I recently showed in “The Griffiths double cone group is isomorphic to the triple” (see the arXiv preprint here) that the fundamental group of the Griffiths space is isomorphic to that of the harmonic archipelago. Jeremy Brazas kindly invited me to give a roadmap of the essential ideas through a series of guest posts on his blog. The first of the three posts can be accessed here. Enjoy!
In this post I’ll present some results and ideas from the paper The number of homomorphisms from the Hawaiian earring group, 2019. First I’ll give some motivating background and historical remarks, and then state the main result and concepts used in its proof.
The question “How many?” is an old one, and the ideas advanced by Cantor gave this question meaning in the infinite setting. Where previously an answer of “Infinitely many” was as descriptive as necessary, one can now ask the followup question “Which infinite?” When the answer is “as many as is naively allowable” then you get a sense of abundance. For example one might wonder how many different ultrafilters exist on an infinite set . There are collections of subsets of . Pospisil showed that there are ultrafilters on (see Remark on bicompact spaces, 1937) and since we have the greatest number imaginable.
When the question is “How many group homomorphisms are there from to ?”, a very naive upper bound is the total number of functions from to , namely . For example there are group homomorphisms from to and this is the same as the number of functions from to . One can see this by picking a basis for as a vector space over and considering all possible maps from basis elements to .
When considering other groups the answer can become more subtle. Recall that the Hawaiian earring is the space where is the circle of radius centered at the point . Its fundamental group, which we denote , is uncountable and not free; in fact any homomorphism to a free group will have finitely generated image. For each the group has a homomorphic retraction to a free subgroup of rank which is induced by the topological retraction map which takes all points which are not in the outermost circles of to the point .
A group is n-slender if for every homomorphism there exists some for which (see posts Automatic continuity parts 2 and 3). Right-angled Artin groups, Baumslag-Solitar groups, and torsion-free word hyperbolic groups are examples of n-slender groups, and groups containing torsion or a subgroup isomorphic to are not n-slender. A rough intuitive way of understanding an n-slender group is that homomorphisms from to can only be superficial- they cannot involve the infinitely deep parts of .
Curiously there exist models of Zermelo-Fraenkel set theory plus the axiom of dependent choices in which if is not n-slender we have (see Deeply concatenable subgroups might never be free, with Shelah). In particular, any homomorphism which witnesses that a group of cardinality less than is not n-slender is somehow “nonconstructive.” From this one suspects that such a homomorphism probably relies on performing many choices and is therefore not unique in witnessing n-slenderness (compare the construction of a non-principal ultrafilter). It turns out that there are as many homomorphisms from to a small cardinality group which is not n-slender:
Theorem If is a group with then
The requirement that is essential here, since is not n-slender (consider the identity map), , and (this latter equality follows from Corollary 2.11 of Eda’s Free -products and fundamental groups of subspaces of the plane, 1998). It was previously known that if is a nontrivial finite group then the number of surjections from to is (see Conner and Spencer, Anomalous behavior of the Hawaiian earring group, 2005). From more recent results one sees more generally that if is a compact Hausdorff topological group there exist many homomorphisms from to (using theorems from Tlas, Big free groups are almost free, 2015 and Zastrow, The non-abelian Specker group is free, 2000).
It is straightforward to show that if is n-slender then . The challenge lies in producing many homomorphisms from the existence of just one nonconstructive homomorphism. The trick for this lies in using the Harmonic archipelago (this space appeared in Bogley, Sieradski, Universal path spaces). This space can be imagined as taking the disc and from each subdisc you form a hill of height . This gives a space whose fundamental group is uncountable and onto which can be naturally surjected.
First, one shows that if and is not n-slender then there exists a nontrivial homomorphism (this fact was told to me by Greg Conner). Next one shows that there exist -many homomorphisms for which differs for distinct . This is the most difficult part of the argument and involves manipulation of infinitary words. Once one has shown this the theorem follows.
In this installment I’ll present some newer automatic continuity results in the spirit of Specker’s theorem. Recall that a group is cm-slender (respectively lcH-slender, n-slender) if every abstract group homomorphism from a completely metrizable topological group (resp. locally compact Hausdorff topological group, the Hawaiian earring group) to has open kernel (see Part 2). A group satisfying any of these slenderness notions must be both torsion-free and not contain as a subgroup.
On initially understanding Higman’s proof that free groups are n-slender, one notices that the proof can be applied to a wider range of groups. The proof utilizes the fact that the natural length function on a free group satisfies certain nice properties, together with a diagonalization argument. We’ll say a function is a length function provided , and . A length function is uniformly monotone (abbrev. u.m.) provided there exists some for which implies . Clearly a u.m. length function cannot exist if has torsion or more generally if has a nontrivial element which has infinitely many roots (i.e. for infinitely many there exists for which ). In the case of a free group one can use constant . The existence of a u.m. length function implies that a group is n-slender by Higman’s argument. Dudley (in Continuity of homomorphisms, 1961) used a different sort of length function (which he called a norm) to prove slenderness results about free (abelian) groups.
One can show that torsion-free word hyperbolic groups always have a u.m. length function (see Torsion-free word hyperbolic groups are noncommutatively slender, 2016), and are therefore n-slender. A u.m. length function argument also shows that the class of n-slender groups is closed under taking graph products (not just free products or direct sums).
By taking a still more general approach one can prove many more automatic continuity theorems. In the following three paragraphs I’ll describe some results with Greg Conner in A note on automatic continuity, 2019. Given a group , subset , and we let . A limiting sequence pair for a group is a pair of sequences and with and such that for every natural number :
A group with a limiting sequence pair is cm-, lcH-, and n-slender (see Theorem A). This gives a plethora of automatic continuity results since each of the following has a limiting sequence pair (see Theorem B):
- groups with a Dudley norm (see above in this post)
- groups with uniformly monotone length function
- countable torsion-free groups with finite roots
- Baumslag-Solitar groups
- Thompson’s group
A few other such groups could be mentioned, but this list gives a sense of the breadth of the class of slender groups and of the applicability of the notion of limiting sequence pair. In this paper, it is also shown that the class of cm-slender groups, the class of lcH-slender groups, and the class of n-slender groups are closed under taking graph products.
A class of groups missing from this list is the torsion-free one-relator groups. By strengthening a lemma due to Newman (Some results on one-relator groups, 1968, Lemma 2) and inductively applying a result of Nakamura (Atomic properties of the Hawaiian earring group for HNN extensions, 2015) I show that such groups are n-slender (Root extraction in one-relator groups and slenderness, 2018). This proves a conjecture in Nakamura’s paper.
Some other groups which are reasonably close to being free groups are also slender. For example, if is a group in which every countable subgroup is free (so-called -free) then is cm-, lcH- and n-slender (see Automatic continuity of -free groups, to appear). With Ilya Kazachkov we also show that various types of braid groups are slender (On preservation of automatic continuity, 2019). More recently with Oleg Bogopolski (An atomic property for acylindrically hyperbolic groups) we prove that for any abstract homomorphism with an acylindrically hyperbolic group and either completely metrizable, locally compact Hausdorff, or the Hawaiian earring group there exists a neighborhood of identity of the domain which maps into the set of elliptic elements. From this, one can deduce that a torsion-free relatively hyperbolic group over a collection of n-slender groups is also n-slender, and similarly for a cm- or lcH-slender group provided the group is of cardinality less than .
So far we have seen some examples of discontinuous homomorphisms between topological groups. In this contrasting post we will review some automatic continuity theorems. Our focus in this and the subsequent post will be on automatic continuity results of a particular strong flavor: maps with open kernels. For a very nice survey which includes other forms of automatic continuity (including measure theoretic) one may consult Rosendal’s Automatic continuity of group homomorphisms, 2009.
One classical motivating theorem, which provided a pattern that has been used time and again in later results, is due to Ernst Specker (see Additive Gruppen von folgen Ganzer Zahlen, 1950):
Theorem For each homomorphism there exists some such that , where is projection to the first coordinates.
The group can be considered a topological group by giving each coordinate group a discrete topology and the group the product (Tychonov) topology. The theorem can be equivalently read as: any homomorphism from the topological group to the discrete group has open kernel (and is therefore continuous). One nice corollary to Specker’s theorem is the fact that the group is not free abelian (originally this was shown by Reinhold Baer in Abelian groups without elements of finite order, 1937). To derive the corollary you note that is of cardinality so that if it were free abelian it would be of rank (i.e. a free abelian generating set would have this cardinality). This implies that the number of elements in is , but Specker’s theorem easily implies that is countable.
Specker’s theorem can be proved by contradiction using a nice diagonalization argument. A generalization of Specker’s result was proved several years later by Sasiada (Proof that every countable and reduced torsion-free abelian group is slender, 1959):
Theorem Let be an abelian group of cardinality less than . Then every homomorphism has open kernel if and only if contains no torsion and no subgroup isomorphic to .
The strongest result in this line is due to Nunke (see Slender groups, 1961): if is abelian then every homomorphism from to has open kernel if and only if does not contain torsion or include a subgroup isomorphic to , or the p-adic integers for any prime . Abelian groups satisfying either of these equivalent properties are called slender.
Leaving abelian groups, one can consider automatic continuity with regard to fundamental groups. The prototypical space with wild fundamental group is the Hawaiian earring. This space can be seen as a shrinking wedge of countably infinitely many circles. More formally we let denote the circle in the plane centered at point of radius . The Hawaiian earring is the subspace . Although this space superficially looks like a countably infinite bouquet of circles, its fundamental group (which we denote ) is not free and is in fact uncountable. The space can be retracted to the wedge of the outer circles by mapping all points in which are not on this outer wedge to the point . This continuous function induces a homomorphic retraction from the group to a subgroup which is isomorphic to a free group of rank . There is a complementary subgroup which is the image of the retraction given by mapping the outer circles to the point . One has the free product decomposition .
The group can be understood as a nonabelian fundamental group analogue of . The infinite torus has fundamental group isomorphic to , and the homomorphic retractions can be induced by topological retractions of to where is some distinguished point.
We give some definitions which hearken back to Specker’s theorem: a group is noncommutatively slender (abbrev. n-slender) if for every homomorphism there exists some for which (see Eda’s Free -products and noncommutatively slender groups, 1992). Alternatively is n-slender if for each homomorphism from to the kernel includes for some . We’ll say a group is completely metrizable slender (abbrev. cm-slender) if any abstract group homomorphism from a completely metrizable topological group to has kernel which is open. Analogously define a group to be locally compact Hausdorff slender (abbrev. lcH-slender) if every abstract group homomorphism whose domain is a locally compact Hausdorff topological group and whose codomain is must have open kernel.
Graham Higman proved that free groups are n-slender, though he did not use this precise terminology (see Theorem 1 of Unrestricted free products and varieties of topological groups, 1952). By using a modification of Higman’s proof, Dudley showed that free (abelian) groups are n-, cm-, and lcH-slender (Continuity of homomorphisms, 1962). I’ll finish this post by providing a mini-catalogue of related results:
- If then for any abstract homomorphism with locally compact either is open or for some and (Morris and Nickolas, Locally compact group topologies on an algebraic free product of groups, 1976, Theorem 3). This immediately implies that the class of lcH-slender groups is closed under taking free products.
- Same as Morris and Nickolas, except is a completely metrizable topological group (Slutsky, Automatic continuity for homomorphisms into free products, 2013, Theorem 5.6). As a result, the cm-slender groups are closed under free products.
- If and is an abstract homomorphism then for some , and we have (Eda, Atomic property of the fundamental groups of the Hawaiian earring and wild locally path-connected spaces, 2011, Theorem 1.3). In particular the class of n-slender groups is closed under free products; this latter fact, as well as the closure of the class of n-slender groups under direct sums, was proved earlier by Eda (Free -products and noncommutatively slender groups, 1992, Theorem 3.6). In this latter paper it was also shown that the abelian n-slender groups are precisely the slender groups (Theorem 3.3).
- Baumslag-Solitar groups are n-slender (Nakamura, Atomic properties of the Hawaiian earring group for HNN extensions, 2015).
In the next post I’ll provide some other slenderness results due to myself and coauthors.
Given two mathematical objects and and a function between them, it is common to ask whether some or all of the mathematical structure of the objects is respected by the function . If the objects and are simply abstract sets then of course any function preserves the structure. When and are groups then the question as to whether preserves the underlying mathematical structure is resolved by considering whether for all ; that is, one checks whether is a homomorphism. More generally when the objects are rings, modules, universal algebras, or relational structures the relevant question is whether is a homomorphism (or possibly a strong homomorphism). When the objects are topological spaces one can ask whether is continuous. As objects become more laden with structure, questions regarding such a map can become more interesting or complicated.
Suppose the objects are topological groups- that is groups having a topology for which multiplication the multiplication function given by and the inverse function given by are continuous. One can ask whether preservation of the algebraic structure implies preservation of the topological structure: Is a group homomorphism continuous? It would be unimaginable to expect each group homomorphism to be continuous, irrespective of the domain and the codomain. As a cheap example one can take and to be the same group but with having a finer topology (more open sets) than . The identity map is an isomorphism but is obviously not continuous. More concretely we can let and be the additive group on the real numbers and endow with the standard topology and with the discrete topology.
For a more interesting example one can take the domain and codomain to again be the additive group on the reals and endow both with the standard topology. By considering as a vector space over the field and selecting a basis including , we can express the group isomorphism . We may project to the coordinate associated with the basis element , giving us a homomorphic retraction . This homomorphism is obviously not continuous since the domain is connected and the image is totally disconnected. More generally one can prove (see Srivastava’s A Course on Borel Sets, Exercise 3.5.11) that any continuous (or Borel measurable, or even Baire measurable) homomorphism from to itself is given by multiplication by a constant. Thus one obtains many examples of discontinuous automorphisms of by permuting the coordinates of .
For another example, suppose that the mathematical structures are path connected topological spaces and and we have an abstract group homomorphism . It is natural to ask whether there exists a continuous function such that equals . This gives a slightly different sense of continuity of a homomorphism. In the next post we will give some concrete examples of cases where homomorphisms are “automatically continuous.”
In this post I give an inadequate review of the axiom of choice and explain the results contained in the paper On quasi-isometry and choice.
Some proofs in mathematics are nonconstructive in the sense that no obvious algorithm or pattern can be used to produce them. When one is working with a well-ordered set then such a specified transfinite algorithm can usually be exhibited, but the production of such a well-order is itself often nonconstructive. The axiom of choice is the assertion that given a nonempty collection of nonempty sets there exists a function such that , and is equivalent (modulo the Zermelo-Fraenkel axioms ZF) to the assertion that every set can be well-ordered.
In its earliest history (late 19th-early 20th century) the axiom was met with skepticism. To the mathematician used to explicit constructions, the ability to create such an abstract function, or alternatively to produce a well-ordering of arbitrary sets, is a tremendous leap in intellectual certitude. Worse still, the axiom produces objects in classical geometry which seem pathological. For example, one can construct subsets of which are not Lebesgue measurable and do not have the property of Baire. The Banach-Tarski paradox is another striking example of such pathology. The axiom gained sympathy and acceptance as mathematicians realized that some earlier theorems were proved using its implicit use. Also, many propositions which one would want to be true are demonstrably equivalent to the axiom of choice. These include Tychonov’s Theorem (due to Kelley), every vector space has a basis (Blass), every connected simplicial graph contains a maximal tree, and Zorn’s Lemma.
A less well known statement equivalent to choice comes from coarse geometry. If and are metric spaces, a function is a quasi-isometry if there exists such that and for all we have , where is the closed neighborhood . Metric space is accordingly quasi-isometric to provided such a function exists. This notion is used very frequently in geometric group theory, where selecting different finite generating sets for a finitely generated group may produce nonisometric Cayley graphs which are nevertheless quasi-isometric. A composition of quasi-isometries is easily a quasi-isometry and the identity map is a quasi-isometry, so the relation is transitive and reflexive.
It is a standard exercise to show that quasi-isometry is also symmetric, but it is quickly realized that the axiom of choice is blatantly used in the “going backwards” map. With some thought, one can cook up a situation in which a quasi-isometry going in the other direction produces a choice function. Thus the symmetry of the quasi-isometry relation is equivalent (modulo ZF) to the axiom of choice. This is somewhat harder to see if both spaces are geodesic hyperbolic.
Recall that a metric space is geodesic if for any two points there is an isometric embedding with and (the image of which is a geodesic segment). Geodesic segments need not be unique, but a choice of geodesic segment for points will be denoted . A geodesic space is –hyperbolic if for any we have and is hyperbolic if it is -hyperbolic for some . In the paper On quasi-isometry and choice it is shown that the claim that quasi-isometry is symmetric between geodesic hyperbolic spaces is equivalent to the axiom of choice (see Theorem 1). This theorem is sharp since if both of the geodesic hyperbolic spaces are -hyperbolic (that is, are -trees) then symmetry follows from ZF alone, with no axiom of choice required (Theorem 3). A so-called Bottleneck Theorem of Jason Fox Manning also implies the axiom of choice (Theorem 4). I give a very brief sketch of how the proof goes.
Given a collection of nonempty sets one can produce a tree-like graph having at most two edges between vertices and no loop (edge from a vertex to itself). The graph is -hyperbolic, has a root and one infinite “arm” protruding from the root for each . If is a quasi-isometry from a simplicial tree to then one can prune to a subtree from which the restriction can be used to define a choice function. The main difficulty in the proof lies in showing that the pruned satisfies key properties which make a selection from each unique. Chasing natural number parameters is required.
First order logic has provided useful tools to mathematicians over the last several decades. We give some general motivation for interest in the first order theory of a structure, give some illuminating examples, and discuss some new results which are spelled out in Limiting theories of substructures.
One of the overarching themes in mathematics is to understand when two mathematical structures are essentially “the same.” This sameness is often characterized by the existence of a bijection between the elements of the structures which preserves the essential characteristics of the structures. Homeomorphisms of topological spaces, isometries of metric spaces, isomorphisms of groups (rings, fields, algebras, etc.), and order preserving bijections between ordered spaces are examples. It is useful to produce characteristics (invariants) of a structure which are preserved under a map which witnesses sameness. Invariants give sufficient conditions from which one may assert that two structures are not the same. For example, a topological space with two path components cannot be homeomorphic to a space which has one. Having a point named is not a homeomorphism invariant. The most naïve invariant of a structure is its cardinality, for if sameness is characterized via a bijection then structures of differing cardinality cannot be the same.
Another invariant of a structure is its first order theory. Without getting into the gory details of languages, well-formed formulas and sentences, the first order theory of a structure is essentially the set of finitary logical sentences which are true of the structure. It will hopefully suffice to give some examples of structures and first order sentences satisfied by them. We let denote the first order theory of a structure .
We consider a group as a set with binary operation , unary inverse operation , and nullary operation which satisfies the standard requirements that define groups: , , and . For abelian groups the binary operation is often denoted and the nullary operation is often denoted . The two element abelian group satisfies the requirements for a group as well as the sentences
Sentence 1 states that all elements have order at most two, sentence 2 states that there are at least two elements, and sentence 3 states that there are not three elements. Any group of exactly two elements is isomorphic to . More generally the first order theory of a finite group totally determines its isomorphism class: if is a finite group and then is isomorphic to . This is because one can spell out the multiplication table of a group using a finitary sentence. Letting and be defined by , the following sentence totally determines up to group isomorphism:
We have seen that first order logical sentences can express ideas like “there exist elements” or “there do not exist elements.” Some other concepts that can be expressed using first order logic are: “every element is twice an element” (which would allow you to distinguish the group from ), “there exists a non trivial element of order two” (which allows you to distinguish from ), and “the group is nilpotent of height .” Thus, many algebraic properties may be expressed using first order logic and it might seem that first order logic can be used to distinguish any two nonisomorphic groups. It turns out that this latter sentiment is not only false but extraordinarily false. We’ll state a corollary to what is called the Upward Löwenheim-Skolem Theorem:
Corollary If a structure utilizes only countably many non-logical symbols and has countably-infinitely many elements then for every infinite cardinal number there exists a structure having exactly elements such that .
Continuing our group theory examples, we note that groups have only finitely many non-logical symbols- the binary operation symbol, the inverse symbol, and the group identity symbol. Thus by this corollary, for any countably infinite group there exist groups of all infinite cardinalities which have the same first order theory (most of which are clearly not isomorphic to .) For example, there exist uncountable groups which have the same first order theory as the infinite cyclic group . This highlights that the property “being infinite cyclic” cannot be expressed using first order logic, since cyclic groups must be countable.
We move to a discussion of direct limits and of how the theories of a system of substructures can limit to the theory of the large structure. Recall that a directed set is a partially ordered set such that for any there exists with . Let be a structure ( is the set of elements, is the set of functions and is the set of relations). We’ll say for the purposes of this discussion that a collection of substructures of has direct limit if and implies . In such a situation, since the non-logical symbols in are the same for as for its distinguished substructures , it is natural to compare to for each . It might be that is precisely the same as each , but it is often the case that the theory of the large structure is different from that of each of the substructures. For example, if has infinitely many elements and each of the has only finitely many elements, then since first order sentences can distinguish a finite structure from an infinite. Even so, it may be that the theories get closer and closer to in a way that we make explicit.
Let be the set of all first order sentences which can be made using functions in and/or relations in . Thus and for any it is either the case that or that , and similar statements hold for . Let
Clearly the inclusion holds, and if we write and say that the limit exists. We illustrate with some examples.
Example 1 Consider the structure where is the standard ordering on and if and only if is even. The finite substructure whose set of elements is satisfies the sentence if and only if is even. Letting under the natural ordering it is clear that is the direct limit of the . We see that is strictly included in and the limit does not exist.
Example 2 Consider the abelian group structure . Let be an enumeration of the prime numbers and for each let be the subgroup generated by . Again is the direct limit of the . Each is isomorphic to the group and so certainly the limit exists since is constant.
Example 3 Consider the structure where is the standard ordering on . Let . Clearly is a directed set under set inclusion, and letting be the substructure having as its elements it is clear that is the direct limit of . Each structure , and also , is a nonempty dense linear order with no first or last point, and it is classically known that all such structures have the same first order theory. Thus exists.
So far we have seen an example where the first order theory of the substructures did not have a limit, an example where the first order theory of the substructures did have a limit which was not equal to the theory of the large structure (since ), and an example where the first order theories of the substructures have a limit which is equal to the theory of the large structure. We give a theorem which gives sufficient conditions under which the theories of substructures has a limit, and under which the limit of their theories is equal to the theory of the large structure (see Theorem A in Limiting theories of substructures):
Theorem Suppose is a collection of substructures of such that is the direct limit of .
(1) If for each finite set of constants there exists an such that for and there exists an automorphism fixing each and such that the limit exists.
(2) If in addition to (1) for each finite set of constants there exists an such that for there exists an automorphism fixing each and such that then .
What we have stated is a bit weaker than the statement of Theorem A in Limiting theories, but it is sufficient for our discussion. The theorem is proved in a more general setting, by induction on the number of quantifiers used in a sentence. This result can obviously be applied only in settings where structures have many automorphisms, that is, structures where there are automorphisms which can fix some prescribed finite subset and move some other finite subset into a designated structure. For example, in a structure which is simply an infinite set (no non-logical symbols), the hypotheses of both conditions (1) and (2) apply to the substructure net of finite subsets and we get that the theory of an infinite set is the limit of the theories of increasingly large finite sets. Similar conclusions regarding infinite abelian groups of prime exponent and infinite rank free abelian groups can be made. Moreover, the fact that the theory of a finite set is never equal to that of an infinite yields a new proof of the classical result that the theory of an infinite set is not finitely axiomatizable (this can otherwise be proved using either the Compactness Theorem or quantifier elimination). Similarly, the first order theory of an infinite abelian group of prime exponent and the first order theory of an infinite rank free abelian group are not finitely axiomatizable.
In this third and final installment I’ll provide some more theorems regarding the fundamental group and first homology that can be derived using descriptive set theory. We have already seen in Part 2 that there are quick and easy dichotomies that can be drawn using either selection theorems from logic or a diagonalization using the Baire category theorem. By working a little harder we get the following results, which can be interpreted as compactness theorems (see On definable subgroups of the fundamental group Theorem B, and On subgroups of first homology Theorem 4, respectively):
Theorem 3.1 If is a Peano continuum there does not exist a strictly increasing sequence of analytic normal subgroups of such that .
Theorem 3.2 If is a Peano continuum with of cardinality there exists and such that any loop of diameter is of commutator length .
Theorem 3.1 implies that for a Peano continuum, if there is no finite subset such that the normal closure is , then for any countable we again have . There exist Peano continua with uncountable fundamental group which is normally generated by one element, and there exist Peano continua which are not finitely (ergot not countably) normally generated. Theorem 3.1 has numerous other applications to first homology (see On subgroups of first homology). For Theorem 3.2 we recall that the commutator length of an element of a group is the smallest number such that may be written as a product of commutators, and defined to be provided is not in the derived subgroup of . One can understand Theorem 3.2 by its contrapositive: if there are arbitrarily small loops of arbitrarily high commutator length, then first homology is of cardinality . This is a strengthening of a result of Greg Conner and myself: a Peano continuum has first homology of cardinality if there exist arbitrarily small loops of infinite commutator length (see On the first homology of Peano continua).
These ideas also lend themselves to very simple characterizations of topologically defined subgroups of the fundamental group. For example, the shape kernel of a path connected space is the intersection of the kernels of all maps to nerves over . Thus, the shape kernel represents the set of loops which unavoidably perish when one maps to a simplicial complex. One gets the following characterization of this subgroup under some mild assumptions (see Definable, Theorem 5.1):
Theorem 3.3 If is a metric space which is path connected, locally path connected then the shape kernel is the intersection of all open normal subgroups of .
This gives one a very short, clean proof of the following (ibid Theorem 5.4):
Theorem 3.4 If is a metric space which is path connected, locally path connected then the shape kernel is equal to the Spanier group.
This is a slightly weaker result than that of Brazas and Fabel (see their work Thick Spanier groups and the first shape group), where they give the same conclusion and instead of “metric” they assume “paracompact.”
One can cook up easy examples of fundamental groups which have a “nonconstructive” decomposition into a direct sum. For example, by taking a countably infinite product of projective places one obtains a Peano continuum which has fundamental group isomorphic to . It is not difficult to find subgroups and such that and is not Borel, analytic, or anything close to nice. The situation with free product decompositions is much nicer (Definable, Theorem C):
Theorem 3.5 Suppose is path connected locally path connected Polish and with each nontrivial. The following hold:
- Each retraction map has analytic kernel.
- Each is of cardinality or .
- The map has analytic kernel.
Local path connectedness cannot be dropped from the hypotheses, else one can violate the first three conditions.
Certain decompositions into products also behave nicely. Recall that a group is noncommutatively slender group if for any homomorphism from the fundamental group of the Hawaiian earring there exists such that (here is the induced retraction to the outer circles). In other words, any homomorphism from the fundamental group of the Hawaiian earring to a noncommutatively slender group kills all circles of sufficiently small size. Free groups, free abelian groups, Thompson’s group , Baumslag-Solitar groups and torsion-free word hyperbolic groups are examples of noncommutatively slender groups (see A note on automatic continuity). We have the following:
Theorem 3.6 If is a path connected, locally path connected -Lindelof metric space and is a collection of nontrivial noncommutatively slender groups with then there is no epimorphism .
If one assumes the generalized continuum hypothesis then the noncommutative slenderness of the groups can be dropped, since the product would simply be of higher cardinality than the set of all loops in the space . However the noncommutative slenderness cannot be dropped in general: in a universe where you can take to be the countably infinite product of projective planes and notice that .
I’ll conclude by noting that extra set theoretic assumptions can enhance the results that one derives. Hypotheses about analyticity of a subgroup of a fundamental group can be relaxed (in the statement of Theorem 3.1, for example) to be or or whatever other semi-algebra is guaranteed to be in the algebra of subsets satisfying the property of Baire.
In this installment I’ll give a brief review of some fundamental concepts of descriptive set theory. For further information I recommend A Course on Borel Sets by Srivastava, or Classical Descriptive Set Theory by Kechris. Some results about subgroups are then outlined.
Descriptive set theory is the study of subsets of a nice space that have a topological description. The nice space is generally Polish (separable, completely metrizable.) One of the initial motivating questions in descriptive set theory was the continuum hypothesis- the question of whether there exists an uncountable set of cardinality less than . Cantor was able to show that a closed subset of the real line cannot be a counterexample to the continuum hypothesis (this follows from the Cantor-Bendixson theorem). What the proof really shows is that no Polish space can be a counterexample to the continuum hypothesis (topology hadn’t come into its own at this point, and so Polish spaces as such weren’t yet a subject of study). To describe further results will require more definitions.
Recall that a -algebra on a set is a subset of the powerset of which contains and is closed under countable unions and under complementation. By considering the intersection of all -algebras containing a collection of subsets of , we see that for any collection of subsets of there is a minimal -algebra containing that collection. If is a topological space we let denote the class of Borel sets– the smallest -algebra containing the open subsets of . Thus open sets, closed sets, countable unions of closed sets, etc., are Borel sets. When is a metric space, the open sets are a countable union of closed sets and the closed sets are a countable intersection of open sets. In this case the Borel sets can be arranged into a nice hierarchy of complexity classes. Let denote the collection of open sets (the topology) of , denote the collection of closed subsets and denote the set of clopen subsets. For we let denote the collection of countable unions of sets in , denote the collection of countable intersections of sets in , and . A transfinite induction shows that , and is seen by proving that is a -algebra. These complexity classes arrange nicely into an array, with inclusions going left:
For example .
Polish spaces are either finite (and therefore discrete), countable, or of cardinality . In case a Polish space is countable we know that every subset is a as a countable union of singletons, and thus the Borel hierarchy stabilizes and is not especially interesting. In case a Polish space is uncountable we get a non-stabilizing hierarchy, with strict containments going left (i.e. .) In a Polish space, all Borel sets are either countable (including finite) or contain a homeomorph of the Cantor set. For example, no Borel subset of can violate the continuum hypothesis.
While Borel sets are closed under many set-theoretic operations, they are not closed under continuous images. There exist Polish spaces , a closed subset , and a continuous function such that is not Borel in . Such a set is called analytic, and the class of analytic sets is denoted . The class consisting of complements of analytic sets is denoted , and we denote . Since the identity map is continuous we see that every Borel set in a Polish space is . That all sets are Borel is a theorem of Souslin (Sur une definition des ensembles B sans nombres transfinis, C. R. Acad. Sciences, Paris, 164 (1917), 88-91.) The notation suggests that we will define a hierarchy, in this case we shall only use subscripts in the natural numbers and not in all of . If have been defined for all we let denote the class of continuous images of sets in , is the class of complements of sets in and . A similar arrangement, with strict inclusions in case the Polish space is uncountable, holds for these so called projective pointclasses:
As some points of interest, sets are either countable or include a homeomorph of the Cantor set, and so are never a counterexample to the continuum hypothesis. The , and therefore also the , subsets of are always Lebesgue measurable. The sets may be countable, of cardinality , or of cardinality and there exist models of ZFC modelling in which there exists a set of cardinality . Other strange phenomena happen further up the hierarchy in various models of ZFC.
Let’s move on to some information on subgroups. We shall assume is a path connected Polish space for what follows and sometimes add additional hypotheses when stating theorems. Recall that a subgroup is open, closed, Borel, etc. provided the set of loops is an open, closed, Borel, etc. subset of the loop space under the metric. We first notice that many of the very natural subgroups of are analytic (we state a specialization of Theorem 3.10 in On definable subgroups of the fundamental group):
Proposition 2.1 Let be a continuous function between Polish spaces. The following hold:
- If is then is .
- If is then is .
- The subgroups and are analytic in .
- If are then so are and .
- Countable subgroups of are .
- If is then so is .
- If is then so is any conjugate of .
- If is a reduced word in the free group and the groups are then so is the subgroup .
- If are then so is the subgroup .
- If is then each countable index term of the derived series and each term of the lower central series is .
Here are some more theorems (also from On definable subgroups of the fundamental group):
Proposition 2.2 Let .
- If is then the index is either or .
- If is then the index is either or .
Theorem 2.3 Suppose is locally path connected. The following groups are of cardinality or , and in case is compact they are of cardinality or are finitely generated:
- for any (derived series)
- for any (lower central series)
- where is the normal subgroup generated by squares of elements, cubes of elements, or -th powers of elements, or is simply any normal subgroup
In case is compact then countability of the fundamental group is equivalent to being finitely presented.
Some general remarks about these results are in order. Proposition 2.1 is verified using direct computation of topological complexity. Proposition 2.2 follows from a theorem of Burgess and a theorem of Silver. Theorem 2.3 is proved using Baire category. I give an example in Definable subgroups of a compact path connected subspace of for which there exists a model of ZFC which has and normal subgroup which is and for which is of cardinality . Thus in an appropriate model an intermediate cardinality can obtain and the hypothesis “locally path connected” in Theorem 2.3 (4 ) cannot be relaxed. The quotients in Theorem 2.3 are countable precisely when the normal subgroup is open, or in other words, when the subgroup has a covering space.
I’ll mention one more result (Theorem C from On subgroups of first homology). Recall that the Hawaiian earring is the subspace , with the circle centered at point of radius . We’ll say a set is true (respectively ) provided it is not (resp. ) and that a set is true provided it is not Borel.
Theorem 2.4 The fundamental group has normal subgroups of the following types:
- true for
- true for each
These normal subgroups can be selected so that the quotient groups are of exponent .
The fundamental group is a useful tool in understanding a topological space. In this installment I’ll present a very brief review of the concept of fundamental group, review the sup metric on the space of loops and introduce the notion of topological descriptions of subgroups of the fundamental group.
Recall that the fundamental group of a space with distinguished point , denoted , is the set of all loops based at modulo homotopy rel the endpoints. The binary operation is defined by loop concatenation, the trivial element is the homotopy class of the constant loop at , and inverses are given by taking the “backwards” loop. As any loop based at must stay within the path component of , fundamental groups are most frequently used in path connected spaces. For the remainder of this discussion I shall assume path connectedness of all spaces mentioned for which a fundamental group is computed, unless stated otherwise. Given a path in from to one obtains a natural isomorphism between and , and thus one often drops the distinguished point and speaks of the fundamental group , which is well defined up to isomorphism. A continuous map induces a homomorphism defined by letting . The fundamental group is a homotopy invariant and so if spaces and are homotopy equivalent there is an isomorphism given by the functions witnessing homotopy equivalence.
Example The fundamental group of a space consisting of one point is isomorphic to the trivial group, as is the fundamental group of any contractible space.
Example The fundamental group of the circle is isomorphic to the group . Intuitively, declare one of the directions around the circle to be positive and given a loop ask the net number of times that the loop has gone around the circle in the positive direction.
The fundamental group factors through products, meaning that . Thus one can compute the fundamental group of a two dimensional torus by noting that .
Many of the spaces that interest topologists are metrizable, and a metric on induces a topology on the set of loops which underlies the fundamental group. More precisely, let be a metric on the space and be a distinguished point. Let denote the set of all loops based at . One can topologize using the sup metric which defines the distance between loops to be . It is important to remember that we are not topologizing but rather the set .
Now one can define a subgroup to be open (respectively closed) if the set is open (resp. closed) as a subset of . Change of base point isomorphisms preserve open-ness and closed-ness. As is the case in topological groups, all open subgroups are also closed (the complement of an open subgroup is a union of left cosets of the group, which are also open subsets of .) If is also assumed to be separable then is also separable, and so any open subgroup of is of at most countable index by noticing that the set of left cosets gives a partition of into nonempty open sets. We recall that a space is semi-locally simply connected if for every and open neighborhood of there exists a neighborhood of such that the map induced by inclusion is the trivial map (here, might not be path connected). Among path connected, locally path connected spaces the condition of semi-local simple connectivity is equivalent to having a universal cover. We have the following (see On definable subgroups of the fundamental group, Proposition 2.10):
Fact If is metric and locally path connected the following are equivalent:
- The trivial subgroup of is open
- All subgroups of are clopen
- is semilocally simply connected
This shows, among other things, that if we want to find topologically fascinating subgroups in the fundamental group of a locally path connected space we need to consider a “wild” space (i.e. one that has no universal cover). As a matter of interest, we can say something a bit stronger (see Chapter 6 in my dissertation Subgroups and quotients of fundamental groups, Theorems 188.8.131.52 and 184.108.40.206):
Theorem If is metric, locally path connected and the following are equivalent:
- is open
- There exists a covering space associated to (i.e. )
To gain information about how sits inside of the loop space , it helps to have at least one further assumption on . If was assumed to be a complete metric then the sup metric on is also complete. Thus if is a separable completely metrizable space (a Polish space) we have an accompanying separable completely metrizable topology on . Descriptive set theory is the field concerned with understanding “topological complication” of subsets of a Polish space. Given a description of a subgroup using logic and/or topology it may be possible to know something about the index , and if is normal we might gain insight into the quotient .