What does “wildness” really refer to?
I’ll post about higher dimensional wildness soon but to avoid getting too general, I’m going to focus on a more specific question: what does it mean for a space to have a wild fundamental group?
If you surveyed mathematicians with this question most responses would likely say:
has a wild fundamental group if is not semilocally simply connected.
You might get a few answers of “not locally contractible,” which is a good answer to a more general question but that would be a deflection from our current focus on . A space can easily fail to be locally contractible but still have tame 1-dimensional homotopy. The first answer is reasonable because we often study fundamental groups using covering space theory and, assuming locally path-connectivity, has a universal (i.e. simply connected) covering space if and only if is semilocally simply connected. But does failing to be semilocally simply connected really imply that the topology interacts with the algebra of in any kind of wild fashion?
The answer is: a lot of the time, but not always.
I’ll detail both parts of this answer but I think the not always part of the answer involves some fun topology. First, some quick and possibly skip-able review:
Definition: A space is semilocally simply connected at if there exists an open neighborhood such that the inclusion induces the trivial homomorphism on fundamental groups. We say is semilocally simply connected if it is semilocally simply connected at all of its points.
Intuition: Despite the complicated name, the idea behind being “semilocally simply connected” is pretty straightforward. The triviality of the homomorphism simply means that every loop in based at can be contracted in (but not necessarily in ). For instance, let be a cone of height over a circle in the xy-plane. Then the open set is a annular neighborhood of the base circle and is therefore homotopy equivalent to a circle. Any loop in is null homotopic in but a loop going around the base circle once is non-trivial in since you must use the top of the cone to contract it. In this case, is trivial even though is not.
Again, this property is a key assumption in classical covering space theory: If is path connected and locally path connected, then has a simply connected (i.e universal) covering space if and only if is semilocally simply connected.
Let’s analyze the situation where a space fails to have this property.
Definition: The topological 1-wild set of is the subspace
Hence, for locally path-connected , the set is precisely the topological obstruction to the existence of a universal covering space. If isn’t locally path connected, remember that we have this handy dandy tool.
- if and only if is semilocally simply connected. Included are all simply connected spaces, CW-complexes, polyhedra, and manifolds.
- If is the Hawaiian earring with join point , then is a singleton.
- The space pictured below has a shrinking sequence of Hawaiian earrings attached to the points . The wild set is , which is not discrete.Notice that this space is wild at because every neighborhood of that point contains all but finitely many Hawaiian earrings.
- If is the dyadic arc-space pictured below as the union of the base-arc and countably many semi-circles, then is an interval and thus has uncountably many points.
- It is possible that ! Examples include the Menger Curve, Sierpinski Triangle, and Sierpinski Carpet.
In all of the above examples, is closed in ; this is not a coincidence. Proving it’s true in general is a nice exercise to include in a course on fundamental groups.
Exercise: Prove that if is locally path-connected, then is closed in .
In fact, even in the non-locally path-connected case, something can be said.
More general exercise: Show that if is a map from a locally path-connected space , then is closed in .
An inquisitive reader may wonder in what ways the space is a kind of homotopy invariant of . Some partial answers are published, but I promised myself not to go down the rabbit hole of writing about this yet.
What is algebraic wildness?
In particular, does , imply that the fundamental group is wild?
At first glance, it seems like it might. Surely, arbitrarily small non-trivial loops are going to result in complicated algebra, right? To find out, let’s clarify what kind of property “wildness” should be for .
- It can’t be a purely group theoretic property since, by CW-approximation, every group is the fundamental group of some 2-dimensional CW-complex. I don’t care how complicated your group is or how hard the Whitehead conjecture is, the fundamental group of a CW-complex does not count as being wild.
- An algebraically “wild” fundamental group should admit at least one (possibly trivial) element that is represented by an infinite concatenation of non-contractible loops.
- It doesn’t just have to happen at the basepoint. Because of path-conjugation, wildness might occur at any point of .
Therefore, since algebraic wildness is really about the natural presence of infinitary operations like infinite sums in calculus/analysis (and in contrast with binary, trinary, or other finitary operations that come for free in ordinary groups), we’ll use the more descriptive word “infinitary” instead of “wild.”
Definition: A fundamental group is infinitary at if there exists a loop based at , and a closed set with such that has infinitely many components, and for each component of , the loop is not null-homotopic. If is not infinitary at , then we say it is finitary at . If is finitary at all points of , we say the group is finitary. If there is at least one point, where is infinitary, we say the group is infinitary.
Equivalent but more practical definition: A fundamental group is infinitary at if and only if there exists a map from the Hawaiian earring such that restricted to each circle of is not a null-homotopic loop.
Proof of Equivalence: If you start with the first definition, the loop satisfies and thus induces a map . However, because has a countably infinite number of components, . Moreover, each loop induces the restriction of to a unique circle of . Conversely, if we start with a map , let be the image of the wild point. If we take to be the loop which is the image of on the -th circle, consider the infinite concatenation defined as on , and . The set now satisfies the first definition.
Definition: The algebraic 1-wild set of a space is the subspace
Exercise: Construct a locally path-connected space such that is not sequentially closed.
Of course, topological 1-wildness (arbitrarily small non-contractible loops) and algebraic 1-wildness (a shrinking sequence of non-contractible loops) should feel closely related – they both involve the existence of arbitrarily small non-contractible loops at some point.
A lot of the time, they’re the same
The next theorem shows where the two kinds of wildness agree.
Theorem: If is semilocally simply connected at , then is finitary at . The converse is true if is first countable at .
Proof. Suppose is semilocally simply connected at and is a map. We must show that applied to at least one circle of is null-homotopic. Find an open neighborhood of such that every loop in based at is null-homotopic in . Write as the usual union of circles of radius . By the continuity of , there is a neighborhood of such that . Given the topology of , we may find an such that for all . Therefore, maps to a loop in , which must be null-homotopic in . This completes the first direction.
For the converse, suppose is first countable at and that is not semilocally simply connected at . Let be a neighborhood base at . By assumption, for each , there is a loop based at , which is not null-homotopic in .
Define a function to be on . To check the continuity of , we really only need to check the continuity of at : If is a neighborhood of , find with . Then it is clear that . For find a neighborhood of in such that . Now is a neighborhood of such that . Since is continuous an non-trivial on each circle, is infinitary.
Corollary: For any space , we have with equality if is first countable.
So, for first countable spaces, the two notions of wildness are the same. Of course, even with some non-first countable spaces like very large CW-complexes, we still have equivalence.
This all suggests that to find a difference between topological and algebraic wildness, we need to dig into difference between nets and sequences of loops.
but not always
General Topology permits all sorts of phenomenon, including interesting spaces that are not first countable. To conclude this post, I’m going to describe a space that isn’t semilocally simply connected (and doesn’t have a universal covering space) but whose fundamental group is not wild, i.e. is finitary.
Let be the first uncountable ordinal and be it’s successor, the first compact uncountable ordinal. Here, denote the maximal point of and we take it to be the basepoint. The key topological fact we’ll need to remember is that there does not exist any sequence in that converges to . Let
be the reduced suspension with canonical basepoint . For each countable ordinal , the image of will result in a unique circle in . These circles will all be joined at . However, the topology of this space is not the one you’d give to a wedge of CW-complexes. To help visualize this monster, consider the first convergent sequence in . This corresponds to the subspace of illustrated below as the sequence of circles converging to the limit circle .
From this subspace, imagine building inductively by creating larger and larger bouquet’s of circles parameterized by countable ordinals. In the limit, the circles will “converge” to in the sense that if is an open neighborhood of , then there exists a such that .
Sorry, Not Sorry: The space is compact and it’s locally path connected at , but it’s not locally path-connected at all of its points. I take no responsibility for this. Countable limit ordinals like are to blame. Fortunately, we’re interested in the topology around so there is no harm in sweeping this under the rug. If you’re unhappy about it, remember that you can take the locally path-connected coreflection without any loss of homotopy/homology group data. The downside to doing so is that the coreflection is not compact. So it goes.
Observation 1: is not semilocally simply connected. In particular, .
Proof. Since is a quotient space of and is compact, for every neighborhood of , there is a countable ordinal such that . Also, retracts onto each circle . Therefore, the loop traversing is contained in and is not contractible in . Since ordinals are totally path disconnected, every path component of is an open interval. Therefore, is semilocally simply connected at every point in .
Observation 2: is finitary, i.e. .
Proof. Since , if contains a point, it must be . Consider any map and suppose, to obtain a contradiction, that is not null-homotopic for each circle , of . In order for this to happen, it must be the case that, for each , there exists some arc of the circle that maps onto some circle .
In particular, we may find such that is the image of the point in the circle (this is the point on furthest from/antipodal to ). Since in , the continuity of tells us that the sequence converges to in .
Maybe you already see the problem. Remember the one thing I said we’d need to use about ? There is no sequence of countable ordinals converging to . But if is an arbitrary neighborhood of in , then we may take to be the open neighborhood of , which is the image of in . The only way for the sequence to eventually be in is for the sequence of countable ordinals to eventually be inside . This implies ; a contradiction of the topology of . Since is not in , this set must be empty.
Having made Observations 1 and 2, we conclude that even though fails to have a simply connected covering space, it’s fundamental group is completely tame. In fact, from a fundamental group perspective, it’s basically the same as an ordinary wedge of circles , which is a CW-complex! Intuitively, the neighborhood base at is so large (more precisely, the “tightness” is so large) that shrinking sequences of loops don’t actually converge to the constant loop at and this sequential convergence is exactly what is required to have an infinitary fundamental group.
If you put together the ideas behind the proofs of Observations 1 and 2, it’s not too hard to prove the following more detailed results:
Theorem: The continuous identity function from the ordinary (i.e. CW) wedge of circles to is a weak homotopy equivalence.
Corollary: is canonically isomorphic to the free group on uncountably many generators.
Here’s a fun exercise I’ll leave you with.
Challenge Exercise: Construct a locally path-connected space for which is not closed.