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GREGORY aur 8 2021. 28, January : eto fkreso ooopim from Ker of and kernels Earring, of Hawaiian section the of group fundamental where nqept-itn rpry( property fibration Hurewicz -lifting using unique by groups fundamental uncountable Abstract. steivrelmto udmna ruso oyerlapp polyhedral of groups fundamental of of inverse the is rmplhda prxmtosof approximations polyhedral from h e fpt-opnnso h nes limit, inverse the of lim path-components of the the n pc mle that implies space ing o oeigsae.I atclr ecnie h inverse the consider we of particular, spaces covering In of sequence spaces. covering for fteivrelmtcnb xrse ymaso h eie i derived the of means by lim expressed be can limit inverse the of X osqec forcmuaini htpath-connectednes that is computation our of consequence A π sa plcto eso that show we application an As . 1 F ( b X ←− ←− stecnnclivrelmto nt akfe groups, free rank finite of limit inverse canonical the is NES IISO COVERINGS OF LIMITS INVERSE sucutbe ocruvn hsdffiut,w express we difficulty, this circumvent To uncountable. is ) 1 hc s oee,ntrosydffiutt opt when compute to difficult notoriously however, is, which , nti ae edvlpanwapoc otesuyof study the to approach new a develop we paper this In n lim and ←− 1 π ple osqecso onal rusarising groups countable of sequences to applied N EA PAVE PETAR AND 1 1 1. ( OFAGHERFORT WOLFGANG , X e supplements ) Introduction X itn spaces lifting ti nw httepath-connectivity the that known is It . 1 X G· . Ker π Z F b 1 SI ˇ o hr)a replacement a as short) for ( ( F b X to C ´ = ) 4 nˇ in ) A . F b X π e 2 1 6= ntrso the of terms in , A UTSKENT CURTIS , ( X ( G· F b hr ˇ where ) Ker steinter- the is ) uvdgop [23], groups curved . ahmtc Depart- Mathematics . roximations vrelimit nverse n rnsN1-0083, grants and ena effec- an been y ihthe with s B ii fa of limit lsses[20], systems al (1 rus The groups. e flift- of s G tlgop of groups ntal ntl pre- finitely r ,n π uncountable sthe is ) 1 ( ( F X b ), op of roups ) 3 , 2 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ and in many other situations. The fundamental difference between the two cases is of a local nature. If the of a Peano X is uncountable, then by [27] it is not semilocally simply connected at some point. As a consequence most of the of π1(X) do not correspond to a covering space over X, which represents a major obstacle for a geometric study of these groups. This work is part of a wider program to study fun- damental groups of ‘wild’ Peano continua where the role of covering spaces is taken by more general fibrations with the unique path-. These fibrations were introduced by Spanier [28, Chapter 2] who developed much of the theory of covering spaces within this more general setting. Since the term ‘Hurewicz fibration with the unique path-lifting property’ is some- what impractical we call them lifting spaces and the corresponding maps from the total space to the are called lifting projections. The main advantage of lifting spaces over covering spaces is that the for- mer are closed with respect to composition and arbitrary inverse limits - see [28, Section II.2]. Most notably, the of a sequence of cov- ering spaces over X is always a lifting space over X (and is not a covering , unless the sequence is eventually constant). If X is semilocally simply-connected (e.g., a CW-complex), and the sequence is not eventually constant, then the limit lifting space is path-disconnected (see Corollary 2.2). This is a geometric reflection of the fact that the fundamental group of the base is countable while the fibre of the lifting projection is uncountable. However, if X is not semilocally simply-connected, then its fundamental group is uncountable by Shelah’s theorem and the limit space can be path- connected or not path-connected, depending on the interplay between π1(X) and the sequence of subgroups corresponding to the coverings in the inverse sequence. The path-components of an inverse limit of covering spaces are classically determined by the derived inverse limit functor lim1 applied to a sequence of ←− subgroups of π (X). Difficulties arise in the computation of lim1 for inverse 1 ←− sequences of uncountable groups, which make this an ineffective approach to determining path-connectivity (see discussion at the of Section 2 and Examples 3.7-3.11 in Section 3) . In Section 3, we consider inverse sequences of coverings over some polyhedral expansion of the base space X and study the path-connectedness of the limit. We prove the following result, which allows one to work with inverse sequences of countable groups and thus avoid the difficulties of computing lim1 for uncountable groups. ←−

Theorem 1 (Theorem 3.4). Every inverse limit of covering projections over a Peano continuum X is homeomorphic to an inverse limit of covering pro- jections over a polyhedral expansion of X. UNCOUNTABLEGROUPS 3

The main result of Section 3, Theorem 3.5, completely describes the funda- mental group and the set of path components of an inverse limit of coverings over some polyhedral expansions of a Peano continuum. The statement is quite technical, but it leads to the following important consequence which characterizes path-connectedness of the limit space. Corollary 2 (Corollary 3.6). Let p: X → X be the inverse limit of covering maps over a Peano continuum X. Then X is path-connected if, only if, the inverse sequence of fundamental groups,e π1(Xi), satisfies the Mittag-Leffler property and the natural e e π (X ) π (X) −→ lim 1 i 1 ←− π1(Xi)! is surjective, where Xi → Xi are covering projectionse of a polyhedral expan- sion {Xi} of X. e Even for the well-studied group G, apart from the fact that G contains the of countable rank and is thus dense in its shape group F , very little is known about the size of G in F . The above corollary leads to interesting algebraic applications concerning the of the fundamentalb group into the shape group which we considerb in Section 4.

Given groups G and A, we let KerA(G) be the intersection of kernels of all homomorphisms from G to A. If G is a free group and B(1,n) is a Baumslag-Solitar group, then KerZ(G) is exactly the commutator of G and KerB(1,n)(G) is exactly the second derived subgroup of G, see [11]. However, F is locally free but non-free and the corresponding KerZ(F ) prop- erly contains the commutator subgroup of F . In fact, KerZ(F ) is sufficiently large to beb a supplement of G in F while KerB(1,n)(F ) is not. b b b Theorem 3 (Theorem 4.4 and Theoremb 4.5). Theb group F is equal to the internal product of its subgroups G and KerZ(F ), while the internal product of G and KerB(1,n)(F ) is a proper subgroup of F . b b b b 2. Preliminaries on inverse limits of spaces and groups

The main object of our study are lifting spaces that arise as inverse limits of covering spaces, so let us consider the following sequence of regular covering maps : Xi → X, together with the inverse limit map p: X → X:

f˜1 f˜2 e X1 o X2 o X3 o ···· o Xe

p1 p2 p3 p e e e e     X X X ······ X 4 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´

For each i we identify π1(Xi) with its image pi∗(π1(Xi)) which is a of π1(X). By standard covering space theory the bonding maps f˜i : Xi+1 → Xi are coveringe maps and we have a decreasinge sequence of normal subgroups of π1(X) e e π1(X) D π1(X1) D π1(X2) D · · ·

Note that the converse is not truee for generale spaces, because a subgroup of π1(X) may not be a covering subgroup, i.e. it does not necessarily come from a covering of X. However, if X is locally path-connected, then by [28, Theorem II, 5.12] every decreasing sequence of normal covering subgroups of π1(X) uniquely determines an inverse sequence of regular coverings over X. Let F (p) denote the fibre of the lifting projection p. Then F (p) is the inverse limit of the fibres F (pi), and so by [28, Theorem II, 6.2] it can be naturally identified with the inverse limit of the quotients F (p) = lim(π (X)/π (X )). ←− 1 1 i Since all bonding maps in the inverse sequencee are fibrations, it is well- known (see [4, 22]) that the homotopy groups of π1(X) can be expressed in terms of the homotopy groups of Xi: e π (X) = lim π (X )= π (X ), π (X) = lim 1π (X ) 1 ←− 1 i e 1 i 0 ←− 1 i i \ and π (X) =eπ (X) for ne > 1. Here lime denotese the inverse limite functor n n ←− on groups and lim1 is its first (see [19, Section 11.3]). For ←− commutativee groups these functors can be defined as and of the homomorphism ϕ: π1(Xi) → π1(Xi), given as

ϕ(u1, u2, u3,...) = (u1 − f˜1∗(u2), u2 − f˜2∗(u3),...) Q e Q e and so they fit in the ϕ 0 → lim π (X ) → π (X ) −→ π (X ) → lim 1π (X ) → 0. ←− 1 i 1 i 1 i ←− 1 i The non-commutative caseY is more delicate:Y in that case lim 1π (X ) is de- e e e ←−e 1 i fined as the quotient of π1(Xi) under the equivalence given as e ˜ −1 (ui) ∼ (vi) ⇔ (vi) =Q (xiuifi∗(xi+1) ) for some (xi) ∈ π1(Xi) e Y (see [17] for detailed treatment of the non-commutative case). e The values of lim1 are notoriously hard to compute. Here we will be only ←− interested whether lim1 of a sequence is trivial, and this can be settled if ←− we can show that the inverse sequence satisfies the Mittag-Leffler condition, which we now define. In an inverse sequence of groups

G1 ←− G2 ←− G3 ←− · · · UNCOUNTABLEGROUPS 5 for a fixed j the image of the homomorphism Gi → Gj decreases as i goes toward infinity. The inverse sequence is said to satisfy the Mittag-Leffler con- dition if for every j the sequence {Im(Gi → Gj) | i = j, j + 1,...} stabilizes. Clearly, sequences with epimorphic bonding maps satisfy the Mittag-Leffler condition. On the other hand, if the bonding maps are , then the sequence satisfies he Mittag-Leffler condition if, and only if, it is eventually constant. The following result is proved in [17] (see also [19, Theorem 11.3.2]):

Proposition 2.1. If an inverse sequence {Gi}, of groups, satisfies the Mittag-Leffler condition, then lim1 G = {1}. Conversely, if lim1 G = {1} ←− i ←− i and each Gi is countable, then {Gi} satisfies the Mittag-Leffler condition.

As we explained before, the sequence of fundamental groups induced by an inverse sequence of coverings satisfies Mittag-Leffler condition only if it is constant from some point on. Thus we get immediately the following corollary.

Corollary 2.2. Let X be the inverse limit of an inverse sequence of covering maps pi : Xi → X and assume that for some i the group π1(Xi) is countable. Then either the sequencee is eventually constant (which implies that p: X → X is a coveringe map), or X is not path-connected. e e In particular an inverse limit of coverings over a countable CW-complex is either a covering or its totale space is not path-connected.

This last statement is not surprising if one considers the tail of the exact homotopy sequence of the fibration p: X → X, shown in the following diagram together with the above mentioned identifications: e p∗ ∂ 1 / π1(X) / π1(X) / π0(F (p)) / π0(X) / ∗

e e 1 / lim π (X ) / π (X) / lim(π (X)/π (X )) / lim1 π (X ) / ∗ ←− 1 i 1 ←− 1 1 i 1 i

If the sequencee is not eventually constant, theneπ0(F (p)) is uncountable.e But if X is a countable CW-complex, then π1(X) is a countable group and so ∂ cannot be surjective.

However, if π1(X) is uncountable, then ∂ can be surjective. For example, the lifting projection over the infinite product of circles, obtained by ’un- wrapping’ one circle at a time, as in (S1)∞ ← (R × S1 × S1 × . . .) ← (R × R × S1 × . . .) ←······← R∞ has a path-connected total space in spite of the fact that it is not eventually constant. Another, less obvious example is a sequence of 2-fold coverings 6 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ over the Hawaiian earring whose limit space is path-connected (see [9, Sec- tion 2.7]). In both cases we have decreasing sequences of uncountable fun- damental groups that do not satisfy the Mittag-Leffler condition but their derived inverse limits are nonetheless trivial. Of course, one can easily find examples of inverse limits of coverings of the Hawaiian whose total space has uncountably many path-components. Distinguishing between different cases is hindered by the difficulties in the explicit computation of the derived limit functor for decreasing sequences of uncountable groups. We present one approach to this problem in the follow- ing section, while in the last section we consider the algebraic implications of the path-(dis)connectedness of the limit space.

3. Inverse limits of coverings over an expansion of X

Our approach to reducing the complexity of computing the derived limit functor to the more manageable countable case is to consider a polyhedral expansion {Xi} of X and represent a lifting projection p: X → X as an in- verse limit of covering maps over the polyhedral expansion, i.e., find covering maps pi : Xi → Xi that satisfy the following diagram. e

e f˜1 f˜2 X1 o X2 o X3 o · · · o X

p1 p2 p3 p e e e e     X1 o X2 o X3 o · · · o X f1 f1

There are several standard methods by which a compact can be represented as a limit of an inverse sequence of polyhedra (see Mardesic- Segal [26]). The one that best suits our purposes is by nerves of coverings, which we briefly recall (see [26, Appendix 1] for details). Every metric compactum X admits arbitrarily fine finite open coverings. Moreover, for every finite open covering U = {U1,...,Un} of X there exists a subordinated partition of unity {ρi : X → [0, 1] | i = 1,...,n}. Let N(U) be the nerve of the covering U, i.e., the simplicial complex whose vertices are elements of U, and whose simplices are spanned by elements of U with non-empty intersection. Then the formula n f(x) := ρi(x) · Ui i X=1 defines a map f : X → |N(U)| (where |N(U)| is the geometric realization of the nerve). It is well known that the choice of the partition of unity does not affect the homotopy class of f, so the f∗ : π1(X) → π1 |N(U)| depends only on the U.  UNCOUNTABLEGROUPS 7

Lemma 3.1. If each element of U is path-connected, then f∗ : π1(X) → π1 |N(U)| is surjective.

Proof. Without loss of generality we may assume that the partition of unity subordinated to U is reduced in the sense that for every i there exists xi ∈ Ui, such that ρi(xi) = 1. For every pair of intersecting sets Ui,Uj ∈U we may choose a path in Ui ∪ Vj between xi and xj. These paths determine a map g : |N(U)(1)| → X from the 1-skeleton of the nerve to X. One can check that, for every 1-simplex σ in N(U)(1), the image f(g(σ)) is contained in the open star of σ in N(U), which implies that f ◦ g is homotopic to the (1) (1) inclusion i: |N(U) | ֒→ |N(U)|. Since i∗ : π1(|N(U) |) → π1(|N(U)|) is surjective, f∗ is also surjective. 

If U ′ is a covering of X that refines U (i.e. every element of U ′ is contained in some element of U), then there is an obvious simplicial map N(U ′) → N(U). By iterating the refinements we obtain an inverse system of polyhedra

|N(U1))| ←− |N(U2))| ←− |N(U3))| ←− . . . whose limit is X. Moreover, if X is locally path-connected, then we may choose covers of X whose elements are path-connected. As an interesting aside, Lemma 3.1 immediately implies the following well-known fact, which also follows immediately from the Hahn-Mazurkiewicz theorem and results of Krasinkiewicz [24, Theorems 4.1 and 4.2]. Corollary 3.2. Every Peano continuum X can be represented as the limit of an inverse system of finite polyhedra X o X o X o · · · o lim X = X 1 2 3 ←− i such that the homomorphisms π1(X) → π1(Xi) induced by the projection maps are surjective.

We do not know whether every inverse limit of coverings over a path- connected and locally path-connected base X can be derived from an inverse limit of coverings over some expansion of X. However, if we assume that X is compact, we may use the approximation by nerves of coverings to prove the following theorem. Lemma 3.3. Let X be a Peano continuum and g : X → Y be a continuous map into a polyhedron Y . If U is a cover of X by path connected open sets such that the preimage of the open star of any vertex of Y is contained in an element of U, then Ker(g∗) ⊂ π1 X; 2 U .

Proof. If we denote by f : X → |U| the map from X to the realization of the nerve of U, then the kernel of the homomorphism f∗ : π1(X) → π1(|U|) is π1(X; 2 U), see [8, Lemma 3.3]. Since the preimage of the open star of any vertex of Y is contained in an element of U, there exists a map h : Y → |U| such that f∗ = h∗ ◦ g∗. Thus Ker(g∗) ⊂ ker(f∗)= π1 X; 2 U .   8 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´

Let U be an open cover of X. We will use π1(X; U) to denote the U-Spanier subgroup, i.e. π1(X; U) is the subgroup of π1(X,x0) generated by

[α ∗ β ∗ α] | α : (I, 0) → (X,x0) and Im(β) ⊂ U for some U ∈U . Theorem 3.4. If X is a Peano continuum, then every inverse sequence of covering projections over X is homeomorphic to a pull-back of an inverse sequence of covering maps over a given polyhedral expansion of X. As a consequence, every inverse limit of covering maps over X is homeomor- phic to an inverse limit of covering maps over a given polyhedral expansion of X.

Proof. Let

X o X o X o · · · o lim X = X 1 2 3 ←− i be a polyhedral expansion of X, such that the homomorphisms π1(X) → π1(Xi) are surjective. Let p: X → X be an arbitrary covering map. Then p evenly covers elements of some open cover U of X and π1(X) ⊇ π1(X; U), where the latter denotes the U-Spaniere subgroup of π1(X). Since X is compact we may assume that U is finite. Choose another finite covere V of X by path connected open sets whose double 2 V (set of all unions of two elements of V) refines U. By Theorem 5 of [26] there exists an i such that the preimage of the open star of any vertex of Xi is contained in an element of V. Then by Lemma 3.3, we have

π1(X) ⊇ π1(X; U) ⊇ π1(X; 2 V) ⊇ Ker π1(X) → π1(Xi) . Therefore, if  e X o X1 o X2 o X3 o · · · is an inverse sequence of coverings over X we may find for each i some index e e e j(i) such that π1(Xi) ⊇ Ker π1(X) → π1(Xj(i)) . Denote K := Ker π (X) →π (X ) and let X → X be the covering i e 1 1 j(i) j(i) j(i) map that corresponds to the subgroup Im π1(Xi) → π1(Xj(i)) . By the ˜ lifting theorem, there is a fibre-preserving map fi : Xi → Xj(i) for which the following diagram commutes e e f˜i Xi / Xj(i)

e   X / Xj(i) UNCOUNTABLEGROUPS 9

To compute the restriction of f˜i on the fibres we examine the following with exact rows

1 / π1(Xi)/Ki / π1(X)/Ki / π1(X)/π1(Xi) / 1

∼ e = e    1 / π1(Xj(i)) / π1(Xj(i)) / π1(Xj(i))/π1(Xj(i)) / 1

By the short five-lemma we conclude that f˜i induces an

π1(X)/π1(Xi) → π1(Xj(i))/π1(Xj(i)), and hence a between the fibres of respective covering maps. In other words, the above diagrame is a pull-back of covering maps. By repeating the above argument for all terms in an inverse sequence of covering maps over X we obtain an inverse sequence of covering maps over the given expansion of X with the same limit lifting projection. 

The above theorem shows that for most cases of interest it is sufficient to consider lifting projections over X that are inverse limits of covering maps over some expansion of X as in

f˜1 f˜2 X1 o X2 o X3 o ···· o X

p1 p2 p3 p e e e e     X1 o X2 o X3 o ······ o X f1 f2

Suppose that Xi are polyhedra and that the maps X → Xi induce epi- between respective fundamental groups (e.g. if X is locally path-connected). As in Section 2, we may compare the tail of the exact homotopy sequence of the fibration p with the derived exact sequence of the inverse limit functor to obtain the following diagram

1 / π (X) / π (X) / lim(π (X )/π (X )) / π (X) / ∗ 1 1 ←− 1 i 1 i 0 ϕ e ι e ψe    1 / lim π (X ) / lim π (X ) / lim(π (X )/π (X )) / lim1 π (X ) / ∗ ←− 1 i ←− 1 i ←− 1 i 1 i ←− 1 i The inverse limits lim π (X ) and lim π (X ) are independent of the choice e ←− 1 i ←− 1 i e e of a polyhedral expansion for X. That two polyhedral expansions give pro- isomorphic inverse limit groups is provede in Mardesic-Segal [26, Ch 2] and the pro-isomorphism can be lifted to a pro-isomorphism of the corresponding covering subgroups via the homotopy lifting criterion. The limit lim π (X ) ←− 1 i is usually called the Cechˇ fundamental group (or the first shape group) of X, and is denoted byπ ˇ (X). Since X is not necessarily compact, lim π (X ) is 1 ←− 1 i e e 10 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ not necessarily the first shape group of X. Regardless, we will still denote lim π (X ) byπ ˇ (X). The kernel of the induced map ι: π (X) → πˇ (X) ←− 1 i 1 1 1 is called the shape kernel of X and denotede ShKer(X). Note that by the exactnesse of the abovee diagram we have that Ker(π (X) → lim π (X )) = Ker(π (X) → lim π (X )) = ShKer(X). 1 ←− 1 i 1 ←− 1 i

Clearly, ψ : πe (X) → lime1 π (X ) is a surjective between homo- 0 ←− 1 i geneous sets. Thus, in order to determine π0(X) we need to compute the fibres of ψ.e From the exactnesse of the second row we deduce that lim1 π (X ) is the set of cosets of the action ofe the groupπ ˇ (X)/πˇ (X) ←− 1 i 1 1 on lim(π (X )/π (X )). Then a straightforward diagram chasing shows that ←− 1 i 1 i the fibrese of ψ can be naturally identified with the cosets of the actione of ι(π1(X)) onπ ˇ1(X)e/πˇ1(X). Theorem 3.5. Let X be a path- that admits a polyhedral e expansion fi : X → Xi such that the induced homomorphisms fi∗ are sur- jective and let p: X → X be the inverse limit of a sequence of coverings pi : Xi → Xi as described above. Then the fundamental group of X is deter- mined by the exacte sequence of groups e e 1 → ShKer(X) → π1(X) → πˇ1(X), and the set of path-components of X is determined by the exact sequence of e e groups and based homogeneous sets ι e π (X) −→ πˇ (X)/πˇ (X) −→ π (X) −→ lim1π (X ) −→ ∗ 1 1 1 0 ←− 1 i

The main advantage of Theoreme 3.5 with respecte to the descripe tion of π0(X) given in Section 2 is that all limits and derived limits are taken over inverse sequences of countable groups. e Corollary 3.6. Let p: X → X be the inverse limit of covering maps as in the above theorem. Then X is path-connected if, only if, the inverse sequence e π (X ) ← π (X ) ← π (X ) ←··· 1 1e 1 2 1 3 satisfies the Mittag-Leffler property and the natural homomorphism e e e πˇ (X) π (X ) π (X) −→ 1 ∼ lim 1 i 1 = ←− πˇ1(X) π1(Xi)! is surjective. e e Proof. Theorem 3.5 implies that π (X) is trivial if, and only if lim1π (X ) 0 ←− 1 i is trivial and π1(X) −→ πˇ1(X)/πˇ1(X) is surjective. Since the fundamen- tal groups of polyhedra are countable,e then by Proposition 2.1 the inversee 1 sequence is Mittag-Leffler if and onlye if lim π1(Xi) is trivial.

e UNCOUNTABLEGROUPS 11

Observe that the isomorphism betweenπ ˇ (X)/πˇ (X) and lim π (X )/π (X ) 1 1 ←− 1 i 1 i follows form the Mittag-Leffler property and does not hold in general.   e e Let us consider a couple of examples. Example 3.7. We have already mentioned the inverse sequence of covering maps over the infinite product of circles (S1)∞ ← (R × S1 × S1 × . . .) ← (R × R × S1 × . . .) ←··· It can be replaced by the following inverse sequence of covering maps over polyhedral approximations of (S1)∞ (with horizontal maps the usual pro- jections)

R o R × R o R × R × R o · · · o RN

    S1 o S1 × S1 o S1 × S1 × S1 o · · · o (S1)N The fundamental groups of the products of copies of R are trivial so the cor- responding sequence is Mittag-Leffler. By Corollary 3.6 the path-connectedness of RN is equivalent to the π (S1)N = lim π (S1)n =π ˇ (S1)N . 1 ←− 1 1 Example 3.8. Let T ∞ be the infinite product of circles endowed with the CW-topology, i.e. the of the sequence of spaces ∞ S1 ֒→ S1 × S1 ֒→ S1 × S1 × S1 ֒→··· T ∞ ∞ Clearly, the fundamental group of T is k=1 Z. For each i ∈ N let p : X → T ∞ be the covering map that corresponds to the subgroup π (X )= i i L 1 i ∞ Z. By Corollary 2.2 the inverse limit X := lim X is not path- k=i ←− i connected.e e L e e Remark 3.9. The last result is somewhat counter-intuitive, as one could i−1 ∞ argue that Xi ≈ R × T and that the inverse limit of the coverings is simply the product R∞ but the computation reveals that the geometry of the inverse limite must be different. We leave this as an (easy) exercise for the reader. As a hint, note that the fibre of X → T ∞ is uncountable while the fundamental group of T ∞ is countable. e Example 3.10. Consider the squaring lifting projection p: H∞ → H over the Hawaiian earring H described in [9, Section 2.7]. The map p is ob- tained as a limit of a sequence of 2-fold covering projectionse and, although it resembles at first sight the construction of the dyadic , we have been able to prove by a geometric argument that its total space H∞ is path- connected. We are going to show how this result is reflected in the algebraic computation of the set of path-components of H∞. e As explained in [9, Section 3.3] we can obtain the projection p as follows. e Let Hi be a wedge of i circles, so that H1 ← H2 ← . . . ← H is the standard 12 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ polyhedral expansion of the Hawaiian earring. For each i let pi : Hi → Hi i 1 i be the 2 -fold covering projection, obtained as the restriction to Hi ⊂ (S ) of the squaring map e 1 i 1 i 2 2 (S ) → (S ) , (z1,...,zi) 7→ (z1 ,...,zi ). It is then easy to check that p is the inverse limit of the sequence of coverings p , in particular H∞ = lim H . i ←− i The sequence of groupse π1(eHi) has surjective bonding maps, so by Corol- lary 3.6 the path-connectedness of H∞ is equivalent to the surjectivity of  N the homomorphism π1(H) →e(Z2) . But the latter is obvious, because any N sequence (ai) ∈ (Z2) , where ai ∈ {e0, 1} can be obtained as the image of the loop that winds around the i-th circle exactly when ai = 1. Example 3.11. We are going to show that a minor modification of covering maps in the inverse sequence described in example 3.7 yields a completely different limit space. This is surprising and very difficult to see geometrically. Let us define two covering projections with fibre Z. The first is the standard covering exponential map from the real line to the circle e: R → S1, e(t) := e2πit and the second combines the exponential covering with the two-fold covering of the circle given by the squaring map f : R × S1 → S1 × S1, f(t, z) := (e2πit + z2, z), 1 1 1 Note that the induced homomorphism f∗ : π1(R × S ) → π1(S × S ) sends 1 1 1 the generator of π1(R×S ) =∼ Z to the element (2, 1) ∈ π1(S ×S ) =∼ Z⊕Z, therefore f can be characterized as the covering of S1 × S1 corresponding to the cyclic subgroup of Z × Z generated by the element (2, 1). The following diagram depicts two inverse sequences of covering maps over (S1)N. . . . .

  R × R × R × S1 ×··· R × R × (R × S1) ×···

1×1×e×1×... 1×1×f×...   R × R × S1 × S1 ×··· R × (R × S1) × S1 ×···

1×e×1×1×... 1×f×1×...   R × S1 × S1 × S1 ×··· (R × S1) × S1 × S1 ×···

e×1×1×1×... f×1×1×...   S1 × S1 × S1 × S1 ×··· S1 × S1 × S1 × S1 ×··· UNCOUNTABLEGROUPS 13

The first sequence was already considered in Example 3.7 where we showed that its inverse limit is RN, which is, of course, path-connected. The second is also a sequence of Z-covering maps and the form of the total spaces suggest that its limit is RN as well. However, it is not difficult to check that there does not exist a map RN → (S1)N that factors through all terms in the sequence. Thus, the question is what is the inverse limit of the second sequence of coverings? To compute the set of path components of the inverse limit of the second sequence we compare it with the following sequence over a polyhedral ex- pansion of (S1)N:

f2 f3 R × S1 o R × R × S1 o R × R × R × S1 o · · ·

p1 p2 p3    S1 × S1 o S1 × S1 × S1 o S1 × S1 × S1 × S1 o · · ·

i 1 i−1 1 The maps fi : R × S → R × S are defined as a product of a projection on the first i-components with the square map (i.e. two-fold covering map) 1 i 1 1 i+1 on S . Furthermore, the maps pi : R × S → (S ) are covering maps corresponding to respectively the cyclic subgroup generated by the element i i−1 i+1 1 i+1 (2 , 2 ,..., 2, 1) ∈ Z = π1((S ) ). It is easy to check that the diagram commutes and that the i-th term in the original sequence is the pull-back of pi along the projection map as in the diagram

Ri × S1 × . . . / Ri × S1

pi   (S1)N / (S1)i+1

It follows that the inverse limit of the original sequence and the inverse limit of the sequence of covering over the polyhedral expansion coincide. We may now Theorem 3.5 to determine the set of components of the total space of the limit. In fact, the inverse sequence of the fundamental groups of the covering spaces is

2 2 2 ZZo o Z o Z o · · · which is not Mittag-Leffler, therefore its derived inverse limit is non-trivial (it is actually an uncountable ). As a consequence, the total space of the inverse limit has uncountably many components. Indeed, by a closer examination of the inverse limit of coverings over the polyhedral expansion of (S1)N we can conclude that the total space of the limit in the second case is homeomorphic to the product of the dyadic solenoid Sol2 with RN. 14 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´

4. Algebraic applications

We begin by describing a construction of inverse sequences of covering pro- jections whose limits correspond to meaningful subgroups of the fundamental group. Given a countable CW-complex X and a continuous map f : X → S1 let p: X → X be the covering map obtained as a pullback of the universal covering of S1 along f, as in the following diagram e X / R

p e e   X / S1 f It is easy to check that p is the covering map that corresponds to the kernel of f∗, that is to say 1 π1(X) =∼ Im(p∗ : π1(X) → π1(X)) = Ker(f∗ : π1(X) → π1(S )). Clearly, p is a regular covering whose fibres can be naturally identified with the infinitee Z,e so we often say that p is the Z-covering, corre- sponding to the map f : X → S1 (or rather, to its homotopy class). Since S1 is an Eilenberg-MacLane space of type K(Z, 1), there is a bijection 1 [X,S ] = Hom(π1(X), Z). Thus, we may also say that p is the Z-covering corresponding to a given homomorphism ϕ: π1(X) → Z, in the sense that π1(X) = Ker ϕ. Note, that every non-trivial subgroup of Z is isomorphic to Z, so we may assume without loss of generality that ϕ is surjective. e

Given a sequence of homomorphisms ϕ1, ϕ2, ϕ3,... : π1(X) → Z, let n Φn := (ϕ1,...,ϕn): π1(X) → Z and let pn : Xn → X be the covering projection whose fundamental group is

Kn := Ker Φn = Ker ϕ1 ∩ . . . ∩ Ker ϕn. e Thus we obtain an inverse sequence of coverings

f1 f2 X1 o X2 o X3 o · · ·

p1 p2 p3 e e e    X X X · · · where each fn is a covering projection. To compute the fibre of fn note that have a short exact sequence

0 → Kn−1/Kn −→ π1(X)/Kn −→ π1(X)/Kn−1 → 0 where the second and third group are subgroups of Zn and Zn−1 respectively. From this we deduce that Kn−1/Kn is either trivial or isomorphic to Z, therefore all fn are either trivial coverings (identity maps) or Z-coverings. UNCOUNTABLEGROUPS 15

Since Hom(π1(X), Z) is countable, we may apply the above construction to the sequence of all homomorphisms from π1(X) to Z. The limit of the resulting inverse sequence of coverings is a lifting projection p: X → X whose fundamental group is b b π1(X)= Ker ϕ. → ϕ: π1\(X) Z b Note that if π1(X) is finitely generated (e.g., if the 1-skeleton of X is fi- nite), then p is a covering projection. In fact, in that case Hom(π1(X), Z)= Hom(H1(X), Z) is also finitely generated and the tower of coverings is actu- ally finite (i.e.,b all but finitely many coverings in the sequence are trivial). Alternatively, p can be obtained as a covering of X that corresponds to the kernel of the natural homomorphism π1(X) → FH1(X), where FH1(X) denotes the maximalb free abelian quotient of H1(X). What is the algebraic meaning of the intersection of kernels of homomor- phisms to some group? It is well-known that the commutator subgroup ′ Fn = [Fn, Fn] of the free group on n generators Fn consists of all words in Fn for which the sum of exponents of each letter equals 0. This description is not intrinsic, as it requires to choose a basis for the free group. An equiv- alent description without reference to a basis is the following: if F is a free group (on any set of generators), then F ′ = Ker(f). → f : \F Z This approach was used in Cannon-Conner [2, Section 4] to describe the big commutator subgroup of the fundamental group of the Hawaiian earring G = π1(H): BC(G)= Ker(f). G→ f :\ Z The big commutator subgroup is much larger than the usual commutator subgroup G′ but shares many interesting properties with the latter. One can consider intersections of homomorphisms to other groups as well, for example finite or torsion-free groups. In [11] we gave an intrinsic de- ′′ scription of he second commutator subgroup Fn using homomorphisms to the Baumslag-Solitar group B(1,n). Let us introduce the following functor for arbitrary groups G and A:

KerA(G) := Ker(f). → f : \G A Note that KerA(G) is a fully characteristic subgroup of G. More generally, if ϕ: G → H is a homomorphism and if x ∈ KerA(G), then for every ψ : H → A we have that ψ ◦ ϕ(x) = 0, therefore ϕ(x) ∈ KerA(H). In categorical terms the correspondence G 7→ KerA(G) is a covariant functor. It turns 16 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ out that many characteristic subgroups can be described as intersection of kernels to some group A. In fact we may view KerA(G) as the part of G that cannot be represented in a product of copies of the group A. For example KerZ(G) = 0 if, and only if, G is residually free-abelian. ′ ′′ We mentioned before that KerZ(F ) = F and KerB(1,n)(F ) = F for every free group F . Recall that a torsion- A is slender if every homomorphism ϕ: ZN → A can be factored through some finite rank free abelian group, i.e., there exists a homomorphism ϕ′ : Zn → A so that the following diagram commutes

N ϕ Z ❇ / A ❇❇ ⑥⑥> ❇❇ ⑥⑥ pr ❇❇ ⑥⑥ ′ n ❇! ⑥⑥ ϕ Zn

(see Fuchs [16, Ch. XIII]) where prn is the projection map onto the first n factors of ZN. Free abelian groups are slender, subgroups and extensions of slender groups are also slender. A reduced abelian group is slender if, and only if, it does not contain a subgroup isomorphic to the group ZN or to a group of p-adic for some prime p (cf. Fuchs [16]). Katsuya Eda [14] extended this concept to arbitrary groups by defining a group A to be non-commutatively slender (nc-slender) if every homomor- phism ϕ: G → A from the Hawaiian earring group G to A can be factored through some free (non-commutative) group of finite rank as in ϕ G / A ❅ > ❅❅ ⑦⑦ ❅❅ ⑦⑦ ❅ ⑦ ′ prn ❅❅ ⑦ ϕ ⑦⑦ Fn where prn is the homomorphism induced by collapsing all but the first n loops of the Hawaiian earring. Lemma 4.1. If A is slender, then Ker (ZN) = lim Ker (Zn). A ←− A Similarly, if A is nc-slender, then Ker (F ) = lim Ker (F ) and Ker (G)= G∩ lim Ker (F ). A ←− A n A ←− A n

N n N Proof. The naturalb projections prn : Z → Z induce an isomorphism Z → lim Zn. By functoriality of Ker , the homomorphisms Ker (ZN) → Ker (Zn) ←− A A A N induced by prn are coherent and induce a homomorphism KerA(Z ) → lim Ker (Zn). This homomorphism is injective, because it is a restriction ←− A of the isomorphism ZN → lim Zn. To prove surjectivity, consider an ele- ←− ment (x ) ∈ lim Ker (Zn)= ZN. We need only show that (x ) ∈ Ker (ZN). i ←− A i A UNCOUNTABLEGROUPS 17

Fix a homorphism ϕ: ZN → A. Since A is slender, ϕ can be factored as ′ ′ n ′ ϕ = ϕ ◦ prn for some n and some ϕ : Z → A. Then ϕ((xi)) = ϕ (xn) which is trivial since x ∈ Ker (Zn). Thus Ker (ZN) → lim Ker (Zn) is n A A ←− A surjective as well. In [13], the authors show that if A is nc-slender then every homomorphism from F to A also factors through a projection to Fn for some n. Then the proof for the non-commutatively slender cases follows analogously.  b As a consequence, A-kernels of F with respect to a slender group A give rise to inverse limits of coverings over the Hawaiian earring. Indeed, recall that the Hawaiian earring H can beb represented as inverse limit of a sequence

···← Xn ← Xn+1 ←···← H, where Xn is a wedge of n circles and the bonding maps are the obvious projections. For each n the fundamental group of Xn is Fn, the free group on n generators and there is the universal covering projection qn : Xn → Xn = Xn/Fn. The limit of the resulting inverse sequence of coverings is a fibration with unique path lifting property b b q : H = lim X → H = lim X . ←− n ←− n By [8, Section 4] the groupbF acts freelyb and transitively on the fibres of q so we may consider the quotient of H with respect to any (normal) subgroup of F . b b Proposition 4.2. b If A is any nc-slender group, then the projection

qA : H/ KerA(F ) → H

′ can be obtained as the inverse limitb of coveringsb qn : Xn/ KerA(Fn) → Xn, and is therefore a fibration with unique path-lifting property. b Proof. For each n we have a commutative diagram of the form

H / Xn

b b   H/ KerA(F ) / Xn/ KerA(Fn)

′ qA q b b b n   H / Xn

′ where for each n the map qn is a covering projection with fibre Fn/ KerA(Fn). These coverings form an inverse sequence and by naturality we get a mapping 18 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ to the inverse limit

f H/ Ker (F ) / lim X / Ker (F ) A ←− n A n

qN q  b b b   H lim X ←− n

To prove our claim it is sufficient to show that the restriction of f to the fibres f : F / Ker (F ) → lim F / Ker (F ) A ←− n A n is an isomorphism. Consider the following inverse sequence of short exact sequences b b

1 / KerA(F1) / F1 / F1/ KerA(F1) / 1 O O O

1 / KerA(F2) / F2 / F2/ KerA(F2) / 1 O O O

1 / KerA(F3) / F3 / F3/ KerA(F3) / 1 O O O

......

Each projection Fn+1 → Fn is a split surjection, so by functoriality of KerA all bonding homomorphisms in the first column are surjective. By [19, Sec- tion 11.3] the resulting sequence of inverse limits is also exact. Moreover, by Lemma 4.1 lim Ker (F ) = Ker F so we have the following short exact ←− A n A sequence b 1 / Ker (F ) / F / lim F / Ker (F ) / 1, A ←− n A n  which implies that theb projectionsb F → Fn induce the isomorphism F / Ker (F ) ∼ lim F / Ker (F ) A = ←−b n A n as claimed.   b b

In order to prove the main results of this section we will need an algebraic lemma that is reminiscent of Theorem 3.5. Let G be a subgroup of the inverse limit of a sequence of groups {Gi} such that all projections G → Gi are surjective. Furthermore, let Hi be a decreasing sequence of subgroups of G and for each i let Hi be the image of Hi in Gi. Observe that G/Hi =∼ Gi/Hi (a bijection for arbitrary groups Hi, and a if Hi UNCOUNTABLEGROUPS 19 are normal subgroups of G). We can fit the above data in a commutative diagram with exact rows:

1 / H / G / lim G/H / lim 1H / • i ←− i i T    1 / lim H / lim G / lim G /H / lim 1H / • ←− i ←− i ←− i i i The following lemma is proved by elementary diagram chasing. Lemma 4.3. Ker( H → lim H ) = Ker(G → lim G ), thus we have an i ←− i ←− i exact sequence T 1 −→ Ker(G → lim G ) −→ H −→ lim H ←− i i ←− i Moreover lim 1H → lim 1H is a surjection of\ homogeneous sets whose fibres ←− i ←− i can be naturally identified with the cokernel of the homomorphism G → (lim G )/(lim H ) ←− i ←− i

As a consequence, lim 1H is trivial iff lim 1H is trivial and G → (lim G )/(lim H ) ←− i ←− i ←− i ←− i is surjective.

We have previously considered the inverse sequence of wedges of circles

···← Xn ← Xn+1 ←···← H, converging to the Hawaiian earring. Since π1(Xn)= Fn, the free group on n generators Fn, the inverse sequence of spaces gives rise to a homomorphism from the Hawaiian earring group G := π1(H) to the inverse limit of finite rank free groups F := lim F . By a result of Higman [21], see also [2] and ←− n [15], the homomorphism G → F is injective, so the Hawaiian earring group can be viewed as ab subgroup of F . It is known that both the group G and its index in F are uncountable, andb that G can be viewed as a dense subgroup of F with respect to the inverseb limit topology on F . The following theorem gives a moreb precise description of the relation between the two groups. The proofb is a combination of algebraic and geometricb techniques developed in the previous sections. Theorem 4.4. The group F is equal to the internal product of its subgroups G and KerZ(F ), i.e., F = G · KerZ(F ). b Proof. We areb going tob apply Lemmab 4.3 to the following data: let Gi be the free group on i generators Fi, G = G, and Hi the kernel of the i G → π1(Xi) → H1(Xi) =∼ Z . Then lim Gi = F . Observe that the image ←− ′ of Hi in π1(Xi)= Fi is precisely the commutator subgroup Fi = KerZ(Fi), since G maps surjectively onto π1(Xi). The bondingb maps in the inverse 20 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ sequence of groups {H } are surjective, therefore lim1 H is trivial. More- i ←− i over, by Lemma 4.1 lim H = lim Ker (F ) = Ker (F ). Thus, by Lemma ←− i ←− Z i Z 4.3 the formula F = G · KerZ(F ) holds if, and only if, the derived inverse limit lim1 H is trivial. b ←− i b b It is well-known that the limit lim1 H is precisely the set of path-components ←− i of the inverse limit of covering maps that correspond to the sequence of kernels of the homomorphisms {G → H1(Xi)} (see [4, 22], as well as our discussion in Section 2). Analogously as in the examples in Section 3, we can represent this lifting projection as a limit of covering spaces over the approximations of the Hawaiian earrings by finite wedges of circles Xi. The covering space over Xi corresponding to the commutator subgroup Fi can be identified with the integral 1-dimensional grid in Ri, i.e.

i Xi = {(x1,...,xi) ∈ R | xj ∈/ Z for at most one index j}.

Observe thate the bonding maps in the inverse system

X1 ←− X2 ←− X3 ←− · · · are retractions, and so theire inversee limite X is the 1-dimensional integral grid in RN. Alternatively, we may describe X by the following pullback diagram e e X / RN

N e e    H / (S1)N

Clearly, X is path-connected, which completes the proof of our claim. 

Note thate there one could also consider the commutator subgroup F ′ of F , which is however much smaller than lim F ′ . In fact, F > G · F ′. ←− n b b Continuing the previous line of thought we may ask whetherb bF can be ob- tained by adding some other term of the derived series of F to the group G. As before, we replace F ′′ with a suitable inverse limit group.b We already mentioned that the second derived group of a free group Fbcan be described ′′ as a kernel, F = KerB(bF ) (see [11, Theorem 1]) for any solvable, deficiency 1 group B that is not virtually abelian. By work of Wilson [31], any solv- able deficiency 1 group is isomorphic to a Baumslag-Solitar group B(1,m) for some m. Since B(1,m) is nc-slender (see [5]), Lemma 4.1 implies that Ker (F ) = lim Ker (F ) for every group B that is solvable, of deficiency 1 B ←− B n and is not virtually abelian. As well, we have

b ′′ F ≤ KerB(F ) ≤ KerZ(F ).

b b b UNCOUNTABLEGROUPS 21

In the next theorem we reverse the reasoning and use our methods to show that this subgroup of F is too small to generate, together with the Hawaiian earring group the entire group F . b Theorem 4.5. F 6= G · KerBb(F ) for every group B that is solvable, of deficiency 1 and is not virtually abelian. b b Proof. Let B be a solvable deficiency 1 group that is not virtually abelian. Then B is isomorphic to a Baumslag-Solitar group B(1,m) and, for a free ′′ group of rank n, we have KerB(Fn)= Fn = KerZ(KerZ(Fn)). We may repeat almost verbatim the algebraic part of the proof of the previous theorem and obtain that F = G · KerB(F ) if, and only if, the inverse limit Y of the sequence of coverings over the Hawaiian earring H, determined by the ′′ kernels of homomorphismsb G → Fbi/Fi , is path-connected. Thus, in ordere to prove our claim, we must show that Y is not path-connected. We will use the same notation as in the proof of the previous theorem. Let e X be the integral grid in RN and let p: X → H be the lifting projection obtained as the limit of the sequence e e X1 o X2 o X3 o · · · o X

p1 p2 p3 p e e e e     X1 o X2 o X3 o · · · o H

′ where π1(Xn)= Fn. Let Yn be a cover of Xn corresponding to the subgroup ′ ′′ π1(Xn) = Fn and let Y be the resulting inverse limit. We will con- e e e struct a continuous surjection from Y to a path-disconnected space. Since e ′ e π1(X2)= F2 is a countably generated free group, we can find a sequence of homomorphisms fi : π1(X2) → Z suche that e n n−1 ∞ e ′′ Ker(fi) ( Ker(fi) and Ker(fi)= F2 . i i i \=1 \=1 \=1 n ′ fi Let Zn be the cover of X2 corresponding to the subgroup Ker π1(X2) −→ i=1 Z . The covers Z′ form an inverse sequence, whose limitT Z′ := lim Z′ is e n e ←−e n ′ path-disconnected by Corollary 2.2. As a consequence, the pullback of Z e e e along the projection p: X → X2 e ′ e e Z o Z

e e X2 o p X

e e 22 G.CONNER,W.HERFORT,C.KENT,ANDP.PAVESIˇ C´ yields a lifting projection Z → X, whose total space is also path-disconnected. Note that Z can be alternatively obtained as inverse limit of a sequence Z = lim Z , where Z is thee pull-backe of Z′ along the projection p : X → ←− n n n n n e X2. Since commutator subgroups are characteristic we have pn∗(π1(Yn)) ≤ e e ′ e e −1 ′ e π1(Y2) ≤ π1(Z ), thus π1(Yn) ≤ π1(Zn) = pn∗ (π1(Z )). It follows that Yn coverse Zn for every n and we obtain the following diagram: e e e e e e e Y2 o e Y3 o · · · o Yn−1 o Yn o Yn+1 o · · · o Y

e e e e e e       Z2 o Z3 o · · · o Zn−1 o Zn o Zn+1 o · · · o Z

e e e e e e       X2 o X3 o · · · o Xn−1 o Xn o Xn+1 o · · · o X

Bye construction,e the fibre of thee lifting projectione Ye→ X maps surjectivelye to the fibre of Z → X. Since every point in Z (respectively Y ) is connected by a path to a point in the fibre, it follows, thate the projectioe n Y → Z is surjective, thereforee eY is not path-connected.e e  e e e References [1] A. K. Bousfield and D. M. Kan. Homotopy limits, completions and localizations. Lec- ture Notes in Mathematics, Vol. 304. Springer-Verlag, Berlin-New York, 1972. [2] J. Cannon, G. Conner, The combinatorial structure of the Hawaiian earring group, Top. Appl. 106 (2000), 225-271. [3] J. Cannon, G. Conner, On the fundamental groups of one-dimensional spaces, Top. Appl. 153 (2006), 2648-2672. [4] Joel M. Cohen. Homotopy groups of inverse limits. In Proceedings of the Advanced Study Institute on Algebraic Topology (Aarhus Univ., Aarhus, 1970), Vol. I, pages 29– 43. Various Publ. Ser., No. 13, 1970. [5] Gregory R. Conner and Samuel M. Corson. A note on automatic continuity. Proc. Amer. Math. Soc., 147(3):1255–1268, 2019. [6] G.R. Conner and K. Eda. Fundamental groups having the whole information of spaces. Topology Appl., 146/147:317–328, 2005. [7] G. Conner and M. Meilstrup. Deforestation of Peano continua and minimal deforma- tion retracts. Topology Appl., 159(15):3253–3262, 2012. [8] Gregory R. Conner and Petar Paveˇsi´c. General theory of lifting spaces. arXiv e-prints, page arXiv:2008.11267, August 2020. [9] G. Conner, W. Herfort, P. Paveˇsi´c, Some anomalous examples of lifting spaces, Top. Appl. 239 (2018), 234-243. [10] G. Conner, W. Herfort, P. Paveˇsi´c, Geometry of compact lifting spaces, Monatsh Math (2020), https://doi.org/10.1007/s00605-020-01460-1. [11] G. Conner, C. Kent, W. Herfort, P. Paveˇsi´c, Recognizing the Second Derived Group of Free Groups, J. 516 (2018), 396-400. [12] G. Conner, C. Kent, W. Herfort, P. Paveˇsi´c, Path space and covering fibrations, preprint. [13] G. Conner, C. Kent, W. Herfort, P. Paveˇsi´c, Inverse Limit Slender, preprint. UNCOUNTABLEGROUPS 23

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