aa r X i v : . [ m a t h . GN ] S e p ON DEFINABLE SUBGROUPS OF THE FUNDAMENTALGROUP
SAMUEL M. CORSON
Abstract.
We present several new theorems concerning the first fundamen-tal group of a path connected metric space. Among the results proven arestrengthenings of the main theorems of [Sh2] and [CoCo]. A compactnesstheorem for the fundamental group of a Peano continuum is given. A usefulcharacterization for the shape kernel of a locally path connected space is pre-sented, along with a very succinct proof of the fact that for such a space theSpanier and shape kernel subgroups coincide (see [BF]). We also show that afree decomposition of the fundamental group of a locally path connected Pol-ish space cannot be nonconstructive. Numerous other results and examplesillustrating the sharpness of our theorems are provided. Introduction
One way to understand a path connected topological space is to analyze itsfundamental group. Fundamental groups are a homotopy invariant and provide auseful tool for distinguishing homotopy equivalence classes. In understanding thefundamental group it is useful to study subgroups that are definable in terms oftopology or logic or some combination of the two. The focus of this paper is thestudy of the first fundamental group of metric spaces, and certain of its subgroups.Assumptions about separability and generalizations of separability, local path con-nectedness, and compactness will figure prominently in our study. Throughout thispaper we take the simplifying assumption that all spaces for which a fundamentalgroup is computed are path connected.In Section 2 we give preliminary definitions and then define the characterizationof subgroups of the fundamental group via the topology of the loop space. For G ≤ π ( X, x ) we study the relationship of G to π ( X, x ) by looking at how ⋃ G sits in the loop space L x . This very simple idea yields a diversity of theorems. Thetheory of open and closed subgroups which is developed in this section gives proofsof many results later in the paper (e.g. Theorems 5.1, 5.4, 6.5; Corollary 6.3).In Section 3 we introduce concepts related to separable completely metrizablespaces- Polish spaces. Some terminology and tools of descriptive set theory areintroduced, including the class of analytic (denoted Σ ) sets and generalizationsthereof (what we call nice classes of sets). The class of subgroups of type Σ enjoys many closure properties (detailed in Theorem 3.10). Certain techniques ofdescriptive set theory yield new theorems regarding the fundamental group andsome of its quotients (see Theorems 3.13 and 3.14), of which the following gives ashort catalogue of applications: Mathematics Subject Classification.
Primary 14F35; Secondary 03E15.
Key words and phrases. fundamental group, metric spaces, descriptive set theory.
Theorem A.
Suppose X is a locally path connected, connected Polish space. Thefollowing groups are of cardinality 2 ℵ or ≤ ℵ , and in case X is compact they areof cardinality 2 ℵ or are finitely generated:(1) π ( X ) (2) π ( X )/( π ( X )) ( α ) for any α < ω (derived series)(3) π ( X )/( π ( X )) n for any n ∈ ω (lower central series)(4) π ( X )/ N where N is the normal subgroup generated by squares of elements,cubes of elements, or n -th powers of elements.In case X is compact then countability of the fundamental group is equivalent tobeing finitely presented.The compact case of part (1) is the main result of [Sh2] and part (3) with n = for P ): Theorem B. If X is a Peano continuum there does not exist a strictly increasinginfinite sequence of P normal subgroups { G n } n ∈ ω of π ( X ) such that ⋃ n ∈ ω G n = π ( X ) .In Section 4 we present an application of Theorem B. We define a comonstergroup to be a group which is not the normal subgroup closure of any finite subsetand a κ -comonster group is not the normal subgroup closure of any set of cardinality < κ . Theorem B implies that if the fundamental group of a Peano continuum isco-monster then it is ℵ -comonster (Theorem 4.2). Examples are presented of thisphenomenon and a curious tie to finitely presented groups is drawn (Theorem 4.4).In Section 5 we compute the complexity of some commonly used subgroups ofthe fundamental group. The shape kernel is shown to be closed, and if the space islocally path connected the shape kernel is the intersection of all clopen subgroups(Theorem 5.1). The Spanier subgroup is shown to be equal to the shape kernel incase the space is locally path connected, and is shown to be Σ in case the space iscompact (Theorem 5.4). That the Spanier and shape kernel subgroups coincide forlocally path connected paracompact Hausdorff spaces is a recent theorem of Brazasand Fabel [BF]. Though our theorem is slightly less general, the proof is rathershorter than that of [BF].In Section 6 we introduce n-slender groups (see [E1]), give such groups an al-ternative characterization, and present some theorems using the theory of openand closed subgroups. Among the results of the section is the fact that a locallypath connected separable metric space cannot have fundamental group that is anuncountable free product of nontrivial groups (Corollary 6.8). We also prove thefollowing: Theorem C.
Suppose X is locally path connected Polish and π ( X ) ≃ ∗ i ∈ I G i witheach G i nontrivial. The following hold:(1) card ( I ) ≤ ℵ (2) Each retraction map r j ∶ ∗ i ∈ I G i → G j has analytic kernel.(3) Each G j is of cardinality ≤ ℵ or 2 ℵ .(4) The map ∗ i ∈ I G i → ⊕ i ∈ I G i has analytic kernel.This theorem can be interpreted to mean that no free decomposition of a funda-mental group as in the hypotheses can be non-constructive. This is rather surprising N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 3 in light of the fact that a direct sum decomposition of the fundamental group canbe non-constructive (see discussion in Section 6).In Section 7 we give a brief discussion of what are called nice pointclasses (theseagree with the P that is found in some of the theorems stated so far). We discusswhich pointclasses can be assumed to be nice (consistent with the standard axiomsof set theory), and hence to which subgroups we can consistently apply the theoremsof Sections 3 and 4.Our discussions will not avoid references to abstract combinatorial set theory.This should not be surprising as topology is ‘visual set theory.’ Also, many facetsof descriptive set theory are directly influenced by the model of set theory in whichone is working. We will keep our references to set theory simple, doing little beyondillustrating the sharpness of our results until the discussion in Section 7.Let ZFC denote the Zermelo-Fraenkel axioms of set theory including the axiomof choice. We assume
ZFC throughout this paper, and also assume that
ZFC isconsistent so as to avoid repetition of the phase “if
ZFC is consistent then thereexists a model . . . ” The parameter κ will be used for infinite cardinals, κ + denotesthe successor cardinal. Let CH denote the continuum hypothesis: ℵ + = ℵ , and GCH denote the generalized continuum hypothesis: (∀ κ )[ κ + = κ ] .2. Preliminaries
In this section we present some of the basic definitions and notation for funda-mental groups in this paper. We then give some lemmas about open and closedsubgroups of the fundamental group which will be used throughout.Given a topological space X and distinguished point x ∈ X we obtain the fun-damental group π ( X, x ) as follows. A loop based at x is a continuous function l ∶ ([ , ] , { , }) → ( X, x ) . Two loops l and l at x are homotopic if there ex-ists a continuous function H ∶ [ , ] × [ , ] → X called a homotopy such that H ( s, ) = l ( s ) , H ( s, ) = l ( s ) and H ( , t ) = H ( , t ) = x for all s, t ∈ [ , ] . Therelation defined by homotopy is an equivalence relation. Letting L x denote the setof all loops at x in X we have the binary operation concatenation, denoted ∗ , on L x defined by l ∗ l ( s ) = ⎧⎪⎪⎨⎪⎪⎩ l ( s ) if s ∈ [ , ] l ( s − ) if s ∈ [ , ] . This definition also works as apartial binary operation on paths, defined whenever the first path ends where thesecond path begins. For specificity, we mean l ∗ ( l ∗ ( ⋯ ∗ ( l n − ∗ l n ) ⋯ ) when wewrite l ∗ l ∗⋯∗ l n . There is also a unary operation − given by l − ( s ) = l ( − s ) . Thefundamental group is the set L x modulo homotopy, the binary operation is givenby [ l ] ∗ [ l ] = [ l ∗ l ] , the equivalence class of the constant loop is the identity andinverses are given by [ l ] − = [ l − ] . Clearly the fundamental group π ( X, x ) is thesame as the fundamental group of π ( C, x ) where C is the path component of x . We shall only consider fundamental groups of spaces which are path connected.
We assume some familiarity with notions in topology such as metrizability andseparability. Let Z be a topological space. A pointclass is a collection P of subsetsof Z that are of a particular topological description, usually in terms of countableunions, countable intersections, complements, or projections. For example, thecollection of open subsets (topology) of Z , the collection of closed sets of Z , and thecollection of countable unions of closed sets of Z are all pointclasses of Z . Anotherexample is the class of Borel subsets of Z . When we restrict our attention to specifictypes of topological spaces, we get more information about sets in pointclasses. SAMUEL M. CORSON
Take ( X, d ) to be a metric space with distinguished point x ∈ X . Topologize L x by the sup metric: the distance between loops l and l is sup s ∈[ , ] d ( l ( s ) , l ( s )) .Since uniform convergence is equivalent to convergence in the compact-open topol-ogy, we may suppress the particular metric d on the space X (since any othercompatible metric gives the same topology on L x ). Definition 2.1.
A subgroup G ≤ π ( X, x ) is of pointclass P if the collection ofloops belonging to elements of G is in the pointclass P in L x . In other words, G ≤ π ( X, x ) is of pointclass P if ⋃ G is in pointclass P in L x .We establish some lemmas. Lemmas 2.2, 2.3, and 2.4 should remind the readerof the analogous facts for topological groups. Lemma 2.2. If G ≤ π ( X, x ) is open and G ≤ H ≤ π ( X, x ) then H is open. Proof.
Let G be open and l ∈ ⋃ H with { l n } n ∈ ω a sequence in L x converging to l .Since l ∗ l − ∈ ⋃ G there exists ǫ > B ( l ∗ l − , ǫ ) ⊆ ⋃ G . The sequence { l ∗ l − n } n ∈ ω is eventually in B ( l ∗ l − , ǫ ) , so that { l ∗ l − n } n ∈ ω is eventually in ⋃ G ⊂ ⋃ H ,so { l − n } n ∈ ω is eventually in ⋃ H , so { l n } n ∈ ω is eventually in ⋃ H . (cid:3) Lemma 2.3. If P is closed under continuous preimages and H ≤ π ( X, x ) is P then:(1) The equivalence relations E, R ⊆ L x × L x defined by l El iff [ l ] H = [ l ] H and l Rl iff H [ l ] = H [ l ] are P .(2) Each equivalence class in E and R is P .By [ l ] H we mean the set of all loops based at x which are homotopic to a loop ofthe form l ∗ l ′ where l ′ ∈ ⋃ H and the definition for H [ l ] is analogous. Proof.
The function L x × L x → L x given by ( l , l ) ↦ ( l ) − ∗ l is continuous and E is the preimage of ⋃ H under this function, so by assumption we have E is P .The proof that R is P is similar. This proves (1). For (2) we notice that for a fixed l ∈ L x the function L x → L x given by l ↦ ( l ) − ∗ l is continuous and the set [ l ] H is the continuous preimage of ⋃ H . (cid:3) Lemma 2.4. If H ≤ π ( X, x ) is open then H is also closed. Proof.
Supposing H is open we have by Lemma 2.3 that the set ⋃ l ∉⋃ H [ l ] H is aunion of open sets in L x , and this is precisely L x ∖ ( ⋃ H ) . (cid:3) We notice that change of basepoint isomorphisms take open (resp. closed) sub-groups to open (resp. closed) subgroups, as seen in the following lemma.
Lemma 2.5.
Let x, y ∈ X and ρ a path from y to x . Let φ ∶ L x → L y be the mapsuch that φ ( l ) = ρ ∗ l ∗ ρ − and ψ ∶ L y → L x be given by ρ − ∗ l ∗ ρ . Then(1) φ and ψ are isometric embeddings and induce isomorphisms φ ∶ π ( X, x ) → π ( X, y ) , and ψ ∶ π ( X, y ) → π ( X, x ) .(2) G ≤ π ( X, x ) is open (resp. closed) iff φ ( G ) is.(3) G ≤ π ( X, x ) is open (resp. closed) iff every conjugate of G is. Proof.
The first part of (1) is clear, and the second is a standard exercise. For (2)suppose G is not open. Let l ∈ L x with [ l ] ∈ G and { l n } n ∈ ω be a sequence of loopssuch that l n → l and [ l n ] ∉ G . Then ρ ∗ l n ∗ ρ − → ρ ∗ l ∗ ρ − and [ ρ ∗ l n ∗ ρ − ] ∉ φ ( G ) ,so φ ( G ) is not open. If φ ( G ) is not open then by the proof for the other directionwe have that ψφ ( G ) = G is not open. N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 5
Suppose that G is not closed and let l ∈ L x be such that [ l ] ∉ G and there existsa sequence { l n } n ∈ ω such that [ l n ] ∈ G and l n → l . Then ρ ∗ l n ∗ ρ − → ρ ∗ l ∗ ρ − and [ ρ ∗ l n ∗ ρ − ] ∈ φ ( G ) and [ ρ ∗ l ∗ ρ − ] ∉ φ ( G ) . Again, for the other direction weconsider the application of the map ψ .The last claim is proved by letting ρ be a loop from x to itself and applying (2). (cid:3) By Lemma 2.5 we may consider open or closed normal subgroups as base pointfree.
Lemma 2.6. If G ⊴ π ( X ) is open there exists an open cover U of X such thatany loop contained entirely in an element of U is in ⋃ G . Proof.
For each point x ∈ X we have G ⊴ π ( X, x ) is open, and the constant loop c at x is in ⋃ G , so we may pick ǫ x > B ( c, ǫ x ) ⊆ ⋃ G . Selecting the ǫ x neighborhood B ( x, ǫ x ) around x gives the desired open cover U = { B ( x, ǫ x )} x ∈ X . (cid:3) The converse to the above lemma is not true in general. The space F in the nextexample will reappear in later examples in this paper. Example 1.
Let F = ⋃ y ∈ K C (( , y ) , y ) ⊆ R where K is a homeomorph of theCantor set that lies in the interval [ , ] and C ( p, r ) denotes the circle centered atpoint p of radius r . The space F can be considered a wedge of 2 ℵ many circlesof diameter ≥ ( , ) correspond to theelements of a Cantor set. This space is compact and the fundamental group iseasily seen to be isomorphic to the free group of rank continuum F ( ℵ ) (witha free generating set corresponding to a set of loops that go exactly once aroundone of the circles C (( , y ) , y ) ). Let P be any pointset defined on metric spaceswhich is closed under taking continuous preimages. Define a map f ∶ K → L ( , ) by letting f ( y )( t ) = ( y sin ( πt ) , y − y cos ( πt )) . It is clear for y , y ∈ K that d ( f ( y ) , f ( y )) = d ( y , y ) since d ( f ( y )( s ) , f ( y )( s )) is maximized precisely at s = and d ( f ( y )( ) , f ( y )( )) = d ( y , y ) . Then f is an embedding of K . Theimage f ( K ) gives a set of loops which freely generate the fundamental group. If G ≤ π ( F, ( , )) is of pointclass P then f − ( ⋃ G ) is as well.For any ∅ ≠ S ⊆ K we have a subgroup: ι ∗ ( π ( ⋃ y ∈ S C (( , y ) , y ) , ( , ))) ≤ π ( F, ( , )) freely generated by the loops in f ( S ) . The normal closure G = ⟨⟨ ι ∗ ( π ( ⋃ y ∈ S C (( , y ) , y ) , ( , )))⟩⟩ ≤ π ( F, ( , )) does not contain any elements of form [ f ( y )] where y ∈ K ∖ S since G is the kernelof the retraction map from π ( F, ( , )) to the free subgroup ι ∗ ( π ( ⋃ y ∈ K ∖ S C (( , y ) , y ) , ( , ))) ≤ π ( F, ( , )) Any loop in F contained in an open ball of radius is nulhomotopic, so there existsan open cover U satisfying the conclusion of Lemma 2.6 for any normal subgroup G ⊴ π ( F, ( , )) . Not every subgroup is open, however, by letting S ⊆ K be notopen and noticing that S = f − ( ⋃ ⟨⟨ ι ∗ ( π ( ⋃ y ∈ S C (( , y ) , y ) , ( , )))⟩⟩) is not open. SAMUEL M. CORSON
We present a partial converse to Lemma 2.6.
Definition 2.7.
A topological space Z is locally path connected if for every z ∈ Z and neighborhood U of z there exists a neighborhood V ⊆ U of z such that V is path connected. Lemma 2.8.
Let X be locally path connected and G ⊴ π ( X ) . If there exists anopen cover U of X such that any loop contained entirely in an element of U is in G then G is open. Proof.
Assume the hypotheses and fix x ∈ X . Let l ∈ ⋃ G ⊆ L x . Cover the image of l with a finite subcollection { U , . . . , U m } ⊆ U , so that the images of each inclusion ι ∗ ∶ π ( U i ) → π ( X ) are in G . Let δ > l by { U , . . . , U m } . Cover l with finitely many open balls { B , . . . , B k } of radius δ . Cover the image of l with finitely many path connected open sets { V , . . . , V q } , each of which is contained in one of the { B , . . . , B k } . Let ǫ be aLebesgue number for the covering { V , . . . , V q } of the image of l . Pick N ∈ ω sufficiently large so that for 0 ≤ n ≤ N − l ([ nN , n + N ]) is containedinside some V j n . Now assuming l ′ ∈ L x is less than distance ǫ from l we have that d ( l ′ ( s ) , l ( s )) < ǫ for all s ∈ [ , ] . For each 1 ≤ n ≤ N − p n be a path in V j n from l ( nN ) to l ′ ( nN ) and let p and p N be the constant path at x . Notice that theloop l ∣[ nN , n + N ] ∗ p n + ∗ ( l ′ ∣[ nN , n + N ]) − ∗ p − n is contained in one of the U i , and sois a representative of an element of G based potentially at a different point. Then l − ∗ l ′ is an element of ⋃ G , so l ′ ∈ ⋃ G . Thus G is open. (cid:3) For the next proposition we recall the following definition.
Definition 2.9.
A topological space Z is semi-locally simply connected if forevery z ∈ Z there exists a neighborhood U of z such that the map induced byinclusion ι ∗ ∶ π ( U, z ) → π ( Z, z ) is the trivial map. For a locally path connectedspace we may obviously select U to be path connected. Proposition 2.10.
Let X be locally path connected in addition to being metriz-able. The following are equivalent:(1) The trivial subgroup of π ( X ) is open.(2) All subgroups of π ( X, x ) are open.(3) X is semi-locally simply connected. Proof.
The implication (1) ⇒ (2) follows from Lemma 2.2. For (2) ⇒ (3) we let x ∈ X be given along with a neighborhood U of x . Since in particular the trivialsubgroup of π ( X, x ) is open and the constant map c ∶ [ , ] → { x } is trivial, wemay select ǫ > B ( c, ǫ ) ⊆ ⋃ [ c ] ⊆ L x , where without loss of generality B ( x, ǫ ) ⊆ U . Now any loop with image in B ( x, ǫ ) must be in B ( c, ǫ ) and thereforenulhomotopic in X .For (3) ⇒ (1) we let U be an open cover of X by path connected open sets U whose inclusion maps induce a trivial map π ( U ) → π ( X ) . Then we are in thesituation of Lemma 2.8 and we see that the trivial subgroup is open, so we aredone. (cid:3) N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 7 Polish Spaces
We present some material which will be specific to dealing with fundamentalgroups of Polish spaces. We give some technical lemmas which will establish clo-sure properties for subgroups of particularly nice types of pointclasses (stated inTheorem 3.10). These will give a sense of the versatility of such subgroups. Wewill then prove a couple of the main results of the paper. Recall the following:
Definition 3.1.
A topological space Z is Polish if it is completely metrizable andseparable.Many commonly used spaces such as the real line R , compact metric spaces,and countable discrete spaces are Polish. Polish spaces are closed under countabledisjoint union and countable products. When X is path connected and Polish thespace L x is also Polish. The space H x of homotopies of loops at x , topologizedby the sup metric, is Polish assuming X is path connected Polish. The followinglemma provides a sense of base point independence as in Lemma 2.5. Lemma 3.2.
Suppose the pointclass P contains the closed sets and is closed undercontinuous images between Polish spaces, finite products, and finite intersections.Let X be Polish and ρ be a path from x to y in X . Letting φ be the map definedin Lemma 2.5, a subgroup G ≤ π ( X, x ) is of type P if and only if φ ( G ) is. Proof.
Assume the hypotheses. We prove the forward direction of the biconditionaland the other direction follows similarly. Let G ≤ π ( X, x ) be of type P . Let D ⊆ L x × H x × L x be defined by D = {( l , H, l ) ∶ H is a homotopy from l to l } . Itis easy to see that D is closed. Since the map l ↦ ρ − ∗ l ∗ ρ is an isometric embeddingfrom L x to L y we have that ρ − ∗ G ∗ ρ is in pointclass P in L y by assumption.Then ( ρ − ∗ G ∗ ρ ) × H y × L y is in pointclass P in L y × H y × L y by hypothesis. Then D ∩ ( ρ − ∗ G ∗ ρ ) × H y × L y is in pointclass P . Letting p ∶ L x × H x × L x → L x be projection to the third coordinate (obviously a continuous map), we have that ⋃ φ ( G ) = p ( D ∩ ( ρ − ∗ G ∗ ρ ) × H y × L y ) is in the pointclass P . (cid:3) For K ⊆ L x let [ K ] ⊆ π ( X, x ) denote the subset of equivalence classes of loopswhich have representatives in K . Lemma 3.3.
Let P and X satisfy the hypotheses of Lemma 3.2. If K ⊆ L x is P then the set ⋃ [ K ] ⊆ L x is P . Proof.
Letting D = {( l , H, l ) ∶ H homotopes l to l } ⊆ L x × H x × L x we have that D is closed and therefore P . The set K is P and therefore so is K × H x × L x . Then ( K × H x × L x ) ∩ D is P , and letting p be projection in the third coordinate wehave p (( K × H x × L x ) ∩ D ) = ⋃ [ K ] is P . (cid:3) Lemma 3.4.
Let P and X satisfy the hypotheses of Lemma 3.2. Assume furtherthat P is closed under countable unions. If K ⊆ L x is P then ⟨[ K ]⟩ is a P subgroupof π ( X, x ) . Proof.
Notice that the inversion map l ↦ l − is an isometry and therefore continu-ous. Thus K − is P , and K ∪ K − is also P . For each n ∈ ω let m n ∶ ∏ n − i = L x → L x be given by ( l , . . . , l n − ) ↦ l ∗ l ∗ ⋯ ∗ l n − . This is clearly a continuous map.Each m n ( ∏ n − i = ( K ∪ K − )) is of type P . Thus ⋃ ∞ n = m n ( ∏ n − i = ( K ∪ K − )) is P . ByLemma 3.3 we have that ⋃ [ ⋃ ∞ n = m n ( ∏ n − i = ( K ∪ K − ))] is P . We are done since ⋃ ⟨[ K ]⟩ = ⋃ [ ⋃ ∞ n = m n ( ∏ n − i = ( K ∪ K − ))] . (cid:3) SAMUEL M. CORSON
Lemma 3.5.
Let P and X satisfy the hypotheses of Lemma 3.4. If K ⊆ L x is P then the normal closure ⟨⟨[ K ]⟩⟩ is P . Proof.
Let c ∶ L x × L x → L x be given by ( l , l ) ↦ l ∗ l ∗ l − . This is easilycontinuous. We have L x × K is P , and so is c ( L x × K ) . Then ⟨⟨[ K ]⟩⟩ = ⟨[ c ( L x × K )]⟩ is P by Lemma 3.4 . (cid:3) The preceeding lemmas motivate the following:
Definition 3.6.
A pointclass P defined on Polish spaces is nice if it contains theclosed sets, is closed under continuous images and preimages, countable intersec-tions and finite unions. Remark 3.7.
A nice pointclass is also closed under countable products, for if A n ⊆ Z n is of nice pointclass P for each n ∈ ω then ∏ n ∈ ω A n = ⋂ n ∈ ω p − n ( A n ) is P inthe Polish space ∏ n ∈ ω Z n . A nice pointclass is also closed under countable unions,for suppose A n ⊆ Z are P for each n ∈ ω . If ⋃ n ∈ ω A n = ∅ then as ∅ is closed wehave ⋃ n ∈ ω A n is P . Otherwise pick z ∈ ⋃ n ∈ ω A n . Since { z } is closed in Z and P isclosed under finite unions we can assume z ∈ A n for all n ∈ ω . Let ⊔ n ∈ ω Z be thedisjoint union of countably many copies of Z . Let f ∶ ⊔ n ∈ ω Z → ∏ n ∈ ω Z take y n to ( z, z, . . . , z, y, z, z . . . ) (here y is in the n th coordinate) where y n is a copy of y inthe n th copy of Z in the disjoint union. The map f is continuous by the universaland co-universal properties of product and disjoint unions, respectively. The set ∏ n ∈ ω A n is P as we have seen. Letting g ∶ ⊔ n ∈ ω Z → Z map each copy of Z viaidentity we get that g ( f − ( ∏ n ∈ ω A n )) = ⋃ n ∈ ω A n is P .Under set inclusion the smallest nice Polish pointclass is that of the analytic sets(denoted Σ ). If Z is Polish we say Y ⊆ Z is analytic if there exists a Polish space W and a continuous map f ∶ W → Z such that f ( W ) = Y . All Borel sets of a Polishspace are analytic (see [Ke]). Lemma 3.8. If X = ∏ n ∈ ω X n where each X n is metrizable, then the loop space of X is homeomorphic to the product of the loop spaces of the spaces X n and can bemetrized thereby. Proof.
By applying a cutoff metric d n to each space X n we may assume diam ( X n ) ≤ − n . The metric d ({ s n } n ∈ ω , { t n } n ∈ ω ) = ∑ ∞ n = d n ( s n , t n ) is compatible with the prod-uct topology on ∏ n X n . Fix a point x n in each X n and let x = { x n } n ∈ ω ∈ ∏ n X n .The metric d induces the sup metric on the loop space L x so that L x is homeomor-phic with the space ∏ n L x n where the distance between loops { l n } n ∈ ω and { l ′ n } ω is ∑ n sup s ∈ [ , ] d n ( l n ( s ) , l ′ n ( s )) . This follows from the fact that uniform convergenceof a sequence of loops in L x occurs precisely when the loops in each coordinateconverge uniformly. Thus we may metrize L x with the metric defined by the metricon the product ∏ n L x n . (cid:3) We cover some functoriality properties. Recall that if ( X, x ) and ( Y, y ) are twopointed spaces and f ∶ ( X, x ) → ( Y, y ) is a continuous function there is an inducedhomomorphism f ∗ ∶ π ( X, x ) → π ( Y, y ) defined by f ∗ ([ l ]) = [ f ○ l ] . The map f also induces a continuous map f ∶ L x → L y given by l ↦ f ○ l . We also recallthat the wedge ( X, x ) ∨ ( Y, y ) is the topological space obtained by identifying thedistinguished points, which has distinguished point corresponding to the identifiedpoints which we denote x ∨ y . There are obvious inclusion maps from the spaces N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 9 ( X, x ) and ( Y, y ) to the wedge as well as retraction maps from the wedge to thetwo spaces. If X and Y are metrizable, competely metrizable, or separable then sois the wedge. Proposition 3.9.
Assume X and Y are metric spaces. The following closureproperties hold:(1) If f ∶ ( X, x ) → ( Y, y ) is continuous, P is a pointclass closed under continuouspreimages and G ≤ π ( X, x ) is P , then ( f ∗ ) − ( G ) is also P .(2) If G ≤ π ( X, x ) and G ≤ π ( Y, y ) are both of pointclass P and P is closedunder products, then G × G ≤ π ( X × Y, ( x, y )) ≃ π ( X, x ) × π ( Y, y ) is P .(3) If f ∶ ( X, x ) → ( Y, y ) is continuous between Polish spaces and P is nice and G ≤ π ( X, x ) is P then f ∗ ( G ) is P .(4) If G ≤ π ( X, x ) and G ≤ π ( Y, y ) are P , with X and Y Polish and P nice, then the subgroup generated by the images of G and G under theinclusion maps is P in π (( X, x ) ∨ ( Y, y ) , x ∨ y ) . Proof.
For part (1) we notice that ⋃ ( f ∗ ) − ( G ) = f − ( ⋃ G ) . Claim (2) followsfrom Lemma 3.8, and applies to countable products if P is closed under count-able products. For (3) the map f induces the continuous map f from L x to L y by composition. The image of ⋃ G under this map is P because P is nice, and ⋃ f ∗ ( G ) = [ f ( ⋃ G )] . Claim (4) follows immediately, since ⋃ ⟨ ι X ∗ ( G ) ∪ ι Y ∗ ( G )⟩ = ⋃ ⟨[ ι X ( ⋃ G ) ∪ ι Y ( ⋃ G )]⟩ is evidently P . (cid:3) The following theorem gives a catalogue of closure properties for nice subgroups.Recall that the derived series is defined by letting G ( ) = G , G ( α + ) = [ G ( α ) , G ( α ) ] and G ( β ) = ⋂ α < β G ( α ) if β is a limit ordinal. The lower central series is defined byletting G = G and G n + = [ G, G n ] . Theorem 3.10.
Let f ∶ ( X, x ) → ( Y, y ) be a continuous function between Polishspaces and let P be a nice pointclass. The following hold:(1) If H ≤ π ( Y, y ) is P then f − ∗ ( H ) ≤ π ( X, x ) is P .(2) If G ≤ π ( X, x ) is P then f ∗ ( G ) ≤ π ( Y, y ) is P .(3) The subgroups 1 and π ( X, x ) are analytic in π ( X, x ) .(4) If G n ≤ π ( X, x ) are P then so are ⋂ n ∈ ω G n and ⟨ ⋃ n ∈ ω G n ⟩ .(5) Countable subgroups of π ( X, x ) are analytic.(6) If G ≤ π ( X, x ) is P then so is ⟨⟨ G ⟩⟩ .(7) If G ≤ π ( X, x ) is P then so is any conjugate of G .(8) If w ( x , . . . , x k ) is a reduced word in the free group F ( x , . . . , x k ) and thegroups G , . . . , G k ≤ π ( X, x ) are P then so is the subgroup ⟨{ w ( g , g , . . . , g k )} g i ∈ G i ⟩ .(9) If G, H ≤ π ( X, x ) are P then so is the subgroup [ G, H ] .(10) If G ≤ π ( X, x ) is P then each countable index term of the derived series G ( α ) and each term of the lower central series G n is P . Proof.
Claim (1) follows from (1) in Proposition 3.9. Claim (2) is claim (3) inProposition 3.9. For (3) we have that π ( X, x ) is a closed subgroup and 1 is thesubgroup generated by the constant map to x , and so is analytic by Lemma 3.4(since a singleton is closed in L x ). Claim (4) follows from the definition of nicepointclasses and Lemma 3.4. Claim (5) follows from the fact that singletons areclosed in L x and claim (4). Claim (6) is an instance of Lemma 3.5. Claim (7) isan instance of Lemma 3.2. For claim (8) we notice that the map w ∶ ∏ ki = L x → L x given by ( l , . . . , l k ) ↦ w ( l , . . . , l k ) is continuous, and so { w ( l , . . . , l k )} l i ∈ ⋃ G i is a P subset in L x and the claim follows from Lemma 3.4. Claim (9) is an instance of claim(8). For claim (10) we iterate claim (9), applying claim (4) at limit ordinals. (cid:3) We recall some definitions. If Z is a topological space we say that Y ⊆ Z is nowhere dense if the closure Y ⊆ Z has empty interior, Y is meager if it is aunion of countably many nowhere dense sets in Z , Y has the property of Baire (abbreviated BP) if there exists an open set O ⊆ Z such that Y ∆ O = ( Y ∖ O ) ∪ ( O ∖ Y ) is meager, and Y is comeager if Z ∖ Y is meager. We say a pointclass P on Polishspaces has the property of Baire if each set in P has the property of Baire. Forexample, the pointclass of open sets obviously has BP. In fact, the class of analyticsets also has BP (see [Ke]).The following was proven in [P] using a result from [My]. Lemma.
Suppose ≈ is an equivalence relation on the Cantor set { , } ω such thatif α and β differ at exactly one coordinate then α ≈ β fails. If ≈ has BP as a subsetof { , } ω × { , } ω , then ≈ has 2 ℵ equivalence classes. Lemma 3.11.
Let X be Polish. Suppose that G ⊴ K ≤ π ( X, x ) with G ofpointclass P and that K is closed. Suppose also that P has BP and is closedunder continuous preimages in Polish spaces, and that there exist arbitrarily smallloops at x which are in ⋃ K and not in ⋃ G . Then card ( K / G ) = ℵ . Proof.
Assume the hypotheses and let { l n } n ∈ ω be a sequence of loops at x in ⋃ ( K ∖ G ) such that the diameter of l n is ≤ − n . Let l n be the constant loop at x andlet l n be the loop l n . Given an element α ∈ { , } ω we define l α to be the loop l α ( ) ∗ ( l α ( ) ∗ ( l α ( ) ∗ ( ⋯ ))) (which must also be in ⋃ K as K is closed). In otherwords, l α restricted to the interval [ , ] is either the constant loop or l in case α ( ) is 0 or 1 respectively, l α restricted to the interval [ , ] is either the constantloop or l in case α ( ) is 0 or 1 respectively, etc. The function from the Cantor set { , } ω to L x given by α ↦ l α is clearly continuous. For l, l ′ ∈ L x letting l ∼ l ′ ifand only if [ l ] G = [ l ′ ] G , we have by Lemma 2.3 that ∼⊆ L x × L x is of pointclass P .Defining an equivalence relation ≈ on { , } ω so that α ≈ β if and only if l α ∼ l β , wesee that ≈⊆ { , } ω × { , } ω is of pointclass P as a continuous preimage. As P hasBP we know that ≈ has BP. By Lemma 3 we shall be done if we show that if α and β differ at exactly one point then α ≈ β fails. Suppose that α ( n ) ≠ β ( n ) and that α ( m ) = β ( m ) whenever m ≠ n and that l α ≈ l β . Letting without loss of generality α ( n ) = β ( n ) = [( l β ) − ∗ l α ] ∈ G . Let h = l α ( n + ) n + ∗ ( l α ( n + ) n + ∗ ( ⋯ )) and g = l α ( ) ∗ ( l α ( ) ∗ ( ⋯ l α ( n − ) n − ) ⋯ ) . Then [( l β ) − ∗ l α ] = [ h − ∗ g − ∗ g ∗ l n ∗ h ] = [ h − ∗ l n ∗ h ] ∈ G , so by normality of G in K we have [ l n ] ∈ G , a contradiction.Thus there are at least 2 ℵ many elements in K / G by the above lemma, and thereare at most 2 ℵ elements because there are at most 2 ℵ loops at x . (cid:3) Lemma 3.12. If X is metric, locally path connected and G ⊴ π ( X ) then either G is open or there exists y ∈ X and a sequence of loops { l n } n ∈ ω at y with diam ( l n ) ↘ [ l n ] ∉ G . Proof. If G is not open we have by the contrapositive of Lemma 2.8 that there mustexist some point y ∈ X such that for any open neighborhood U of y there is a loop in U which is not in G . We get a sequence of loops { l n } n ∈ ω with diam ( l n ([ , ]) ∪ { y }) ≤ − n and [ l n ] ∉ G . By local path connectedness we may take a subsequence of the l n N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 11 whose base point is close enough to y and join the basepoint to y via a small path,so that the loops may eventually be assumed to have been based at y . Taking asubsequence having all loops based at y gives the desired result (cid:3) Through the remainder of Section 3 we shall assume P is a nice pointclass withBP. Thus for example one can take P = Σ . The discussion of pointclasses will takeplace in Section 7. Theorem 3.13.
Suppose X is locally path connected Polish. If G ⊴ π ( X ) is P then card ( π ( X )/ G ) is either ≤ ℵ (in case G is open) or 2 ℵ (in case G is notopen). Proof. If G is open then the collection of left cosets {[ l ] G } l ∈ L x is a covering of L x by pairwise disjoint open sets, and since L x is separable we know that the collection {[ l ] G } l ∈ L x is countable. Else, by Lemma 3.12 we get a point y ∈ X and a sequenceof loops { l n } n ∈ ω with diam ( l n ) ↘ [ l n ] ∉ G . Considering G as a subgroupof π ( X, y ) we see that G is P since P is nice, and thus we have satisfied thehypotheses of Lemma 3.11 and we are done. (cid:3) The above may be strengthened if X is also compact. Recall that a Peanocontinuum is a path connected, locally path connected compact metrizable space.
Theorem 3.14. If X is a Peano continuum and G ⊴ π ( X, x ) is P then π ( X, x )/ G is either finitely generated (in case G is open) or of cardinality 2 ℵ (in case G is notopen). Proof.
By Theorem 3.13 we need only show that π ( X )/ G is finitely generated if G is open. For this we will use a theorem from [CC] which will require a definition.Let φ ∶ π ( X ) → H be a group homomorphism. We say an open cover U is φ if each element of U is path connected and any loop in the union oftwo elements of U is in the kernel of φ . This property of a cover implies that forany nerve associated with with U there is a homomorphism from the fundamentalgroup of the nerve with the same image as φ . The following is a part of Theorem7.3 in [CC]: Theorem.
Let X be path connected, φ ∶ π ( X ) → H a homomorphism and U a2-set simple cover rel φ . If U is finite then φ ( π ( X )) is finitely generated.Now, assuming G is open we get by Lemma 2.6 an open cover U for X such thatany loop contained in an element of U is in G . Let ǫ > U and let U be a cover of X by open balls of redius ǫ . By local pathconnectedness let U be an open cover of X by path connected sets, each of whichis contained in an element of U . By compactness we may pick U to be finite, andit is clear that U is 2-set simple rel the quotient projection π ( X ) → π ( X )/ G . Weare done by the theorem of Cannon and Conner that is quoted above. (cid:3) The conditions on G and the pointclass P cannot be removed in Theorems 3.13and 3.14 as evidenced by the following example. Example 2.
Let P denote the projective plane and P ω = ∏ ω P . Let x ∈ P andlet x ∈ P ω be the point whose every coordinate is x . By the functoriality of thefundamental group there is a natural isomorphism π ( P ω , x ) ≃ ∏ ω π ( P, x ) . ByLemma 3.8 the loop space L x is homeomorphic to the product ∏ ω L x where theloop space L x at the n th coordinate has diameter ≤ − n . By the coordinatewise isomorphism ∏ ω π ( P, x ) ≃ ∏ ω Z / Z we may regard elements of π ( P ω , x ) as ω sequences of 0s and 1s. By the shrinking of the metrics on the coordinate loopspaces, any g ∈ π ( P ω , x ) whose first n coordinates are 0s has a representative loop l ∈ g with diam ( l ) ≤ − n . The space P ω is a Peano continuum.Let κ be any cardinal satisfying ℵ ≤ κ ≤ ℵ . As ∏ ω Z / Z is a Z / Z vec-tor space, we select a linearly independent set W ⊆ ∏ ω Z / Z such that W ⊇ {( , , , , . . . ) , ( , , , , . . . ) , ( , , , , . . . ) , . . . )} and card ( W ) = κ . Let B be abasis for ∏ ω Z / Z with B ⊇ W . Letting H = ⟨ W ⟩ and G = ⟨ W ∖ B ⟩ we have ∏ ω Z / Z ≃ H ⊕ G . Now G ⊴ π ( P ω , x ) and π ( P ω , x )/ G ≃ H is of cardinality κ .Letting κ = ℵ gives a counterexample to the claim of Lemma 3.11 with the “ G of pointclass P ” and “ P has BP and is closed under continuous preimages in Polishspaces” hypotheses removed. By considering a model of ZFC in which CH fails,we let κ = ℵ and see that Theorem 3.13 can fail if the conditions on G and P areremoved. Similarly removing these conditions can make the conclusion of Theorem3.14 fail. The quotient can be a countable infinitely generated group or a group ofcardinality violating CH if one is in a model of ZFC + ¬ CH .The conclusion of Theorem 3.14 cannot be strengthened by replacing “finitelygenerated” by “finitely presented” by the following basic example. Example 3.
Let X be the bouquet of two circles and H be a 2-generated groupwhich is not finitely presented (for example, the lamplighter group). The fundamen-tal group π ( X ) is the free group of rank 2. Let φ ∶ π ( X ) → H be a homomorphismgiven by taking each of the free generators of π ( X ) to a distinct generator of H .The space X is a semilocally simply connected Peano continuum and ker ( φ ) = G isopen by Proposition 2.10, but π ( X )/ G ≃ H is not finitely presented. Similar ex-amples can be given by replacing the number 2 by any finite number ≥ G be replaced by any other n -generated group which is not finitely presented.We name some of the numerous applications of the above theorems. Theorem A.
Suppose X is a locally path connected Polish space. The followinggroups are of cardinality 2 ℵ or ≤ ℵ , and in case X is compact they are of cardinality2 ℵ or are finitely generated:(1) π ( X ) (2) π ( X )/( π ( X )) ( α ) for any α < ω (derived series)(3) π ( X )/( π ( X )) n for any n ∈ ω (lower central series)(4) π ( X )/ N where N is the normal subgroup generated by squares of elements,cubes of elements, or n -th powers of elementsIn case X is compact then countability of the fundamental group is equivalent tobeing finitely presented. Proof.
The noncompact case in parts (1)-(4) immediately follow from Theorem3.13. For parts (2)-(4) in the compact case we apply Theorem 3.14. That π ( X ) would be finitely presented follows from Theorem 7.3 in [CC] in part (1) assuming X is compact. (cid:3) Part (1) in the compact case is the main result of the papers [Sh2] and [P], andpart (2) with α = N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 13
Lemma 3.15.
Let X be a Polish space. Suppose N ⊴ K ≤ π ( X, x ) is such that ⋃ N = ⋃ ∞ n = N n with each N n closed under inverses and homotopy and containingthe trivial loop, and that K is closed. Assume also that N n ∗ N m ⊆ N n + m . If each N n is P and for each n ∈ ω there exist loops at x of arbitrarily small diameter in ⋃ K not contained in N n , then K / N is of cardinality 2 ℵ . Proof.
Let ǫ > x in ⋃ K of diameter less than ǫ that is not in ⋃ N , since N is P . Forcontradiction we assume that no such loop exists. For each loop in ⋃ K of diameterless than ǫ let φ map that loop to the minimal k such that l ∈ N k . For two loops l , l of radius less than ǫ we have that φ ( l ∗ l ) ≤ φ ( l ) + φ ( l ) and φ ( l ) = φ ( l − ) .Let { l n } n ∈ ω be a sequence of loops such that diam ( l n ) < ǫ − n and that φ ( l ) > φ ( l n ) > n + ∑ n − m = φ ( l m ) . In particular none of the l n is nullhomotopic. Define l α as before for each α in the Cantor set. Abuse notation by letting φ ∶ { , } ω → ω be defined by φ ( α ) = φ ( l α ) .Let E n = { α ∈ { , } ω ∶ l α ∈ N n } . As we are assuming that there is no loopin ⋃ K of diameter less than ǫ that is not in ⋃ N , we have ⋃ ∞ n = E n = { , } ω .We will derive our contradiction if we show that each E n is meager, which wouldimply that { , } ω is meager in itself. Each E n is P , and so has the property ofBaire. Supposing E n is not meager there exists a nonempty open set in which E n is comeager. In particular there exists a basic open set U ( δ , . . . , δ k ) = { α ∈ { , } ω ∶ α ( ) = δ , . . . , α ( k ) = δ k } such that E n ∩ U ( δ , . . . , δ k ) is comeager in U ( δ , . . . , δ k ) .For each p ≥ k + ℶ p ∶ U ( δ , . . . , δ k ) → U ( δ , . . . , δ k ) be the homeomorphism thatchanges the p coordinate. Then U ( δ , . . . , δ k ) ∖ ℶ p ( E n ) is meager for each p ≥ k + α ∈ U ( δ , . . . , δ k ) such that switching finitely many of thecoordinates beyond the k th coordinate gives an element of E n . It cannot be thatthe support of α is finite, for if N ∈ ω is a bound on the support of α (we can assume N > n ), then n ≥ φ ( l α ∗ f N + ) ≥ φ ( l N + ) − φ ( l α ) > N + − n > n , a contradiction.Thus taking a subsequence of the l n , we may assume that α = ( , , . . . ) and that U ( δ , . . . , δ k ) = { , } ω . We assume that this subsequence was the original sequence.Let β k , γ k ∈ { , } ω be given by β k ( m ) = ⎧⎪⎪⎨⎪⎪⎩ m < k m ≥ k and γ k ( m ) = ⎧⎪⎪⎨⎪⎪⎩ m < k m ≥ k .We have that φ ( γ k ) ≥ k and φ ( β k ) ≥ φ ( γ k ) − φ ( α ) ≥ k − φ ( α ) , so that if k = n + β k ∈ E n and on the other hand φ ( β k ) ≥ k − φ ( α ) ≥ ( n + ) − n , a contradiction. (cid:3) This gives the following:
Theorem B. If X is a Peano continuum there does not exist a strictly increasinginfinite sequence of P normal subgroups { G n } n ∈ ω of π ( X ) such that ⋃ n ∈ ω G n = π ( X ) . Proof.
For each n ∈ ω let ⋃ G n = N n in the notation of Lemma 3.15. If π ( X )/ G n is finitely generated for some n , then the sequence { N n } n ∈ ω cannot be strictlyincreasing. Then π ( X )/ G n must be uncountable for each n , so for each n thereexist arbitrarily small loops not in G n by the proof of Theorem 3.13. By picking anappropriate basepoint by local path connectedness, we are done by Lemma 3.15. (cid:3) The necessity of the conditions on P can be easily seen by considering the de-composition π ( P ω , x ) = ( ⊕ ω Z / Z ) ⊕ ( ⊕ ℵ Z / Z ) given in Example 2 and letting G n = ( ⊕ k ≤ n Z / Z ) ⊕ ( ⊕ ℵ Z / Z ) . Example 4.
The dual analog of Theorem B does not hold: there exists a Peanocontinuum with an infinite strictly descending chain of analytic (in fact closed)normal subgroups whose intersection is the trivial subgroup. We again use thespace P ω from Example 2. We change the superscript ω for Q and the group { , } Q remains unchanged since the cardinalities of ω and Q are the same. Givenany subset S ⊆ Q the subgroup of π ( P ω ) corresponding to the subgroup { α ∈ { , } Q ∶ α ( q ) = ⇒ q ∈ S } ≤ { , } Q is closed. For each r ∈ R let G r ≤ π ( P ω ) bethe subgroup { α ∈ { , } Q ∶ α ( q ) = ⇒ q < r } . Then each G r is a closed subgroupand the following hold:(1) r n ↗ r implies ⋃ n ∈ ω G r n < G r (2) r n ↘ r implies ⋂ n ∈ ω G r n = G r (3) ⋂ r ∈ R G r is the trivial subgroupPicking a sequence r n ↘ −∞ gives a strictly descending sequence of normalanalytic subgroups G r n as claimed. The subgroup ⋃ r ∈ R G r cannot be equal to π ( X ) (else we could pick any sequence r n ↗ ∞ and the ascending chain G r n wouldcontradict Theorem B). For example the sequence over Q which is constantly 1 isnot in ⋃ r ∈ R G r .We now address what can happen in the absence of local path connectedness.Before stating the next theorem we quote the famous selection theorems of Silverand Burgess (from [Si] and [Bu] respectively). A set Y ⊆ Z in a Polish space is coanalytic if Z ∖ Y is analytic. The class of coanalytic sets is denoted Π . Theorem. (J. Silver) Suppose E ⊆ Z × Z is a coanalytic equivalence relation on aPolish space Z . Then either there are ≤ ℵ many equivalence classes or there existsa homeomorph of a Cantor set C ⊆ Z such that for distinct x, y ∈ C we have ¬ xEy . Theorem. (J. Burgess) Suppose E ⊆ Z × Z is an analytic equivalence relation on aPolish space Z . Then either there are ≤ ℵ many equivalence classes or there existsa homeomorph of a Cantor set C ⊆ Z such that for distinct x, y ∈ C we have ¬ xEy . Theorem 3.16.
Suppose X is path connected Polish and G ≤ π ( X, x ) .(1) If G is coanalytic then the index π ( X, x ) ∶ G is either ≤ ℵ or 2 ℵ .(2) If G is analytic then the index π ( X, x ) ∶ G is either ≤ ℵ or 2 ℵ . Proof. If G is coanalytic (repesctively analytic) then by Lemma 2.3 the equivalencerelation induced by the left coset partition is coanalytic (respectively analytic).Apply the theorem of Silver (resp. Burgess) to conclude (1) (resp. (2)). (cid:3) Example 5.
We give an instructive example which demonstrates the sharpnessof Theorem 3.16 as well as the sharpness of Theorems 3.13 and 3.14 in a differentsense than that of Example 2. There exists a model M of set theory satisfying(1) ZFC (2) ¬ CH (3) Any subset of { , } ω of cardinality ℵ is coanalytic. N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 15 (see [MaSo]). We consider the space F and the function f ∶ K → L ( , ) used inExample 1 within the model M . Let S ⊆ K be such that card ( K ∖ S ) = ℵ . Then S is analytic in K . Then f ( S ) is analytic in L ( , ) . Then ⟨⟨[ f ( S )]⟩⟩ ≤ π ( F, ( , )) is analytic by Lemma 3.5. The quotient π ( F, ( , ))/⟨⟨[ f ( S )]⟩⟩ is isomorphic to afree group of rank ℵ so that card ( π ( F, ( , ))/⟨⟨[ f ( S )]⟩⟩) = ℵ .This demonstrates that the case ℵ in Theorem 3.16 (2) can obtain in the absenceof CH . It also shows that one cannot hope to extend Theorem 3.16 part (1) to ahigher projective class (since any higher projective class also contains the analyticsets, see Section 7). This also shows that one cannot drop local path connectednessin Theorems 3.13 and 3.14 and obtain the same conclusion.4. Comonster Groups
As an application of the above theory we give the following definition.
Definition 4.1.
We say a group G is comonster if for every finite subset S ⊆ G we have ⟨⟨ S ⟩⟩ ≠ G . More generally G is κ - comonster if for every S ⊆ G with S ofcardinality < κ we have ⟨⟨ S ⟩⟩ ≠ G .Thus comonster groups are ℵ -comonster groups. One easily sees that anyabelian group of cardinality κ > ℵ is κ -comonster. Also, if h ∶ G → H is an epi-morphism with H comonster (respectively κ -comonster), then G is also comonster(resp. κ -comonster).We assume still that P is nice with BP. We have the following: Theorem 4.2.
Let X be a Peano continuum and N ⊴ π ( X ) of type P . If π ( X )/ N is comonster then π ( X )/ N is ℵ -comonster. In particular, if π ( X ) is comonster,then π ( X ) is ℵ -comonster. Proof.
Suppose for contradiction that π ( X )/ N is comonster but not ℵ -comonster.Let S = { g , . . . } ⊆ π ( X ) be a countably infinite set such that ⟨⟨ N ∪ S ⟩⟩ = π ( X ) .The normal groups G n = ⟨⟨ N ∪ { g , . . . , g n }⟩⟩ are easily seen to be P and ⋃ n G n = π ( X ) . On the other hand the sequence G n cannot stabilize since π ( X )/ N iscomonster. Thus one can pick a strictly increasing subsequence of normal P sub-groups whose union is π ( X ) , contradicting Theorem B. (cid:3) Example 6.
Consider the Hawaiian earring E = ⋃ n ∈ ω C (( , n + ) , n + ) . We havean epimorphism h ∶ π ( E ) → ∏ ω Z given by letting the n -th coordinate of h ([ l ]) count the number of times a loop traverses the n -th circle of the infinite wedgethat defines E in an oriented direction. Then π ( E ) is 2 ℵ -comonster, since ∏ ω Z is abelian of cardinality 2 ℵ . Example 7. If X is a one-dimensional Peano continuum with π ( X ) uncountable,then X retracts to a subspace that is homeomorphic to E , so that again π ( X ) is2 ℵ -comonster.Even if π ( X ) is uncountable, it may still be the case that π ( X ) is not comon-ster, as illustrated in the following example. Example 8.
Let Y be a Peano continuum with π ( Y ) ≃ A . Such a Y exists bytaking a finite presentation for A and constructing the finite 2-dimensional CWcomplex by letting loops correspond to generators in the presentation and gluingon the boundary of a disc along a path that gives the relators. Such a space is compact, metrizable, path connected and locally path connected. Thus such a Y is a Peano continuum, and so is X = ∏ ω Y . We have π ( X ) ≃ ∏ ω A . Letting g ∈ ∏ ω A have every entry be the 3-cycle ( ) , we claim that ⟨⟨ g ⟩⟩ = ∏ ω A . Thisdemonstrates that π ( X ) is not comonster.To see that ⟨⟨ g ⟩⟩ = ∏ ω A , notice that all 3-cycles are conjugate (in A ) to eachother. Thus for each h ∈ ∏ ω A whose each entry is a 3-cycle we have h ∈ ⟨⟨ g ⟩⟩ . Each3-cycle is a product of two 3 cycles (if ( abc ) is a 3-cycle then ( abc ) = ( abc ) − ( abc ) − = ( cba )( cba ) ). Since the trivial element in A is a product of two three cycles andeach 5-cycle and each product of two disjoint transpositions ( ab )( cd ) is a productof two 3-cycles then in fact every element in ∏ ω A is a product of exactly twoconjugates of g and we are done.In all the above examples of Peano continua with comonster fundamental group,we used the fact that if the abelianization is uncountable then the fundamentalgroup is comonster. Question 4.3.
Does there exist a Peano continuum whose first homology is trivialand whose fundamental group is comonster?A negative answer would be very interesting as it would imply a theorem forfinitely presented perfect groups (groups with trivial abelianization).
Theorem 4.4.
Suppose the answer to the above question is no. Let P n be theclass of groups whose elements are products of n or fewer commutators. For each n ∈ N there exists k ( n ) ∈ N such that if G ∈ P n is finitely presented there exists aset S ⊆ G with card ( S ) ≤ k ( n ) and ∏ k ( n ) S G = G (each element of G is a productof k ( n ) or fewer conjugates of elements of S ). Proof.
Suppose for contradiction that for some n ∈ N there is no such k ( n ) . Selectfinitely presented groups G m ∈ P n such that for any S ⊆ G m with card ( S ) ≤ m wehave that ∏ m S G m ≠ G m . For each m there is a finite CW complex Y m of dimensionat most two whose fundamental group is isomorphic to G m . Each such Y m is aPeano continuum. Then ∏ m Y m is a Peano continuum, with fundamental groupisomorphic to ∏ m G m . It is easy to see that ∏ m G m ∈ P n and is also comonster. (cid:3) This is adjacent to a question of Wiegold: Does every finitely generated perfectgroup contain an element which normally generates the group?5.
Examples of Topologically Defined Subgroups
We give some standard examples, and introduce some new examples, of sub-groups of the fundamental group which are topologically defined. These are in-tended to illustrate the richness of the theory and give a grab bag of examples towhich to apply the theorems.5.1.
The Shape Kernel.
One well known subgroup of the fundamental group isthe shape kernel. We discuss this subgroup by first giving preliminary definitionstowards defining the shape group and the shape kernel and then prove that theshape kernel is a closed subgroup.We assume some familiarity with geometric simplicial complexes. Given a topo-logical space X and a open cover U of X let N (U) denote the nerve of the cover-that is, the geometric simplicial complex which has a distinct vertex v U for ev-ery U ∈ U and which contains the n -simplex [ v U , v U , . . . , v U n ] if and only if N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 17 U ∩ U ∩ ⋯ ∩ U n ≠ ∅ . If V is an open cover of X that refines U (i.e. for each V ∈ V there is a U ∈ U such that V ⊆ U ) then any map from the vertices of N (V) to the vertices of N (U) such that v V ↦ v U implies V ⊆ U extends to a simplicialmap from N (V) to N (U) .If the topological space has a distinguished basepoint x , then one can distin-guish an element U in an open cover U such that x ∈ U , which in turn givesa distinguished vertex in the nerve N (U) . With this added structure, if V re-fines U with distinguished elements V and U such that V ⊆ U then a simplicialmap as described above extending a vertex assignment satisfying v V → v U , say p ( V ,V ) , ( U ,U ) ∶ ( N (V) , v V ) → ( N (U) , v U ) preserves basepoint and is unique up tobasepoint preserving homotopy. Assuming X is path connected all nerves are con-nected, and the refinement relation on open covers gives an inverse directed system ( π ( N (U) , v U ) , p ( V ,V ) , ( U ,U ) ∗ ) . The shape group of X is defined as the inverse limitˇ π ( X, x ) = lim ← ( π ( N (U) , v U ) , p ( V ,V ) , ( U ,U ) ∗ ) The index of the inverse limit will generally be of uncountable cardinality. As-suming ( X, x ) is also paracompact and Hausdorff we have for each open cover withdistinguished neighborhood of x , (U , U ) , a refinement (V , V ) such that V is theunique element of V containing x . A partition of unity subordinate to V (which nec-essarily exists by our assumption of paracompactness and Hausdorffness) induces abarycentric map f U ∶ ( X, x ) → ( N (V) , v V ) which is unique up to based homotopy.The induced map f U ∗ on the fundamental group π ( X, x ) can be checked to com-mute with the maps of the inverse system in the appropriate way, and since the setof all such open covers V described are cofinal in the inverse system we get a mapΨ ∶ π ( X, x ) → ˇ π ( X, x ) .The natural object used to assess the loss of information when passing fromthe fundamental to the shape group is the shape kernel ker ( Ψ ) . The followingdemonstrates an alternative characterizaion of the shape kernel. Theorem 5.1.
Suppose ( X, x ) is a path connected metrizable space. Then theshape kernel is a closed normal subgroup of π ( X, x ) . If in addition X is locallypath connected then the shape kernel is equal to the following two subgroups:(1) ⋂ f ker ( f ∗ ) where f is taken over all continuous maps to semilocally simplyconnected spaces(2) ⋂ G open, normal G Proof.
It is clear from the definition that the shape kernel is equal to ⋂ f ker ( f ∗ ) where f is taken over all baricentric maps of open covers. Fix a barycentric map f to the nerve N (V) . As N (V) is a geometric simplicial complex it is not difficultfor each loop l ∈ L x to select ǫ > l ′ ∈ L x that is ǫ close to l we have f ○ l is homotopic to f ○ l ′ in N (V) . This shows that ker ( f ∗ ) is open, andtherefore also closed by Lemma 2.4. Then the shape kernel is a closed subgroup asan intersection of closed subgroups.Now suppose that X is also locally path connected. Since each nerve is a geomet-ric simplicial complex, each nerve is also semilocally simply connected. Thus theshape kernel contains the subgroup (1). Furthermore, if f ∶ X → Y is continuouswith Y semilocally simply connected, then we can find an open cover U of X suchthat the image of any loop in an element of U has nulhomotpic image under the map f . This gives an open cover satisfying the criteria of Lemma 2.8 and since X is locally path connected we have that ker ( f ∗ ) is open. Thus subgroup (1) containssubgroup (2).We conclude by proving that subgroup (2) contains the shape kernel. Let G be an open normal subgroup in π ( X ) (since G is open, normal we may consider π ( X, x ) as basepoint free by Lemma 2.5). Let q ∶ π ( X ) → π ( X )/ G be thecanonical quotient homomorphism. We introduce some terms and a theorem givenin [CC].Recall that if φ ∶ π ( Y ) → H is a group homomorphism we say an open cover V of Y by path connected sets is φ provided any loop whose image liesin the union of two elements of V is in ker ( φ ) (as defined in the proof of Theorem3.14). Two paths p and p are V -related if there is some parametrization for p and p such that for all s ∈ [ , ] the points p ( s ) and p ( s ) are in a commonelement of V . To be V -related is not necessarily an equivalence relation; we saythat paths p and p are V -equivalent if they are in the same class under theequivalence class generated by V -relatedness. The following is part (1) of Theorem7.3 in [CC]: Theorem.
Let Y be a path connected topological space, φ ∶ π ( Y ) → H a ho-momorphism and V a 2-set simple cover of Y rel φ . If two loops l, l ′ ∈ L y are V -equivalent then φ ([ l ]) = φ ([ l ′ ]) .By Lemma 2.6 we have an open cover U of X such that each loop in an elementof U is in G . For each z ∈ X we may select V z ∈ U satisfying z ∈ V z . Define r ( z ) = d ( z, X − V z ) . Letting U = { B ( z, r ( z ) )} z ∈ X it is straightforward to check that if for U, U ′ ∈ U we have U ∩ U ′ ≠ ∅ then U ∪ U ′ is contained in an element of U . By localpath connectedness we let U be a refinement of U by path connected open sets.It is easy to see that U is a 2-set simple cover rel q . For each z ∈ X pick a W z ∈ U such that z ∈ W z and let r ( z ) = d ( z, X − W z ) . Letting U = { B ( z, r ( z ) )} z ∈ X onecan check that if U, U ′ , U ′′ ∈ U satisfy U ∩ U ′ ≠ ∅ and U ′ ∩ U ′′ ≠ ∅ then U ∪ U ′ ∪ U ′′ is contained entirely in an element of U . Let U be a refinement of U by pathconnected open sets. Finally for each z ∈ X select a U z ∈ U such that z ∈ U z , let r ( z ) = d ( z, X − U z ) and U = { B ( z, r ( z ) )} z ∈ X . Again, it is straightforward to seethat if U, U ′ ∈ U satisfy U ∩ U ′ ≠ ∅ then U ∪ U ′ is entirely contained in an elementof U . Without loss of generality we can assume U is refined so that x is containedin exactly one element of the cover U .Let b ∶ X → N (U) be a barycentric map associated to some partition of unitysubordinated to U . Then b ( x ) = v U where U ∈ U is unique such that x ∈ U . Wedefine a map f from the 1-skeleton N (U) to X . Let f ( v U ) = x and for all othervertices v U ′ ∈ N (U) simply let f ( v U ′ ) ∈ U ′ . By our choice of U if [ v U ′ , v U ′′ ] is a1-simplex in N (U) then U ′ ∩ U ′′ ≠ ∅ and so there exists a path contained entirelyin an element of U from f ( v U ′ ) to f ( v U ′′ ) . Let f ∣[ v U ′ , v U ′′ ] map via this path.We will be done if we show that ker ( b ∗ ) ≤ G . Suppose now that l ∈ L x is such that [ l ] ∈ ker ( b ∗ ) . Then b ○ l is a loop in N (U) based at v U which is nulhomotopic. Recallthat b has the property that b − ( Star v U ′ ) ⊆ U ′ where Star v U ′ is the open star of thevertex v U ′ . There exists a combinatorial loop p ( v U , v U , v U , . . . , v U n − , v U n = v U ) which is homotopic in N (U) to b ○ l such that b ○ l ( s ) ∈ Star v U k when s ∈ [ kn , k + n ] .Letting l ∶ [ , ] → N (U) be a topological realization of this loop we see that l is U -related to f ○ l . N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 19
By assumption there exists a nulhomotopy of l , and so in particular there existsa combinatorial nulhomotopy of p ( v U , v U , v U , . . . , v U n − , v U n = v U ) . In other words,there exists a finite sequence of combinatorial paths: p = p ( v U , v U , v U , . . . , v U n − , v U n = v U ) p = p ( v U , v U , , v U , , . . . , v U ,n = v U ) p = p ( v U , v U , , v U , , . . . , v U ,n = v U ) ⋮ p m = p ( v U ) such that one obtains p k from p k − by performing one of the following elementarypath homotopies:(1) Exchanging the subpath v U p , v U p + for the subpath v U p assuming U p = U p + ,or vice versa.(2) Exchanging the subpath v U p , v U p + , v U p + for the subpath v U p assuming U p + = U p , or vice versa.(3) Exchanging the subpath v U p , v U p + , v U p + for the subpath v U p , v U p + assum-ing [ v U p , v U p + , v U p + ] is a 2-simplex in N (U) , or vice versa.Letting l k ∶ [ , ] → N (U) be a topological realization of the combinatorial path p k , it is easy to see that f ○ l k is U related to f ○ l k + . By the theorem of Cannonand Conner quoted above, we have that q ([ l ]) = q ([ f ○ l ]) = q ([ f ○ l ]) = ⋯ = q ([ f ○ l m ]) = q ( ) , and so [ l ] ∈ G . (cid:3) As a direct consequence of Theorem 3.16 we get the following: if X is a Polishspace then the quotient of π ( X ) by the shape kernel is of cardinality ≤ ℵ or 2 ℵ .5.2. The Spanier Group.
Another useful subgroup of the fundamental group isthe Spanier group, which we denote π s ( X, x ) (first defined in [Sp]). We give thenecessary definitions for this group, then give some results about the topologicalproperties.Let X be a path connected topological space and x ∈ X . If U is a collectionof open subsets of X we define π (U , x ) to be the subgroup of π ( X, x ) generatedby loops of the form ρ ∗ l ∗ ρ − where ρ ( ) = x and l is a loop based at ρ ( ) andcontained in some element of U . This subgroup is easily seen to be normal. TheSpanier group is defined to be π s ( X, x ) = ⋂ U π (U , x ) where the parameter U istaken over all open covers. The first of the following two lemmas does not assumemetrizability of X . It is proven in [FZ] as Proposition 4.8. We provide our ownproof for completeness. Lemma 5.2. π s ( X, x ) is contained in the shape kernel. Proof.
Let b be a barycentric map from X to some nerve. Since a nerve is semilo-cally simply connected we have an open cover U of X such that any loop containedin an element of U is in ker ( b ∗ ) . Obviously π (U , x ) ≤ ker ( b ∗ ) and taking theappropriate intersections gives the claim. (cid:3) Lemma 5.3.
Let X be a metric space, U an open cover of X and x ∈ X .(1) If X is locally path connected then π (U , x ) is open.(2) If X is Polish and U is countable then π (U , x ) is analytic. Proof.
Assume the hypotheses for part (1). The open cover U is such that anyloop contained in an element thereof (considering loops to be base point free) is anelement of π (U) (we switch here to a basepoint free notation for emphasis). Thenby Lemma 2.8 we have that π (U) is an open subgroup.Assume the hypotheses for part (2). Let L x,U,n = { l ∈ L x ∶ ( ∀ s ∈ [ , ])[ l ( s ) = l ( − s )] ∧ ( ∀ s ∈ [ , ])[ d ( l ( s ) , X − U ) ≥ n ]} where U ∈ U and n ∈ ω . It is clear that L x,U,n is closed as a subset of L x . The set ⋃ U ∈U ,n ∈ ω L x,U,n is a countable union ofclosed sets (and therefore analytic). Then π (U , x ) = ⟨⟨ ⋃ U ∈U ,n ∈ ω L x,U,n ⟩⟩ is analyticby Lemma 3.5. (cid:3) That the shape kernel is equal to the Spanier group for all locally path connected,path connected paracompact Hausdorff spaces was recently shown in [BF]. Part(1) of the following theorem gives a rather short proof of a slightly less general fact.
Theorem 5.4.
The following hold:(1) If X is a locally path connected metric space then π s ( X, x ) is equal to theshape kernel, and in particular closed.(2) If X is a compact metric space then π s ( X, x ) is analytic. Proof. (1) Assume the hypotheses. That the Spanier group is contained in theshape kernel was proved in Lemma 5.2. That the shape kernel is contained in theSpanier group follows from characterization (2) of Theorem 5.1 and from Lemma5.3 part (1).For (2) we assume the hypotheses. As X is a compact metric space there exists asequence {U n } n ∈ ω of finite open covers such that U n + refines U n and which is cofinalin the inverse directed system of open covers. Thus π s ( X, x ) = ⋂ n ∈ ω π (U n , x ) isanalytic as a countable intersection of analytic subgroups (Lemma 5.3 part (2) andTheorem 3.10 part (4)). (cid:3) Subgroups reflecting local behavior.
We give a couple of subgroups thatcan be thought of as indicating local behavior. First, recall that a space X is homotopically Hausdorff at x if each loop based at x which can be homotopedinto any neighborhood of x is nulhomotopic. This notion has found many uses (forexample in [BS] and [FZ]). If X is a Polish space, let L x,n be the set of all loopsgiven by l ∈ L x,n if and only if ( ∀ s ∈ [ , ])[ d ( l ( s ) , x ) ≤ n ] . Then L x,n is clearlya closed subset of L x , so the subgroup ⟨[ L x,n ]⟩ is analytic by Lemma 3.4. Thesubgroup ⋂ n ∈ ω ⟨[ L x,n ]⟩ is trivial if and only if X is homotopically Hausdorff at x .This subgroup is analytic and can be thought of as the indicator subgroup for theproperty.We give another example of a subgroup reflacting local behavior. If X is compact,metrizable and path connected, then it is easy to see that the cone over X , C X = X × [ , ]/ X × { } , is also compact, metrizable and path connected. We shall consider X as a subset of C X by identifying X with X × { } .Let S ⊆ X be nonempty. Fixing a metric on C X we let Y n,S ⊆ C X be given by Y n,S = X ∪ (C X ∖ B ( S, n )) . Let f n,S be the inclusion map from X to Y n,S . Then f n,S is a continuous map to a compact metric space, and ker ( f n,S ∗ ) is analytic.Since Y n,S = Y n,S there is no generality lost in assuming that S is compact. Also,the choice of metric on C X does not change ⋃ n ker ( f n,S ∗ ) (by compactness). Let N ( S ) denote the normal subgroup ⋃ n ker ( f n,S ∗ ) . This subgroup is intended toconvey a sense of the importance of the subspace S in the fundamental group of X . N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 21
If the subgroup N ( S ) is all of π ( X ) then the points of S carry little significancein the fundamental group. If N ( S ) is trivial, then the points of S can be thoughtof as holding importance. If S ⊆ S ′ then Y n,S ⊇ Y n,S ′ and so N ( S ′ ) ≤ N ( S ) . Example 9.
Let X be compact, metrizable and path connected. Letting S = X we get that for every n ∈ ω ∖ { } , the path component in Y n,S including all elementsof X is simply the subset X . Thus any nulhomotopy of a loop in X taking placein Y n,S must in fact already take place in X , so N ( S ) is trivial. Example 10.
Let S ⊆ X be a compactum such that any map f ∶ S → X can behomotoped to have image disjoint from S . Then given x ∈ X and a loop l ∈ L x thereis a homotopy of l to a loop ρ ∗ l ′ ∗ ρ − such that l ′ is a loop with image disjoint from S . By compactness there is some positive distance between S and the image of l ′ ,and so l ′ can be nulhomotoped in Y n,S for some n , so that l is also nulhomotopicin Y n,S . Then N ( S ) = π ( X ) . Example 11.
Let X = S and S = { x } be any singleton. For each n ∈ ω ∖ { } thereis a superset Z ⊇ Y n,S such that Z strongly deformation retracts to the set X , sothat N ( S ) is trivial. This holds true as well if X is a wedge of finitely many circlesand x is the wedge point by the same proof. Lemma 5.5. If r ∶ X → Y is a retraction with Y ⊃ S then the monomorphisminduced by inclusion π ( Y ) → π ( X ) induces a monomorphism π ( Y )/ N Y ( S ) → π ( X )/ N X ( S ) (here we use the subscript to denote the ambient space). Proof.
This follows from the fact that the retraction r extends to a retraction R ofthe cones R ∶ C ( X ) → C ( Y ) given by R ( x, t ) = ( r ( x ) , t ) where t ∈ [ , ] . (cid:3) Example 12.
Let E be the Hawaiian earring (see Example 6) and S = { x } where x is the wedge point. The wedge Y m of the outer m circles is a retract of X andeach N Y m ( S ) is trivial by the previous example. Then N E ( S ) has no elements ofthe canonical free group retracts. Then N E ( S ) is trivial by the standard fact thatthe Hawaiian earring fundamental group injects naturally into the inverse limit ofthe canonical free subgroups. Example 13.
Consider the Hawaiian earring E again and S = { x } with x any otherpoint in E besides the wedge point. Then for some n ∈ ω ∖ { } the ball B ( x, n ) doesnot intersect any other circle on the Hawaiian earring besides that on which x lies.Then N ( S ) contains the kernel of the retraction induced homomorphism r ∗ where r fixes the circle on which x lies and takes all other points to the wedge point. Onthe other hand, N ( S ) must be precisely the kernel of the induced homomorphismby Lemma 5.5.6. N-slenderness, products and free products
In this section we introduce n-slender groups (see [E1]). This will require anunderstanding of the fundamental group of the Hawaiian earring E . Recall thatthe Hawaiian earring is the compact subspace E = ⋃ n ∈ ω C (( , n + ) , n + ) of R ,where C ( p, r ) is the circle centered at p of radius r . The space E can be thought ofas a shrinking wedge of countably infinitely many circles. The fundamental group π ( E ) has a combinatorial characterization which we describe below.We let { a ± n } ∞ n = be a countably infinite set with formal inverses. A map W ∶ W → { a ± n } ∞ n = from a countable totally ordered set W is a word if for every n ∈ ω the set W − ({ a ± n }) is finite. We say two words U and V are isomorphic, U ≃ V ,provided there is an order isomorphism of the domains of each word f ∶ U → V suchthat U ( t ) = V ( f ( t )) . We identify isomorphic words. The class of isomorphismclasses of words is a set of cardinality continuum which we denote W . For each N ∈ ω define the projection p N to the set of finite words by letting p N ( W ) = W ∣{ t ∈ W ∶ W ( t ) ∈ { a ± n } Nn = } . Define an equivalence relation ∼ on words as follows: givenwords U, V ∈ W we let U ∼ V if for each N ∈ ω we have p N ( U ) = p N ( V ) in thefree group F ({ a , . . . , a N }) . For each word U there is an inverse word U − whosedomain is the totally ordered set U under the reverse order and U − ( t ) = U ( t ) − .Given two words U, V ∈ W we form the concatenation U V by taking the domainof
U V to be the disjoint union of U with V , with order extending that of U and V and placing all elements of U before those of V , and U V ( t ) = ⎧⎪⎪⎨⎪⎪⎩ U ( t ) if t ∈ UV ( t ) if t ∈ V .The set W/ ∼ is endowed with a group structure with binary operation given by [ U ][ V ] = [ U V ] , inverses defined by [ U ] − = [ U − ] and the equivalence class of theempty word being the trivial element.Letting HEG denote the group
W/ ∼ , the free group F ({ a , . . . , a N }) , whichwe shall denote HEG N , may be though of as a subgroup in HEG . The wordmap p N gives a group retraction HEG → HEG N which we also denote p N . Theword map p N given by the restriction p N ( W ) = W ∣{ t ∈ W ∶ W ( t ) ∈ { a ± n } ∞ n = N + } induces another group retraction from HEG to the subgroup
HEG N consistingof those equivalence classes which contain words involving no letters in { a ± n } Nn = .Let p N denote this group retraction. By considering a word W as a concatena-tion of finitely many words in the letters { a ± n } Nn = and finitely many words in theletters { a ± n } ∞ n = N + we obtain an isomorphism HEG ≃ HEG N ∗ HEG N . The ho-momorphism p N corresponds to the topological retraction of E to the subspace ⋃ n ≤ N C (( , n + ) , n + ) and similarly for p N and ⋃ n > N C (( , n + ) , n + ) . We arenow ready for the following definition: Definition 6.1.
A group G is noncommutatively slender (or n-slender) if for eachhomomorphism φ ∶ HEG → G there exists N ∈ ω such that φ = φ ○ p N .This definition was first introduced by K. Eda in [E1]. The additive group on Z was the first nontrivial group known to be n-slender [H], and Eda has shownthat the class of n-slender groups is closed under free-products and direct sums (see[E1]). Torsion-free word hyperbolic groups are known to be n-slender [Co]. For eachinfinite cardinal κ there exists an n-slender group of cardinality κ (for example, thefree group of rank κ ). We give the following alternative characterization of n-slendergroups before moving on to the theorems associated with this section: Lemma 6.2.
A group G is n-slender if and only if for every locally path connectedmetric space X each homomorphism φ ∶ π ( X ) → G has open kernel. Proof.
For the ⇒ direction we suppose that G is n-slender and that φ ∶ π ( X ) → G is a homomorphism, with X a metric path connected, locally path connected space.Letting x ∈ X we claim that for some ǫ > x )in the open ball B ( x, ǫ ) are in the kernel of φ . Were this not the case there wouldexist a sequence of loops { l n } n ∈ ω such that diam ({ x } ∪ l n ([ , ])) ≤ − n and l n isnot in the kernel of φ . By local path connectedness we may pass to a subsequenceand eventually attach the bases of the l n to x via a small path. Thus we may N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 23 assume without loss of generality that the l n are all based at x . Define a map f ∶ E → X by mapping the circle C (( , n + ) , n + ) along the loop l n so that agenerator of π ( C (( , n + ) , n + )) maps to [ l n ] ∈ π ( X, x ) under the restriction f ∗ ∣ π ( C (( , n + ) , n + )) . Now φ ○ f ∗ is a map from HEG to G and so for some N we have n ≥ N implies φ ○ f ∗ ( a n ) =
1. But a n corresponds to one of the twogenerators of π ( C (( , n + ) , n + )) , so that 1 = φ ○ f ∗ ( a n ) = φ ([ l n ]) , a contradiction.Thus such an ǫ must exist, and we get an open cover satisfying the hypotheses ofLemma 2.8 for the subgroup ker ( φ ) , so that ker ( φ ) is open.For the direction ⇐ we let G be a group such that for every locally path connectedmetric space X every homomorphism φ ∶ π ( X ) → G has open kernel. Letting E = X and φ ∶ HEG → G be a homomorphism, ker ( φ ) is an open subgroup of π ( E ) . By Lemma 2.6 there exists some ǫ > B (( , ) , ǫ ) ⊆ E is in ker ( φ ) . Selecting N ∈ ω such that ǫ > N + , we have φ ∣ HEG N is trivial, so that φ = φ ○ p N . Thus G is n-slender. (cid:3) Recall that a space is κ -Lindel¨of if every open cover of the space contains asubcover of cardinality at most κ . It is easily seen that a metric space Z is κ -Lindel¨of if and only if Z has a dense subset of cardinality ≤ κ if and only if Z hasa basis of cardinality ≤ κ . It is also true that if X is a metric κ -Lindel¨of space thenso is L x for each x ∈ X . Lemma 6.2 has the following easy corollary: Corollary 6.3. If X is locally path connected metrizable κ -Lindel¨of and G is n-slender then the image of any homomorphism φ ∶ π ( X ) → G has card ( φ ( G )) ≤ κ . Proof.
Assume the hypotheses. Then ker ( φ ) is open, and by Lemma 2.3 the equiv-alence relation given by left cosets of ker ( φ ) has open equivalence classes. As L x has a dense subset of cardinality ≤ κ , we see that card ( π ( X )/ ker ( φ )) ≤ κ . (cid:3) By considering a wedge of κ circles, each circle of diameter 1, endowed withthe path metric, one has an example of a κ -Lindel¨of space which is completelymetrizable, locally path connected whose fundamental group is free of rank κ . Thusthe conclusion of Corollary 6.3 cannot be strengthened.One can prove other results which give obstructions to surjections from thefundamental group, even when the codomain is not n-slender, such as the following: Theorem 6.4.
Suppose X is a locally path connected κ -Lindel¨of metric space, { G i } i ∈ I is a collection of n-slender groups, and φ ∶ π ( X ) → ∏ i ∈ I G i is a homomor-phism. Then there exists some I ′ ⊆ I with card ( I ′ ) ≤ κ such that ker ( p I ′ ○ φ ) = ker ( φ ) . (Here the map p I ′ ∶ ∏ i ∈ I G i → ∏ i ∈ I ′ G i is projection.)This immediately yields: Theorem 6.5. If X is a locally path connected κ -Lindel¨of metric space and { G i } i ∈ I is a collection of nontrivial n-slender groups with card ( I ) > κ then there is noepimorphism φ ∶ π ( X ) → ∏ i ∈ I G i . Example 14.
Theorem 6.5 need not hold if local path connectedness is dropped.For example there exists a model N of set theory satisfying(1) ZFC (2) 2 ℵ = ℵ (3) ( ∀ κ ≥ ℵ )[ κ = κ + ] (see [Be], 2.19). We consider the space F from Example 1. Since π ( X ) is a freegroup of rank 2 ℵ and the group ∏ ℵ Z is of cardinality 2 ℵ = ℵ , there exists asurjection from π ( X ) to ∏ ℵ Z .Theorem 6.5 also fails in the model N if the hypothesis that the G i are n-slenderis dropped. We consider the space P ω from Example 2 in the model N . We have π ( P ω , x ) ≃ ∏ ω Z / Z ≃ ⊕ ℵ Z / Z ≃ ∏ ℵ Z / Z .That Theorem 6.5 holds is a nontrivial fact, since π ( X ) and ∏ i ∈ I G i can havethe same cardinality (as would happen in the model N above). In a model of ZFC where the generalized continuum hypothesis holds, Theorem 6.5 would holdwithout the local path connectedness assumption or the n-slenderness of the G i simply by noticing thatcard ( π ( X )) ≤ card ( L x ) ≤ card ( X ) ℵ ≤ ( κ ℵ ) ℵ ≤ κ + = κ ≤ card ( I ) < card ( I ) ≤ card ( ∏ i ∈ I G i ) Proof. (of Theorem 6.4) Assume the hypotheses hold and the conclusion fails. Let p j ∶ ∏ i ∈ I G i → G j denote projection to the j -th coordinate. Let x ∈ X and B bea basis for the topology on L x with card (B) ≤ κ and ∅ ∉ B . Pick i ∈ I such thatker ( p i ○ φ ) ≠ ker ( φ ) . Suppose we have defined i α for all α < β < κ + so that for all γ < γ < β we have that ker ( p { i α } α ≤ γ ○ φ ) is a proper superset of ker ( p { i α } α ≤ γ ○ φ ) .Select i β so that ker ( p i β ○ φ ) does not contain ker ( p { i α } α < β ○ φ ) . Such a selection ispossible since we assume that ker ( p I ′ ○ φ ) ≠ ker ( φ ) for all I ′ such that card ( I ′ ) ≤ κ .Now each ker ( p j ○ φ ) is an open subgroup of π ( X, x ) by Lemma 6.2, and so isclosed by Lemma 2.4. As it is clear that ker ( p I ′ ○ φ ) = ⋂ j ∈ I ′ ker ( p j ○ φ ) for any I ′ ⊆ I we know that any ker ( p I ′ ○ φ ) is closed. Pick O ∈ B such that O ∩ ker ( p i ○ φ ) ≠ ∅ and O ∩ ker ( p { i ,i } ○ φ ) = ∅ . For 0 < β < κ + select O β ∈ B such that O β ∩ ker ( p { i α } α ≤ β ○ φ ) ≠ ∅ and O β ∩ ker ( p { i α } α ≤ β + ○ φ ) = ∅ . The O β are pairwise distinct for different indices,so the map κ + → B given by β ↦ O β is an injection, a contradiction. (cid:3) We move on to a result on free products of groups. We first state the followinginstance of Theorem 1.3 in [E2]:
Corollary.
Suppose φ ∶ HEG → ∗ i ∈ I G i is a homomorphism from HEG to a freeproduct. Then there exists N ∈ ω , g ∈ ∗ i ∈ I G i and j ∈ I such that φ ( HEG N ) ≤ gG j g − .We use this to prove the following: Lemma 6.6.
Suppose X is a first countable, locally path connected space and φ ∶ π ( X ) → ∗ i ∈ I G i is a homomorphism. For each x ∈ X there exists a pathconnected neighborhood B x and j ∈ I such that φ ( π ( B x , x )) is contained in aconjugate of G j . Proof.
We first notice that the statement of the lemma actually makes sense, be-cause any change of basepoint would only alter φ by conjugation. Thus we mayassume that the domain of φ is in fact π ( X, x ) . By φ ( π ( B x , x )) we understandthe image of the composition of the map induced by inclusion ι ∶ B x → X with φ .Assuming the lemma is false there exists a sequence of path connected neigh-borhoods { U n } n ∈ ω with ⋂ n ∈ ω U n = { x } and loops { l n } n ∈ ω based at x such that theimage of l n is contained in U n and for every n if φ ([ l n ]) ∈ gG j g − there exists n > n with φ ([ l n ]) ∉ gG j g − . As in the proof of Lemma 6.2 we define a map g ∶ E → X so that the n -th circle in E traces out l n . Then φ ○ g ∗ violates the corollary. (cid:3) N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 25
Lemma 6.6 yields the following theorem, which is similar in flavor to Theorem6.4 but with no mention of n-slender groups:
Theorem 6.7.
Suppose φ ∶ π ( X ) → ∗ i ∈ I G i is a homomorphism, with X a locallypath connected κ -Lindel¨of metric space. Then for some I ′ ⊆ I with card ( I ′ ) ≤ κ wehave φ ( π ( X )) ≤ ∗ i ∈ I ′ G i . Proof.
By the Kurosh subgroup theorem [Ku] we have φ ( π ( X )) = F ( J ) ∗ ( ∗ m ∈ M g m H j m g − m ) where j m ∈ I , H j m ≤ G j m and F ( J ) is a free group generated by J ⊆ ∗ i ∈ I G i . Letting r ∶ F ( J ) ∗ ( ∗ m ∈ M g m H j m g − m ) → F ( J ) be the obvious retraction we notice that as r ○ φ is a map to an n-slender group, ker ( r ○ φ ) is open in π ( X ) by Lemma 6.2. ByLemma 2.6 we have an open cover U of X such that any loop in an element of U isin ker ( r ○ φ ) . Let U be a refinement of U such that U, V ∈ U with U ∩ V ≠ ∅ implies U ∪ V is contained in an element of U . Let U be a refinement of U consisting ofpath connected open sets. Since X is κ -Lindel¨of we may assume card (U) ≤ κ . Thefollowing is a statement of Theorem 7.3 of [CC] part (2) (see proof of Theorem 3.14for definition of 2-set simple): Theorem. If ψ ∶ π ( X ) → H is a group homomorphism and U is a 2-set simplecover rel ψ with nerve N (U) then ψ ( π ( X )) is a factor group of π ( N (U)) .It is clear that U is 2-set simple rel r ○ φ , so F ( J ) is the homomorphic imageof π ( N ) , and since N has only κ -many vertices we know κ ≥ card ( π ( N )) ≥ card ( F ( J )) .We show that M is of cardinality at most κ . Then we will let I = { j m } m ∈ M ∪⋃ m ∈ M I m where g m ∈ ∗ j ∈ I m G j and I be such that F ( J ) ≤ ∗ j ∈ I G j , where each I m is finite and card ( I ) ≤ κ . Thus I ′ = I ∪ I will be of cardinality ≤ κ and clearly φ ( π ( X )) ≤ ∗ i ∈ I ′ G i ..We show card ( M ) ≤ κ by demonstrating that if ψ ∶ π ( X ) → ∗ m ∈ M Γ m is onto,with each group Γ m nontrivial, then card ( M ) ≤ κ . By Lemma 6.6 we can obtaina cover V of X by open balls such that each loop with image in an element of V maps to a conjugate of one of the Γ m . As X is κ -Lindel¨of we may assumecard (V ) ≤ κ . For each B ∈ V select an m B ∈ M such that π ( B ) maps under φ to a conjugate of Γ m B and let M ′ = { m B } B ∈V . Then card ( M ′ ) ≤ κ . We show thatcard ( M ∖ M ′ ) ≤ κ and we shall be done. Let now r ∶ ∗ m ∈ M Γ m → ∗ m ∈ M ∖ M ′ Γ m be theobvious retraction. Refine V to an open cover V by path connected open sets suchthat U ∩ V ≠ ∅ implies U ∪ V is included in some element of V and card (V) ≤ κ .Now the cover V is 2-set simple rel the map r ○ ψ ∶ π ( X ) → ∗ m ∈ M ∖ M ′ Γ m , so theimage of r ○ ψ is a homomorphic image of the nerve of V , and so r ○ ψ has image ofcardinality ≤ κ . Then card ( M ∖ M ′ ) ≤ κ and we are done. (cid:3) The following corollary is immediate:
Corollary 6.8. If X is a locally path connected separable metric space then π ( X ) is not a free product of uncountably many nontrivial groups. More generally if X isa locally path connected κ -Lindel¨of metric space then π ( X ) is not a free productof > κ many nontrivial groups.The comparable result for a compact space holds as well: Theorem 6.9. If X is a Peano continuum and φ ∶ π ( X ) → ∗ i ∈ I G i is a homomor-phism, then for some finite I ′ ⊆ I we have φ ( π ( X )) ≤ ∗ i ∈ I ′ G i . Proof.
The proof runs the same as the proof of Theorem 6.7 except that the imagesof the fundamental groups of the nerves of the covers become finitely generated.Thus F ( J ) is finite rank and the M in the corresponding claim is finite for thesame reason. (cid:3) Lemma 6.6 also yields the following result for Polish spaces:
Theorem C.
Suppose X is locally path connected Polish and π ( X ) ≃ ∗ i ∈ I G i witheach G i nontrivial. The following hold:(1) card ( I ) ≤ ℵ (2) Each retraction map r j ∶ ∗ i ∈ I G i → G j has analytic kernel.(3) Each G j is of cardinality ≤ ℵ or 2 ℵ .(4) The map ∗ i ∈ I G i → ⊕ i ∈ I G i has analytic kernel. Proof.
Part (1) is from Corollary 6.8 . Part (3) follows from part (2) by Theorem3.13. For part (3), we use part (1), and the kernel of the map ∗ i ∈ I G i → ⊕ i ∈ I G i is precisely ⋂ i ∈ I ker ( r i ) where r i is the retraction map to G i . Thus the kernel of ∗ i ∈ I G i → ⊕ i ∈ I G i is a countable intersection of analytic subgroups (by part (2)) andtherefore analytic.It remains to prove part (2). Fix x ∈ X . It suffices to prove that if φ ∶ π ( X, x ) → G ∗ H is an isomorphism and r ∶ G ∗ H → G is the retraction map to G then ker ( r ○ φ ) is analytic. By Lemma 6.6 there exists an open cover U of X by open balls suchthat φ ( π ( B y , y )) is contained in a conjugate of G or in a conjugate of H for each B y ∈ U . Let U be a refinement of U such that any two overlapping elements of U have union contained in an element of U . Let U be a refinement of U by pathconnected open sets, with U countable. Let U G denote those elements of U whichmap under φ composed with the inclusion map into a conjugate of G , and similarlyfor U H . Thus U G ∪ U H = U and it is possible that U G ∩ U H ≠ ∅ .Define π (U , x ) , π (U G , x ) and π (U H , x ) as in subsection 5.2. Notice that U is 2-set simple rel the quotient map q ∶ π ( X, x ) → π ( X, x )/ π (U , x ) . Thenby part (2) of Theorem 7.3 (quoted in our proof of Theorem 6.7) we know that π ( X, x )/ π (U , x ) is countable. Let { w n } n ∈ ω ⊆ G ∗ H be a countable collection suchthat q ( φ − ({ w n } n ∈ ω )) = π ( X, x )/ π (U , x ) . Let { h m } m ∈ ω ⊆ H be those elementsof H which occur in the words { w n } n ∈ ω and let { g n } n ∈ ω be defined by letting g n = r ( w n ) .The normal subgroup φ − (⟨⟨{ h m } m ∈ ω ⟩⟩) is analytic by Theorem 3.10 part (5)and the normal subgroup π (U H , x ) is analytic as well. Thus the normal subgroup φ − (⟨⟨{ h m } m ∈ ω ⟩⟩) π (U H , x ) is analytic by Theorem 3.10 part (4). We shall be done if we prove that ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) = ⟨⟨ H ⟩⟩ The inclusion ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) ≤ ⟨⟨ H ⟩⟩ is self-evident. For the otherinclusion we have G ∗ H = { w n } n ∈ ω φ ( π (U , x )) = { w n } n ∈ ω φ ( π (U H , x )) φ ( π (U G , x )) = { w n } n ∈ ω ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) φ ( π (U G , x )) = { g n } n ∈ ω ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) φ ( π (U G , x )) = ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )){ g n } n ∈ ω φ ( π (U G , x )) N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 27
Let r H ∶ G ∗ H → H be the retraction map and h ∈ H . We know h = ww ′ where w ∈ ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) and w ′ ∈ { g n } n ∈ ω φ ( π (U G , x )) . Now h = r H ( h ) = r H ( w ) r H ( w ′ ) = r H ( w ) , so that H ≤ r H (⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x ))) . The group φ ( π (U H , x )) is generated by elements of form h w with h ∈ H , and the same is ob-viously true of ⟨⟨{ h m } m ∈ ω ⟩⟩ . Thus the subgroup ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) is gen-erated by elements of form h w . Now r H ( h w ) = h r H ( w ) ∈ ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) for any h w ∈ ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) . Thus we have shown that H ≤ r H (⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x ))) ≤ ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) so that ⟨⟨ H ⟩⟩ ≤ ⟨⟨{ h m } m ∈ ω ⟩⟩ φ ( π (U H , x )) and we have the other inclusion. (cid:3) We use the space F from Example 1 to check aspects of sharpness for TheoremC. Since π ( F ) is isomorphic to the free group F ( ℵ ) it is clear that we cannotdrop local path connectedness and still assert part (1) of the conclusion. Local pathconnectedness is also required for parts (2) and (3) by the example that follows. Itseems unlikely that part (4) holds absent local path connectedness. Example 15.
By 2.12 in [Be] there exists a model Q of set theory satisfying(1) ZFC (2) 2 ℵ = ℵ We consider the space F in the model Q . We have π ( F ) ≃ F ( ℵ ) ≃ F ( ℵ ) ∗ F ( ℵ ) . The retraction to F ( ℵ ) cannot have analytic kernel by Theorem 3.16 part(2).It is fascinating to note that the abelian version of Theorem C fails. Recall fromExample 2 that we had subgroups G and H of π ( P ω , x ) such that card ( H ) = ℵ , G was not of nice pointclass with BP and π ( P ω , x ) ≃ H ⊕ G . Also, π ( P ω , x ) ≃ ⊕ ℵ Z / Z so the index of a direct decomposition into nontrivial groups can fail tobe countable even for a Peano continuum.7. Nice Pointclasses
We end with a brief discussion of Polish pointclasses. We define the projectivepointclasses Σ n , Π n , ∆ n for n ∈ ω ∖ { } . We have seen that Σ is the class of analyticsets and Π is the class of coanalytic sets. Let ∆ = Σ ∩ Π . For n ≥ n is the class of continuous images of sets of type Π n − Π n is the class of complements of sets in Σ n ∆ n = Π n ∩ Σ n All Borel sets are easily seen to be of type ∆ . That ∆ is precisely the class ofBorel sets is a theorem of Suslin (see 14.11 in [Ke]). The projective pointclasses sitnaturally in an array Σ Σ ⋯ ∆ ∆ ⋯ Π Π ⋯ with each pointclass containing all pointclasses to the left (e.g. Π ⊆ ∆ ). In anuncountable Polish space all these inclusions are proper. Each ∆ n is a σ -algebra.Each Σ n is closed under taking countable intersections, countable unions and imagesunder continuous maps between Polish spaces. Each Π n is closed under countableunions, countable intersections and continuous preimages.Each Σ n is a nice pointclass and we have already noted that Σ is nice with BP.Unfortunately, the BP status of the other Σ n cannot be decided from ZFC alone.For example, G¨odel furnished a model of set theory, L , in which the following hold:(1) ZFC (2)
GCH (3) There exists a ∆ set which does not have BP.(see [G], [N], [A]). Thus it is consistent with ZFC that Σ is the only nice projectivepointclass with BP. Other models of set theory are less stingy with nice pointclasseshaving BP. We discuss two situations in which the nice pointclasses with BP aremore plentiful: models which satisfy Martin’s axiom and ¬ CH , and a model of settheory constructed by Shelah [Sh1].Martin’s axiom (abbreviated MA ) is a principle of combinatorial set theory. Weshall not give a formal statement of this principle (the interested reader may consult[Fr]), but state a relevant consequence: if Z is a Polish space then the σ -algebraof subsets having BP is closed under unions of index less than 2 ℵ . If CH holdsthen this statement is uninformative ( MA is in fact a trivial consequence of CH ),but the point is that there exists a model of ZFC + MA + ¬ CH (see [ST]).The principle MA affords more applications of some of the theory developed inthis paper. If P is a Polish pointclass we let S κ < ℵ (P) be the closure of P underunions and intersections over indices of cardinality < ℵ . Proposition 7.1.
The following hold:(1) (
ZFC + MA ) If P is nice with BP then so is S κ < ℵ P .(2) ( ZFC + MA + ¬ CH ) S κ < ℵ Σ is nice with BP. Proof.
For (1) we note that MA makes the σ -algebra of Baire property sets closedunder unions and intersections over cardinals less than 2 ℵ . Thus S κ < ℵ P has BP,and is obviously nice.For (2) we recall the theorem of Sierpinski that any Σ set is an ℵ union ofBorel sets (see 38.8 in [Ke]). Then by ¬ CH we have Σ ⊆ S κ < ℵ Σ and we apply(1), using the obvious fact that the operation S κ < ℵ is idempotent. (cid:3) The following proposition illustrates some consequences that can be derived from
ZFC + MA + ¬ CH : Proposition 7.2. ( ZFC + MA + ¬ CH ) Let X be a Peano continuum. Thequotient π ( X )/ G is either finitely generated or of cardinality 2 ℵ for the following G : (1) The center G = Zπ ( X ) .(2) G = ⟨⟨[{ l α } α < ℵ ]⟩⟩ where { l α } α < ℵ is any collection of loops of cardinality ℵ .(3) G = ( π ( X )) ( α ) for any α < ℵ . Proof.
For (1), if G is open then we are done by Theorem 3.14. If G is not openthen by Lemma 3.12 there is a point y ∈ X and sequence of loops { l n } n ∈ ω withdiam ( l n ) ↘ [ l n ] ∉ G . Notice that N DEFINABLE SUBGROUPS OF THE FUNDAMENTAL GROUP 29 l ∈ ⋃ Zπ ( X ) ⇐⇒ ( ∀ l ′ ∈ L y )( ∃ H ∈ H y )[ H homotopes l ∗ l ′ to l ′ ∗ l ] which illustrates that ⋃ Zπ ( X ) is Π . By Proposition 7.1 we know that Σ hasBP and so Π has BP. Since Π is also closed under continuous preimages we havepart (1) by Lemma 3.11.For part (2) we notice that { l α } α < ℵ is S κ < ℵ Σ and so we can directly ap-ply Theorem 3.14 since S κ < ℵ Σ is nice with BP. For part (3) one can prove byinduction over all ordinals less than 2 ℵ that ( π ( X )) ( α ) is of type S κ < ℵ Σ . (cid:3) Very generous applications of our theory can be derived from 7.17 in [Sh1]:
Corollary.
There exists a model R of set theory in which the following hold:(1) ZFC (2) All projective subsets of { , } ω have BP.In R the conclusions of Theorems 3.13, 3.14, and B apply to subgroups definablefrom first-order formulas in conjunction with iterations of countable set operationsapplied to subgroups already known to be of a projective pointclass. We illustratewith an example. In R if X is a Peano continuum and x ∈ X then letting G be thenormal subgroup generated by the set of elements that are not a cube of a centralelement: G = ⟨⟨{ g ∈ π ( X, x ) ∶ ¬ ( ∃ h ∈ Zπ ( X, x ))[ h = g ]}⟩⟩ we have that G is a Σ subgroup and so the quotient π ( X, x )/ G is either finitelygenerated or of cardinality 2 ℵ . References
A. J. W. Addison,
Some consequences of the axiom of constructibility , Fund. Math. 46(1959), 337-357.Be. J. L. Bell,
Boolean-Valued Models and Independence Proofs in Set Theory , OxfordUniversity Press, 2 edition, 1985.Bu. J. P. Burgess,
Equivalences generated by families of Borel sets . Proc. Amer. Math.Soc., 69 (1978), 323-326.BF. J. Brazas and P. Fabel,
Thick Spanier groups and the first shape group , RockyMountain Journal of Mathematics 44 (2014), 1415-1444.BS. W. A. Bogley and A. J. Sieradski,
Universal path spaces , preprint, Oregon StateUniversity and University of Oregon, 1998.CC. J. W. Cannon and G. R. Conner,
On the fundamental groups of one-dimensionalspaces , Topology and its Applications 153 (2006), 2648-2672.CoCo. G. R. Conner and S. M. Corson,
On the first homology of Peano continua , Fund.Math. 232 (2016), 41-48.Co. S. M. Corson,
Torsion-free word hyperbolic groups are noncommutatively slender ,Int. J. Algebra Comut. 26 (2016), 1467-1482.E1. K. Eda,
Free σ -products and noncommutatively slender groups , J. Algebra 148(1992), 243-263.E2. K. Eda, Atomic property of the fundamental groups of the Hawaiian earring andwild locally path-connected spaces , J. Math. Soc. Japan 63 (2011), 769-787.Fr. D. H. Fremlin,
Consequences of Martin’s Axiom , Cambridge University Press, 1984.FZ. H. Fischer and A. Zastrow,
Generalized universal covering spaces and the shapegroup , Fund. Math. 197 (2007), 167-196.G. K. G¨odel,
The consistency of the axiom of choice and of the generalized continuumhypothesis , Ann. Math. Studies 3 (1940).H. G. Higman,
Unrestricted free products and varieties of topological groups , J. LondonMath. Soc. 27 (1952), 73-81.Ke. A. Kechris,
Classical Descriptive Set Theory , Springer-Verlag, 1995.
Ku. A. G. Kurosh,
The Theory of Grops , Vol. 2, Chelsea, 1960.My. J. Mycielski,
Independent sets in topological algebras , Fund. Math. 55 (1964), 139-147.MaSo. D. A. Martin, R. M. Solovay,
Internal Cohen extensions , Ann. Math. Logic 2 (1970),143-178.N. P. S. Novikov,
On the consistency of some propositions of the descriptive theory ofsets (in Russian) , Trudy Mat. Inst. Steklova 38 (1951), 279-316.P. J. Pawlikowski,
The fundamental group of a compact metric space , Proc. of theAmer. Math. Soc. 126 (1998), 3083-3087.Sh1. S. Shelah,
Can you take Solovay’s inaccessible away? , Israel Journal of Mathematics,48 (1), 1-47.Sh2. S. Shelah,
Can the fundamental (homotopy) group of a space be the rationals? , Proc.Amer. Math. Soc. 103 (1988), 627–632.Si. J. H. Silver,
Counting the number of equivalence classes of Borel and coanalyticequivalence relations . Ann. Math. Logic, 18 (1980), 1-28.Sp. E. H. Spanier,
Algebraic Topology , McGraw-Hill, 1966.ST. R. M. Solovay and S. Tennenbaum,
Cohen extensions and Souslin’s problem , Ann.of Math. 94 (1971), 201-245.
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