Let be the path-component of in . Then is an open neighborhood of in . The inclusion induces a homomorphism which factors as where the later homomorphism is trivial. Thus is trivial.
Conversely, suppose is semi-locally simply connected and . Find an open neighborhood of in such that . We can assume is a basic neighborhood, so that is the path-component of an open set of . If is a loop based , then it must have image in . Since is null-homotopic in , it must be null-homotopic when viewed as a loop in . Thus is trivial.
Corollary 14: If is path-connected and semi-locally simply connected, then admits a universal covering .
The composition is essentially a universal covering of the space except it doesn’t exactly satisfy the local triviality part of the definition of a covering map. However, it does have pretty much all of the same lifting properties as a covering map: if is path-connected, locally path-connected, and is a map such that , then there is a unique continuous lift satisfying .
Take the example of the generalized wedge of circles pictured above. This space does not have a universal covering space but it’s -coreflection does. We can conclude that for many non-locally path-connected spaces, there is still a covering theoretic approach to characterizing the structure of the fundamental group – just apply the locally path-connected coreflection first.
For more, see: