Uncountable groups and the geometry of inverse limits of coverings
Gregory Conner, Wolfgang Herfort, Curtis Kent, Petar Pavesic
aa r X i v : . [ m a t h . GN ] J a n UNCOUNTABLE GROUPS AND THE GEOMETRY OFINVERSE LIMITS OF COVERINGS
GREGORY R. CONNER , WOLFGANG HERFORT , CURTIS KENT ,AND PETAR PAVEˇSI ´C Abstract.
In this paper we develop a new approach to the study ofuncountable fundamental groups by using Hurewicz fibrations with theunique path-lifting property ( lifting spaces for short) as a replacementfor covering spaces. In particular, we consider the inverse limit of asequence of covering spaces of X . It is known that the path-connectivityof the inverse limit can be expressed by means of the derived inverse limitfunctor lim ←− , which is, however, notoriously difficult to compute whenthe π ( X ) is uncountable. To circumvent this difficulty, we expressthe set of path-components of the inverse limit, e X , in terms of thefunctors lim ←− and lim ←− applied to sequences of countable groups arisingfrom polyhedral approximations of X .A consequence of our computation is that path-connectedness of lift-ing space implies that π ( e X ) supplements π ( X ) in ˇ π ( X ) where ˇ π ( X )is the inverse limit of fundamental groups of polyhedral approximationsof X . As an application we show that G· Ker Z ( b F ) = b F = G· Ker B (1 ,n ) ( b F ),where b F is the canonical inverse limit of finite rank free groups, G is thefundamental group of the Hawaiian Earring, and Ker A ( b F ) is the inter-section of kernels of homomorphisms from b F to A . Introduction
A famous theorem of Shelah [27] states that the fundamental groups ofPeano continua present a striking dichotomy: they are either finitely pre-sented or uncountable. The first case corresponds to fundamental groups offinite polyhedra and covering space theory has traditionally been an effec-tive geometric approach to the study of the structure of these groups. Thesecond case is by no means exotic either: Peano continua with uncountablefundamental group arise naturally as attractors of dynamical systems [20],as fractal spaces [25, 29], as boundaries of non-positively curved groups [23],
Date : January 28, 2021. Supported by Simons Foundation Collaboration Grant 646221. The second author is grateful for the warm hospitality at the Mathematics Depart-ment of BYU in February 2018. Supported by Simons Foundation Collaboration Grant 587001. Supported by the Slovenian Research Agency program P1-0292 and grants N1-0083,N1-0064. and in many other situations. The fundamental difference between the twocases is of a local nature. If the fundamental group of a Peano continuum X is uncountable, then by [27] it is not semilocally simply connected at somepoint. As a consequence most of the subgroups of π ( X ) do not correspondto a covering space over X , which represents a major obstacle for a geometricstudy 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 spacesis taken by more general fibrations with the unique path-lifting property.These fibrations were introduced by Spanier [28, Chapter 2] who developedmuch of the theory of covering spaces within this more general setting. Sincethe term ‘Hurewicz fibration with the unique path-lifting property’ is some-what impractical we call them lifting spaces and the corresponding mapsfrom the total space to the base 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 inverse limit of a sequence of cov-ering spaces over X is always a lifting space over X (and is not a coveringprojection, unless the sequence is eventually constant). If X is semilocallysimply-connected (e.g., a CW-complex), and the sequence is not eventuallyconstant, then the limit lifting space is path-disconnected (see Corollary2.2).This is a geometric reflection of the fact that the fundamental group of thebase is countable while the fibre of the lifting projection is uncountable.However, if X is not semilocally simply-connected, then its fundamentalgroup is uncountable by Shelah’s theorem and the limit space can be path-connected or not path-connected, depending on the interplay between π ( X )and the sequence of subgroups corresponding to the coverings in the inversesequence.The path-components of an inverse limit of covering spaces are classicallydetermined by the derived inverse limit functor lim ←− applied to a sequence ofsubgroups of π ( X ). Difficulties arise in the computation of lim ←− for inversesequences of uncountable groups, which make this an ineffective approachto determining path-connectivity (see discussion at the end of Section 2 andExamples 3.7-3.11 in Section 3) . In Section 3, we consider inverse sequencesof coverings over some polyhedral expansion of the base space X and studythe path-connectedness of the limit. We prove the following result, whichallows one to work with inverse sequences of countable groups and thus avoidthe difficulties of computing lim ←− for uncountable groups. Theorem 1 (Theorem 3.4) . Every inverse limit of covering projections overa Peano continuum X is homeomorphic to an inverse limit of covering pro-jections over a polyhedral expansion of X . NCOUNTABLE GROUPS 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 coveringsover some polyhedral expansions of a Peano continuum. The statement isquite technical, but it leads to the following important consequence whichcharacterizes path-connectedness of the limit space.
Corollary 2 (Corollary 3.6) . Let p : e X → X be the inverse limit of coveringmaps over a Peano continuum X . Then e X is path-connected if, only if, theinverse sequence of fundamental groups, π ( e X i ) , satisfies the Mittag-Lefflerproperty and the natural homomorphism π ( X ) −→ lim ←− π ( X i ) π ( e X i ) ! is surjective, where e X i → X i are covering projections of a polyhedral expan-sion { X i } of X . Even for the well-studied Hawaiian earring group G , apart from the factthat G contains the free group of countable rank and is thus dense in itsshape group b F , very little is known about the size of G in b F . The abovecorollary leads to interesting algebraic applications concerning the image ofthe fundamental group into the shape group which we consider in Section 4.Given groups G and A , we let Ker A ( G ) be the intersection of kernels ofall homomorphisms from G to A . If G is a free group and B (1 , n ) is aBaumslag-Solitar group, then Ker Z ( G ) is exactly the commutator subgroupof G and Ker B (1 ,n ) ( G ) is exactly the second derived subgroup of G , see [11].However, b F is locally free but non-free and the corresponding Ker Z ( b F ) prop-erly contains the commutator subgroup of b F . In fact, Ker Z ( b F ) is sufficientlylarge to be a supplement of G in b F while Ker B (1 ,n ) ( b F ) is not. Theorem 3 (Theorem 4.4 and Theorem 4.5) . The group b F is equal to theinternal product of its subgroups G and Ker Z ( b F ) , while the internal productof G and Ker B (1 ,n ) ( b F ) is a proper subgroup of b F . Preliminaries on inverse limits of spaces and groups
The main object of our study are lifting spaces that arise as inverse limits ofcovering spaces, so let us consider the following sequence of regular coveringmaps p i : e X i → X , together with the inverse limit map p : e X → X : e X p (cid:15) (cid:15) e X p (cid:15) (cid:15) ˜ f o o e X p (cid:15) (cid:15) ˜ f o o · · · · o o e X o o p (cid:15) (cid:15) X X X · · · · · · X G. CONNER, W. HERFORT, C. KENT, AND P. PAVEˇSI´C
For each i we identify π ( e X i ) with its image p i ∗ ( π ( e X i )) which is a normalsubgroup of π ( X ). By standard covering space theory the bonding maps˜ f i : e X i +1 → e X i are covering maps and we have a decreasing sequence ofnormal subgroups of π ( X ) π ( X ) D π ( e X ) D π ( e X ) D · · · Note that the converse is not true for general spaces, because a subgroupof π ( X ) may not be a covering subgroup, i.e. it does not necessarily comefrom a covering of X . However, if X is locally path-connected, then by [28,Theorem II, 5.12] every decreasing sequence of normal covering subgroupsof π ( 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 inverselimit of the fibres F ( p i ), and so by [28, Theorem II, 6.2] it can be naturallyidentified with the inverse limit of the quotients F ( p ) = lim ←− ( π ( X ) /π ( e X i )) . Since all bonding maps in the inverse sequence are fibrations, it is well-known (see [4, 22]) that the homotopy groups of π ( e X ) can be expressed interms of the homotopy groups of e X i : π ( e X ) = lim ←− π ( e X i ) = \ i π ( e X i ) , π ( e X ) = lim ←− π ( e X i )and π n ( e X ) = π n ( X ) for n >
1. Here lim ←− denotes the inverse limit functoron groups and lim ←− is its first derived functor (see [19, Section 11.3]). Forcommutative groups these functors can be defined as kernel and cokernel ofthe homomorphism ϕ : Q π ( e X i ) → Q π ( e X i ), given as ϕ ( u , u , u , . . . ) = ( u − ˜ f ∗ ( u ) , u − ˜ f ∗ ( u ) , . . . )and so they fit in the exact sequence0 → lim ←− π ( e X i ) → Y π ( e X i ) ϕ −→ Y π ( e X i ) → lim ←− π ( e X i ) → . The non-commutative case is more delicate: in that case lim ←− π ( e X i ) is de-fined as the quotient of Q π ( e X i ) under the equivalence relation given as( u i ) ∼ ( v i ) ⇔ ( v i ) = ( x i u i ˜ f i ∗ ( x i +1 ) − ) for some ( x i ) ∈ Y π ( e X i )(see [17] for detailed treatment of the non-commutative case).The values of lim ←− are notoriously hard to compute. Here we will be onlyinterested whether lim ←− of a sequence is trivial, and this can be settled ifwe can show that the inverse sequence satisfies the Mittag-Leffler condition ,which we now define. In an inverse sequence of groups G ←− G ←− G ←− · · · NCOUNTABLE GROUPS 5 for a fixed j the image of the homomorphism G i → G j decreases as i goestoward infinity. The inverse sequence is said to satisfy the Mittag-Leffler con-dition if for every j the sequence { Im( G i → G j ) | i = j, j + 1 , . . . } stabilizes.Clearly, sequences with epimorphic bonding maps satisfy the Mittag-Lefflercondition. On the other hand, if the bonding maps are monomorphisms,then the sequence satisfies he Mittag-Leffler condition if, and only if, it iseventually constant. The following result is proved in [17] (see also [19,Theorem 11.3.2]): Proposition 2.1.
If an inverse sequence { G i } , of groups, satisfies theMittag-Leffler condition, then lim ←− G i = { } . Conversely, if lim ←− G i = { } and each G i is countable, then { G i } satisfies the Mittag-Leffler condition. As we explained before, the sequence of fundamental groups induced byan inverse sequence of coverings satisfies Mittag-Leffler condition only if itis constant from some point on. Thus we get immediately the followingcorollary.
Corollary 2.2.
Let e X be the inverse limit of an inverse sequence of coveringmaps p i : e X i → X and assume that for some i the group π ( e X i ) is countable.Then either the sequence is eventually constant (which implies that p : e X → X is a covering map), or e X is not path-connected.In particular an inverse limit of coverings over a countable CW-complex iseither a covering or its total space is not path-connected. This last statement is not surprising if one considers the tail of the exacthomotopy sequence of the fibration p : e X → X , shown in the followingdiagram together with the above mentioned identifications:1 / / π ( e X ) p ∗ / / π ( X ) ∂ / / π ( F ( p )) / / π ( e X ) / / ∗ / / lim ←− π ( e X i ) / / π ( X ) / / lim ←− ( π ( X ) /π ( e X i )) / / lim π ( e X i ) / / ∗ If the sequence is not eventually constant, then π ( F ( p )) is uncountable.But if X is a countable CW-complex, then π ( X ) is a countable group andso ∂ cannot be surjective.However, if π ( 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( S ) ∞ ← ( R × S × S × . . . ) ← ( R × R × S × . . . ) ← · · · · · · ← R ∞ has a path-connected total space in spite of the fact that it is not eventuallyconstant. Another, less obvious example is a sequence of 2-fold coverings G. CONNER, W. HERFORT, C. KENT, AND P. PAVEˇSI´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 theirderived inverse limits are nonetheless trivial.Of course, one can easily find examples of inverse limits of coverings of theHawaiian ring whose total space has uncountably many path-components.Distinguishing between different cases is hindered by the difficulties in theexplicit computation of the derived limit functor for decreasing sequences ofuncountable groups. We present one approach to this problem in the follow-ing section, while in the last section we consider the algebraic implicationsof 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 limitfunctor to the more manageable countable case is to consider a polyhedralexpansion { X i } of X and represent a lifting projection p : e X → X as an in-verse limit of covering maps over the polyhedral expansion, i.e., find coveringmaps p i : e X i → X i that satisfy the following diagram. e X p (cid:15) (cid:15) e X p (cid:15) (cid:15) ˜ f o o e X p (cid:15) (cid:15) ˜ f o o · · · o o e X p (cid:15) (cid:15) o o X X f o o X f o o · · · o o X o o There are several standard methods by which a compact metric space canbe 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 metriccompactum X admits arbitrarily fine finite open coverings. Moreover, forevery finite open covering U = { U , . . . , U n } of X there exists a subordinatedpartition of unity { ρ i : X → [0 , | i = 1 , . . . , n } . Let N ( U ) be the nerve of the covering U , i.e., the simplicial complex whose vertices are elementsof U , and whose simplices are spanned by elements of U with non-emptyintersection. Then the formula f ( x ) := n X i =1 ρ i ( x ) · U i defines a map f : X → | N ( U ) | (where | N ( U ) | is the geometric realizationof the nerve). It is well known that the choice of the partition of unitydoes not affect the homotopy class of f , so the induced homomorphism f ∗ : π ( X ) → π (cid:0) | N ( U ) | (cid:1) depends only on the cover U . NCOUNTABLE GROUPS 7
Lemma 3.1.
If each element of U is path-connected, then f ∗ : π ( X ) → π (cid:0) | N ( U ) | (cid:1) is surjective.Proof. Without loss of generality we may assume that the partition of unitysubordinated to U is reduced in the sense that for every i there exists x i ∈ U i ,such that ρ i ( x i ) = 1. For every pair of intersecting sets U i , U j ∈ U we maychoose a path in U i ∪ V j between x i and x j . These paths determine a map g : | N ( U ) (1) | → X from the 1-skeleton of the nerve to X . One can checkthat, for every 1-simplex σ in N ( U ) (1) , the image f ( g ( σ )) is contained inthe open star of σ in N ( U ), which implies that f ◦ g is homotopic to theinclusion i : | N ( U ) (1) | ֒ → | N ( U ) | . Since i ∗ : π ( | N ( U ) (1) | ) → π ( | N ( U ) | ) issurjective, f ∗ is also surjective. (cid:3) If U ′ is a covering of X that refines U (i.e. every element of U ′ is contained insome 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 ( U )) | ←− | N ( U )) | ←− | N ( U )) | ←− . . . whose limit is X . Moreover, if X is locally path-connected, then we maychoose covers of X whose elements are path-connected. As an interestingaside, Lemma 3.1 immediately implies the following well-known fact, whichalso follows immediately from the Hahn-Mazurkiewicz theorem and resultsof Krasinkiewicz [24, Theorems 4.1 and 4.2]. Corollary 3.2.
Every Peano continuum X can be represented as the limitof an inverse system of finite polyhedra X X o o X o o · · · o o lim ←− X i = X o o such that the homomorphisms π ( X ) → π ( X i ) induced by the projectionmaps 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 inverselimit 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 provethe following theorem. Lemma 3.3.
Let X be a Peano continuum and g : X → Y be a continuousmap into a polyhedron Y . If U is a cover of X by path connected open setssuch that the preimage of the open star of any vertex of Y is contained inan element of U , then Ker( g ∗ ) ⊂ π (cid:0) X ; 2 U (cid:1) .Proof. If we denote by f : X → |U | the map from X to the realization of thenerve of U , then the kernel of the homomorphism f ∗ : π ( X ) → π ( |U | ) is π ( X ; 2 U ), see [8, Lemma 3.3]. Since the preimage of the open star of anyvertex 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 ∗ ) = π (cid:0) X ; 2 U (cid:1) . (cid:3) G. CONNER, W. HERFORT, C. KENT, AND P. PAVEˇSI´C
Let U be an open cover of X . We will use π ( X ; U ) to denote the U -Spaniersubgroup, i.e. π ( X ; U ) is the subgroup of π ( X, x ) generated by (cid:8) [ α ∗ β ∗ α ] | α : ( I, → ( X, x ) and Im( β ) ⊂ U for some U ∈ U (cid:9) . Theorem 3.4. If X is a Peano continuum, then every inverse sequence ofcovering projections over X is homeomorphic to a pull-back of an inversesequence 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 expansionof X .Proof. Let X X o o X o o · · · o o lim ←− X i = X o o be a polyhedral expansion of X , such that the homomorphisms π ( X ) → π ( X i ) are surjective.Let p : e X → X be an arbitrary covering map. Then p evenly covers elementsof some open cover U of X and π ( e X ) ⊇ π ( X ; U ), where the latter denotesthe U -Spanier subgroup of π ( X ). Since X is compact we may assume that U is finite. Choose another finite cover V of X by path connected open setswhose 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 openstar of any vertex of X i is contained in an element of V . Then by Lemma3.3, we have π ( e X ) ⊇ π ( X ; U ) ⊇ π ( X ; 2 V ) ⊇ Ker (cid:0) π ( X ) → π ( X i ) (cid:1) . Therefore, if X e X o o e X o o e X o o · · · o o is an inverse sequence of coverings over X we may find for each i some index j ( i ) such that π ( e X i ) ⊇ Ker (cid:0) π ( X ) → π ( X j ( i ) ) (cid:1) .Denote K i := Ker (cid:0) π ( X ) → π ( X j ( i ) ) (cid:1) and let X j ( i ) → X j ( i ) be the coveringmap that corresponds to the subgroup Im (cid:0) π ( e X i ) → π ( X j ( i ) ) (cid:1) . By thelifting theorem, there is a fibre-preserving map ˜ f i : e X i → X j ( i ) for which thefollowing diagram commutes e X i (cid:15) (cid:15) ˜ f i / / X j ( i ) (cid:15) (cid:15) X / / X j ( i )NCOUNTABLE GROUPS 9 To compute the restriction of ˜ f i on the fibres we examine the followingcommutative diagram with exact rows1 / / π ( e X i ) /K i / / (cid:15) (cid:15) (cid:15) (cid:15) π ( X ) /K i / / ∼ = (cid:15) (cid:15) π ( X ) /π ( e X i ) / / (cid:15) (cid:15) / / π ( X j ( i ) ) / / π ( X j ( i ) ) / / π ( X j ( i ) ) /π ( X j ( i ) ) / / f i induces an isomorphism π ( X ) /π ( e X i ) → π ( X j ( i ) ) /π ( X j ( i ) ) , and hence a bijection between the fibres of respective covering maps. Inother words, the above diagram is a pull-back of covering maps.By repeating the above argument for all terms in an inverse sequence ofcovering maps over X we obtain an inverse sequence of covering maps overthe given expansion of X with the same limit lifting projection. (cid:3) The above theorem shows that for most cases of interest it is sufficient toconsider lifting projections over X that are inverse limits of covering mapsover some expansion of X as in e X p (cid:15) (cid:15) e X p (cid:15) (cid:15) ˜ f o o e X p (cid:15) (cid:15) ˜ f o o · · · · o o e X o o p (cid:15) (cid:15) X X f o o X f o o · · · · · · o o X o o Suppose that X i are polyhedra and that the maps X → X i induce epi-morphisms between respective fundamental groups (e.g. if X is locallypath-connected). As in Section 2, we may compare the tail of the exacthomotopy sequence of the fibration p with the derived exact sequence of theinverse limit functor to obtain the following diagram1 / / π ( e X ) / / ϕ (cid:15) (cid:15) π ( X ) / / ι (cid:15) (cid:15) lim ←− ( π ( X i ) /π ( e X i )) / / π ( e X ) / / ψ (cid:15) (cid:15) ∗ / / lim ←− π ( e X i ) / / lim ←− π ( X i ) / / lim ←− ( π ( X i ) /π ( e X i )) / / lim ←− π ( e X i ) / / ∗ The inverse limits lim ←− π ( X i ) and lim ←− π ( e X i ) are independent of the choiceof a polyhedral expansion for X . That two polyhedral expansions give pro-isomorphic inverse limit groups is proved in Mardesic-Segal [26, Ch 2] andthe pro-isomorphism can be lifted to a pro-isomorphism of the correspondingcovering subgroups via the homotopy lifting criterion. The limit lim ←− π ( X i )is usually called the ˇCech fundamental group (or the first shape group ) of X ,and is denoted by ˇ π ( X ). Since e X is not necessarily compact, lim ←− π ( e X i ) is not necessarily the first shape group of e X . Regardless, we will still denotelim ←− π ( e X i ) by ˇ π ( e X ). The kernel of the induced map ι : π ( X ) → ˇ π ( X )is called the shape kernel of X and denoted ShKer( X ). Note that by theexactness of the above diagram we have thatKer( π ( e X ) → lim ←− π ( e X i )) = Ker( π ( X ) → lim ←− π ( X i )) = ShKer( X ) . Clearly, ψ : π ( e X ) → lim ←− π ( e X i ) is a surjective function between homo-geneous sets. Thus, in order to determine π ( e X ) we need to computethe fibres of ψ . From the exactness of the second row we deduce thatlim ←− π ( e X i ) is the set of cosets of the action of the group ˇ π ( X ) / ˇ π ( e X )on lim ←− ( π ( X i ) /π ( e X i )). Then a straightforward diagram chasing shows thatthe fibres of ψ can be naturally identified with the cosets of the action of ι ( π ( X )) on ˇ π ( X ) / ˇ π ( e X ). Theorem 3.5.
Let X be a path-connected space that admits a polyhedralexpansion f i : X → X i such that the induced homomorphisms f i ∗ are sur-jective and let p : e X → X be the inverse limit of a sequence of coverings p i : e X i → X i as described above. Then the fundamental group of e X is deter-mined by the exact sequence of groups → ShKer( X ) → π ( e X ) → ˇ π ( e X ) , and the set of path-components of e X is determined by the exact sequence ofgroups and based homogeneous sets π ( X ) ι −→ ˇ π ( X ) / ˇ π ( e X ) −→ π ( e X ) −→ lim ←− π ( e X i ) −→ ∗ The main advantage of Theorem 3.5 with respect to the description of π ( e X )given in Section 2 is that all limits and derived limits are taken over inversesequences of countable groups. Corollary 3.6.
Let p : e X → X be the inverse limit of covering maps as inthe above theorem. Then e X is path-connected if, only if, the inverse sequence π ( e X ) ← π ( e X ) ← π ( e X ) ← · · · satisfies the Mittag-Leffler property and the natural homomorphism π ( X ) −→ ˇ π ( X )ˇ π ( e X ) ∼ = lim ←− π ( X i ) π ( e X i ) ! is surjective.Proof. Theorem 3.5 implies that π ( e X ) is trivial if, and only if lim ←− π ( e X i )is trivial and π ( X ) −→ ˇ π ( X ) / ˇ π ( e X ) is surjective. Since the fundamen-tal groups of polyhedra are countable, then by Proposition 2.1 the inversesequence is Mittag-Leffler if and only if lim π ( e X i ) is trivial. NCOUNTABLE GROUPS 11
Observe that the isomorphism between ˇ π ( X ) / ˇ π ( e X ) and lim ←− (cid:0) π ( X i ) /π ( e X i ) (cid:1) follows form the Mittag-Leffler property and does not hold in general. (cid:3) Let us consider a couple of examples.
Example 3.7.
We have already mentioned the inverse sequence of coveringmaps over the infinite product of circles( S ) ∞ ← ( R × S × S × . . . ) ← ( R × R × S × . . . ) ← · · · It can be replaced by the following inverse sequence of covering maps overpolyhedral approximations of ( S ) ∞ (with horizontal maps the usual pro-jections) R (cid:15) (cid:15) R × R o o (cid:15) (cid:15) R × R × R o o (cid:15) (cid:15) · · · o o R N (cid:15) (cid:15) o o S S × S o o S × S × S o o · · · o o ( S ) N o o 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-connectednessof R N is equivalent to the equality π (cid:0) ( S ) N (cid:1) = lim ←− π (cid:0) ( S ) n (cid:1) = ˇ π (cid:0) ( S ) N (cid:1) . Example 3.8.
Let T ∞ be the infinite product of circles endowed with theCW-topology, i.e. the direct limit of the sequence of spaces S ֒ → S × S ֒ → S × S × S ֒ → · · · T ∞ Clearly, the fundamental group of T ∞ is L ∞ k =1 Z . For each i ∈ N let p i : e X i → T ∞ be the covering map that corresponds to the subgroup π ( e X i ) = L ∞ k = i Z . By Corollary 2.2 the inverse limit e X := lim ←− e X i is not path-connected. Remark 3.9.
The last result is somewhat counter-intuitive, as one couldargue that e X i ≈ R i − × T ∞ and that the inverse limit of the coverings issimply the product R ∞ but the computation reveals that the geometry of theinverse limit must be different. We leave this as an (easy) exercise for thereader. As a hint, note that the fibre of e X → T ∞ is uncountable while thefundamental group of T ∞ is countable. Example 3.10.
Consider the squaring lifting projection p : e H ∞ → H overthe 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 projections and, althoughit resembles at first sight the construction of the dyadic solenoid, we havebeen able to prove by a geometric argument that its total space e H ∞ is path-connected. We are going to show how this result is reflected in the algebraiccomputation of the set of path-components of e H ∞ .As explained in [9, Section 3.3] we can obtain the projection p as follows.Let H i be a wedge of i circles, so that H ← H ← . . . ← H is the standard polyhedral expansion of the Hawaiian earring. For each i let p i : e H i → H i be the 2 i -fold covering projection, obtained as the restriction to H i ⊂ ( S ) i of the squaring map( S ) i → ( S ) i , ( z , . . . , z i ) ( z , . . . , z i ) . It is then easy to check that p is the inverse limit of the sequence of coverings p i , in particular e H ∞ = lim ←− e H i .The sequence of groups (cid:0) π ( e H i ) (cid:1) has surjective bonding maps, so by Corol-lary 3.6 the path-connectedness of e H ∞ is equivalent to the surjectivity ofthe homomorphism π ( H ) → ( Z ) N . But the latter is obvious, because anysequence ( a i ) ∈ ( Z ) N , where a i ∈ { , } can be obtained as the image ofthe loop that winds around the i -th circle exactly when a i = 1. Example 3.11.
We are going to show that a minor modification of coveringmaps in the inverse sequence described in example 3.7 yields a completelydifferent limit space. This is surprising and very difficult to see geometrically.Let us define two covering projections with fibre Z . The first is the standardcovering exponential map from the real line to the circle e : R → S , e ( t ) := e πit and the second combines the exponential covering with the two-fold coveringof the circle given by the squaring map f : R × S → S × S , f ( t, z ) := ( e πit + z , z ) , Note that the induced homomorphism f ∗ : π ( R × S ) → π ( S × S ) sendsthe generator of π ( R × S ) ∼ = Z to the element (2 , ∈ π ( S × S ) ∼ = Z ⊕ Z ,therefore f can be characterized as the covering of S × S correspondingto the cyclic subgroup of Z × Z generated by the element (2 , S ) N . ... (cid:15) (cid:15) ... (cid:15) (cid:15) R × R × R × S × · · · × × e × × ... (cid:15) (cid:15) R × R × ( R × S ) × · · · × × f × ... (cid:15) (cid:15) R × R × S × S × · · · × e × × × ... (cid:15) (cid:15) R × ( R × S ) × S × · · · × f × × ... (cid:15) (cid:15) R × S × S × S × · · · e × × × × ... (cid:15) (cid:15) ( R × S ) × S × S × · · · f × × × ... (cid:15) (cid:15) S × S × S × S × · · · S × S × S × S × · · · NCOUNTABLE GROUPS 13
The first sequence was already considered in Example 3.7 where we showedthat its inverse limit is R N , which is, of course, path-connected. The secondis also a sequence of Z -covering maps and the form of the total spaces suggestthat its limit is R N as well. However, it is not difficult to check that theredoes not exist a map R N → ( S ) N that factors through all terms in thesequence. Thus, the question is what is the inverse limit of the secondsequence of coverings?To compute the set of path components of the inverse limit of the secondsequence we compare it with the following sequence over a polyhedral ex-pansion of ( S ) N : R × S p (cid:15) (cid:15) R × R × S p (cid:15) (cid:15) f o o R × R × R × S p (cid:15) (cid:15) f o o · · · o o S × S S × S × S o o S × S × S × S o o · · · o o The maps f i : R i × S → R i − × S are defined as a product of a projectionon the first i -components with the square map (i.e. two-fold covering map)on S . Furthermore, the maps p i : R i × S → ( S ) i +1 are covering mapscorresponding to respectively the cyclic subgroup generated by the element(2 i , i − , . . . , , ∈ Z i +1 = π (( S ) i +1 ). It is easy to check that the diagramcommutes and that the i -th term in the original sequence is the pull-backof p i along the projection map as in the diagram R i × S × . . . (cid:15) (cid:15) / / R i × S p i (cid:15) (cid:15) ( S ) N / / ( S ) i +1 It follows that the inverse limit of the original sequence and the inverse limitof the sequence of covering over the polyhedral expansion coincide. We maynow apply Theorem 3.5 to determine the set of components of the totalspace of the limit. In fact, the inverse sequence of the fundamental groupsof the covering spaces is
Z Z o o Z o o Z o o · · · o o which is not Mittag-Leffler, therefore its derived inverse limit is non-trivial(it is actually an uncountable abelian group). As a consequence, the totalspace of the inverse limit has uncountably many components. Indeed, bya closer examination of the inverse limit of coverings over the polyhedralexpansion of ( S ) N we can conclude that the total space of the limit in thesecond case is homeomorphic to the product of the dyadic solenoid Sol with R N . Algebraic applications
We begin by describing a construction of inverse sequences of covering pro-jections whose limits correspond to meaningful subgroups of the fundamentalgroup. Given a countable CW-complex X and a continuous map f : X → S let p : e X → X be the covering map obtained as a pullback of the universalcovering of S along f , as in the following diagram e X / / p (cid:15) (cid:15) R e (cid:15) (cid:15) X f / / S It is easy to check that p is the covering map that corresponds to the kernelof f ∗ , that is to say π ( e X ) ∼ = Im( p ∗ : π ( e X ) → π ( X )) = Ker( f ∗ : π ( X ) → π ( S )) . Clearly, p is a regular covering whose fibres can be naturally identified withthe infinite cyclic group Z , so we often say that p is the Z -covering, corre-sponding to the map f : X → S (or rather, to its homotopy class). Since S is an Eilenberg-MacLane space of type K ( Z , X, S ] = Hom( π ( X ) , Z ) . Thus, we may also say that p is the Z -covering corresponding to a givenhomomorphism ϕ : π ( X ) → Z , in the sense that π ( e X ) = Ker ϕ . Note,that every non-trivial subgroup of Z is isomorphic to Z , so we may assumewithout loss of generality that ϕ is surjective.Given a sequence of homomorphisms ϕ , ϕ , ϕ , . . . : π ( X ) → Z , letΦ n := ( ϕ , . . . , ϕ n ) : π ( X ) → Z n and let p n : e X n → X be the covering projection whose fundamental group is K n := Ker Φ n = Ker ϕ ∩ . . . ∩ Ker ϕ n . Thus we obtain an inverse sequence of coverings e X p (cid:15) (cid:15) e X p (cid:15) (cid:15) f o o e X p (cid:15) (cid:15) f o o · · · o o X X X · · · where each f n is a covering projection. To compute the fibre of f n note thathave a short exact sequence0 → K n − /K n −→ π ( X ) /K n −→ π ( X ) /K n − → Z n and Z n − respectively.From this we deduce that K n − /K n is either trivial or isomorphic to Z ,therefore all f n are either trivial coverings (identity maps) or Z -coverings. NCOUNTABLE GROUPS 15
Since Hom( π ( X ) , Z ) is countable, we may apply the above constructionto the sequence of all homomorphisms from π ( X ) to Z . The limit of theresulting inverse sequence of coverings is a lifting projection b p : b X → X whose fundamental group is π ( b X ) = \ ϕ : π ( X ) → Z Ker ϕ. Note that if π ( X ) is finitely generated (e.g., if the 1-skeleton of X is fi-nite), then b p is a covering projection. In fact, in that case Hom( π ( X ) , Z ) =Hom( H ( X ) , Z ) is also finitely generated and the tower of coverings is actu-ally finite (i.e., all but finitely many coverings in the sequence are trivial).Alternatively, b p can be obtained as a covering of X that corresponds to thekernel of the natural homomorphism π ( X ) → F H ( X ), where F H ( X )denotes the maximal free abelian quotient of H ( 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 F ′ n = [ F n , F n ] of the free group on n generators F n consists of all words in F n for which the sum of exponents of each letter equals 0. This descriptionis 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 freegroup (on any set of generators), then F ′ = \ f : F → Z Ker( f ) . 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 = π ( H ): BC ( G ) = \ f : G→ Z Ker( f ) . The big commutator subgroup is much larger than the usual commutatorsubgroup 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 F ′′ n using homomorphisms tothe Baumslag-Solitar group B (1 , n ).Let us introduce the following functor for arbitrary groups G and A :Ker A ( G ) := \ f : G → A Ker( f ) . Note that Ker A ( G ) is a fully characteristic subgroup of G . More generally, if ϕ : G → H is a homomorphism and if x ∈ Ker A ( G ), then for every ψ : H → A we have that ψ ◦ ϕ ( x ) = 0, therefore ϕ ( x ) ∈ Ker A ( H ). In categoricalterms the correspondence G Ker A ( G ) is a covariant functor. It turns out that many characteristic subgroups can be described as intersection ofkernels to some group A . In fact we may view Ker A ( G ) as the part of G thatcannot be represented in a product of copies of the group A . For exampleKer Z ( G ) = 0 if, and only if, G is residually free-abelian.We mentioned before that Ker Z ( F ) = F ′ and Ker B (1 ,n ) ( F ) = F ′′ for everyfree group F . Recall that a torsion-free abelian group A is slender if everyhomomorphism ϕ : Z N → A can be factored through some finite rank freeabelian group, i.e., there exists a homomorphism ϕ ′ : Z n → A so that thefollowing diagram commutes Z N ϕ / / pr n ! ! ❇❇❇❇❇❇❇❇ A Z n ϕ ′ > > ⑥⑥⑥⑥⑥⑥⑥⑥ (see Fuchs [16, Ch. XIII]) where pr n is the projection map onto the first n factors of Z N . Free abelian groups are slender, subgroups and extensions ofslender groups are also slender. A reduced abelian group is slender if, andonly if, it does not contain a subgroup isomorphic to the group Z N or to agroup of p -adic integers for some prime p (cf. Fuchs [16]).Katsuya Eda [14] extended this concept to arbitrary groups by defining agroup 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 factoredthrough some free (non-commutative) group of finite rank as in G ϕ / / pr n (cid:31) (cid:31) ❅❅❅❅❅❅❅❅ AF n ϕ ′ > > ⑦⑦⑦⑦⑦⑦⑦⑦ where pr n 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 A ( Z N ) = lim ←− Ker A ( Z n ) . Similarly, if A is nc-slender, then Ker A ( b F ) = lim ←− Ker A ( F n ) and Ker A ( G ) = G ∩ lim ←− Ker A ( F n ) . Proof.
The natural projections pr n : Z N → Z n induce an isomorphism Z N → lim ←− Z n . By functoriality of Ker A , the homomorphisms Ker A ( Z N ) → Ker A ( Z n )induced by pr n are coherent and induce a homomorphism Ker A ( Z N ) → lim ←− Ker A ( Z n ) . This homomorphism is injective, because it is a restrictionof the isomorphism Z N → lim ←− Z n . To prove surjectivity, consider an ele-ment ( x i ) ∈ lim ←− Ker A ( Z n ) = Z N . We need only show that ( x i ) ∈ Ker A ( Z N ). NCOUNTABLE GROUPS 17
Fix a homorphism ϕ : Z N → A . Since A is slender, ϕ can be factored as ϕ = ϕ ′ ◦ pr n for some n and some ϕ ′ : Z n → A . Then ϕ (( x i )) = ϕ ′ ( x n )which is trivial since x n ∈ Ker A ( Z n ). Thus Ker A ( Z N ) → lim ←− Ker A ( Z n ) issurjective as well.In [13], the authors show that if A is nc-slender then every homomorphismfrom b F to A also factors through a projection to F n for some n . Then theproof for the non-commutatively slender cases follows analogously. (cid:3) As a consequence, A -kernels of b F with respect to a slender group A give riseto inverse limits of coverings over the Hawaiian earring. Indeed, recall thatthe Hawaiian earring H can be represented as inverse limit of a sequence · · · ← X n ← X n +1 ← · · · ← H, where X n is a wedge of n circles and the bonding maps are the obviousprojections. For each n the fundamental group of X n is F n , the free groupon n generators and there is the universal covering projection q n : b X n → X n = b X n /F n . The limit of the resulting inverse sequence of coverings is afibration with unique path lifting property q : b H = lim ←− b X n → H = lim ←− X n . By [8, Section 4] the group b F acts freely and transitively on the fibres of q sowe may consider the quotient of b H with respect to any (normal) subgroupof b F . Proposition 4.2. If A is any nc-slender group, then the projection q A : b H/ Ker A ( b F ) → H can be obtained as the inverse limit of coverings q ′ n : b X n / Ker A ( F n ) → X n ,and is therefore a fibration with unique path-lifting property.Proof. For each n we have a commutative diagram of the form b H / / (cid:15) (cid:15) b X n (cid:15) (cid:15) b H/ Ker A ( b F ) / / q A (cid:15) (cid:15) b X n / Ker A ( F n ) q ′ n (cid:15) (cid:15) H / / X n where for each n the map q ′ n is a covering projection with fibre F n / Ker A ( F n ).These coverings form an inverse sequence and by naturality we get a mapping to the inverse limit b H/ Ker A ( b F ) f / / q N (cid:15) (cid:15) lim ←− (cid:0) b X n / Ker A ( F n ) (cid:1) q (cid:15) (cid:15) H lim ←− X n To prove our claim it is sufficient to show that the restriction of f to thefibres f : b F /
Ker A ( b F ) → lim ←− (cid:0) F n / Ker A ( F n ) (cid:1) is an isomorphism. Consider the following inverse sequence of short exactsequences 1 / / Ker A ( F ) / / F / / F / Ker A ( F ) / / / / Ker A ( F ) O O / / F O O / / F / Ker A ( F ) / / O O / / Ker A ( F ) O O / / F O O / / F / Ker A ( F ) / / O O O O ... O O ... O O Each projection F n +1 → F n is a split surjection, so by functoriality of Ker A 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 A ( F n ) = Ker A b F so we have the following short exactsequence1 / / Ker A ( b F ) / / b F / / lim ←− (cid:0) F n / Ker A ( F n ) (cid:1) / / , which implies that the projections b F → F n induce the isomorphism b F /
Ker A ( b F ) ∼ = lim ←− (cid:0) F n / Ker A ( F n ) (cid:1) as claimed. (cid:3) In order to prove the main results of this section we will need an algebraiclemma that is reminiscent of Theorem 3.5. Let G be a subgroup of theinverse limit of a sequence of groups { G i } such that all projections G → G i are surjective. Furthermore, let H i be a decreasing sequence of subgroupsof G and for each i let H i be the image of H i in G i . Observe that G/H i ∼ = G i /H i (a bijection for arbitrary groups H i , and a group isomorphism if H i NCOUNTABLE GROUPS 19 are normal subgroups of G ). We can fit the above data in a commutativediagram with exact rows:1 / / T H i / / (cid:15) (cid:15) G / / (cid:15) (cid:15) lim ←− G/H i / / lim H i / / (cid:15) (cid:15) • / / lim ←− H i / / lim ←− G i / / lim ←− G i /H i / / lim H i / / • The following lemma is proved by elementary diagram chasing.
Lemma 4.3.
Ker( T H i → lim ←− H i ) = Ker( G → lim ←− G i ) , thus we have anexact sequence −→ Ker( G → lim ←− G i ) −→ \ H i −→ lim ←− H i Moreover lim ←− H i → lim ←− H i is a surjection of homogeneous sets whose fibrescan be naturally identified with the cokernel of the homomorphism G → (lim ←− G i ) / (lim ←− H i ) As a consequence, lim ←− H i is trivial iff lim ←− H i is trivial and G → (lim ←− G i ) / (lim ←− H i ) is surjective. We have previously considered the inverse sequence of wedges of circles · · · ← X n ← X n +1 ← · · · ← H, converging to the Hawaiian earring. Since π ( X n ) = F n , the free group on n generators F n , the inverse sequence of spaces gives rise to a homomorphismfrom the Hawaiian earring group G := π ( H ) to the inverse limit of finiterank free groups b F := lim ←− F n . By a result of Higman [21], see also [2] and[15], the homomorphism G → b F is injective, so the Hawaiian earring groupcan be viewed as a subgroup of b F . It is known that both the group G and itsindex in b F are uncountable, and that G can be viewed as a dense subgroupof b F with respect to the inverse limit topology on b F . The following theoremgives a more precise description of the relation between the two groups. Theproof is a combination of algebraic and geometric techniques developed inthe previous sections. Theorem 4.4.
The group b F is equal to the internal product of its subgroups G and Ker Z ( b F ) , i.e., b F = G ·
Ker Z ( b F ) .Proof. We are going to apply Lemma 4.3 to the following data: let G i be thefree group on i generators F i , G = G , and H i the kernel of the epimorphism G → π ( X i ) → H ( X i ) ∼ = Z i . Then lim ←− G i = b F . Observe that the imageof H i in π ( X i ) = F i is precisely the commutator subgroup F ′ i = Ker Z ( F i ),since G maps surjectively onto π ( X i ). The bonding maps in the inverse sequence of groups { H i } are surjective, therefore lim ←− H i is trivial. More-over, by Lemma 4.1 lim ←− H i = lim ←− Ker Z ( F i ) = Ker Z ( b F ). Thus, by Lemma4.3 the formula b F = G ·
Ker Z ( b F ) holds if, and only if, the derived inverselimit lim ←− H i is trivial.It is well-known that the limit lim ←− H i is precisely the set of path-componentsof the inverse limit of covering maps that correspond to the sequence ofkernels of the homomorphisms {G → H ( X i ) } (see [4, 22], as well as ourdiscussion in Section 2). Analogously as in the examples in Section 3, wecan represent this lifting projection as a limit of covering spaces over theapproximations of the Hawaiian earrings by finite wedges of circles X i . Thecovering space over X i corresponding to the commutator subgroup F i canbe identified with the integral 1-dimensional grid in R i , i.e. e X i = { ( x , . . . , x i ) ∈ R i | x j / ∈ Z for at most one index j } . Observe that the bonding maps in the inverse system e X ←− e X ←− e X ←− · · · are retractions, and so their inverse limit e X is the 1-dimensional integralgrid in R N . Alternatively, we may describe e X by the following pullbackdiagram e X / / (cid:15) (cid:15) R N e N (cid:15) (cid:15) H (cid:31) (cid:127) / / ( S ) N Clearly, e X is path-connected, which completes the proof of our claim. (cid:3) Note that there one could also consider the commutator subgroup b F ′ of b F ,which is however much smaller than lim ←− F ′ n . In fact, b F >
G · b F ′ .Continuing the previous line of thought we may ask whether b F can be ob-tained by adding some other term of the derived series of b F to the group G .As before, we replace b F ′′ with a suitable inverse limit group. We alreadymentioned that the second derived group of a free group F can be describedas a kernel, F ′′ = Ker B ( F ) (see [11, Theorem 1]) for any solvable, deficiency1 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 thatKer B ( b F ) = lim ←− Ker B ( F n ) for every group B that is solvable, of deficiency 1and is not virtually abelian. As well, we have b F ′′ ≤ Ker B ( b F ) ≤ Ker Z ( b F ) . NCOUNTABLE GROUPS 21
In the next theorem we reverse the reasoning and use our methods to showthat this subgroup of b F is too small to generate, together with the Hawaiianearring group the entire group b F . Theorem 4.5. b F = G ·
Ker B ( b F ) for every group B that is solvable, ofdeficiency and is not virtually abelian.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 freegroup of rank n , we have Ker B ( F n ) = F ′′ n = Ker Z (Ker Z ( F n )). We may repeatalmost verbatim the algebraic part of the proof of the previous theoremand obtain that b F = G ·
Ker B ( b F ) if, and only if, the inverse limit e Y ofthe sequence of coverings over the Hawaiian earring H , determined by thekernels of homomorphisms G → F i /F ′′ i , is path-connected. Thus, in orderto prove our claim, we must show that e 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 R N and let p : e X → H be the lifting projectionobtained as the limit of the sequence e X p (cid:15) (cid:15) e X p (cid:15) (cid:15) o o e X p (cid:15) (cid:15) o o · · · o o e X p (cid:15) (cid:15) o o X X o o X o o · · · o o H o o where π ( e X n ) = F ′ n . Let e Y n be a cover of e X n corresponding to the subgroup (cid:0) π ( e X n ) (cid:1) ′ = F ′′ n and let e Y be the resulting inverse limit. We will con-struct a continuous surjection from e Y to a path-disconnected space. Since π ( e X ) = F ′ is a countably generated free group, we can find a sequence ofhomomorphisms f i : π ( e X ) → Z such that n \ i =1 Ker( f i ) ( n − \ i =1 Ker( f i ) and ∞ \ i =1 Ker( f i ) = F ′′ . Let e Z ′ n be the cover of e X corresponding to the subgroup n T i =1 Ker (cid:0) π ( e X ) f i −→ Z (cid:1) . The covers e Z ′ n form an inverse sequence, whose limit e Z ′ := lim ←− e Z ′ n ispath-disconnected by Corollary 2.2. As a consequence, the pullback of e Z ′ along the projection p : e X → e X e Z ′ (cid:15) (cid:15) e Z o o (cid:15) (cid:15) e X e X p o o yields a lifting projection e Z → e X , whose total space is also path-disconnected.Note that e Z can be alternatively obtained as inverse limit of a sequence e Z = lim ←− e Z n , where e Z n is the pull-back of e Z ′ n along the projection p n : e X n → e X . Since commutator subgroups are characteristic we have p n ∗ ( π ( e Y n )) ≤ π ( e Y ) ≤ π ( e Z ′ ), thus π ( e Y n ) ≤ π ( e Z n ) = p − n ∗ ( π ( e Z ′ )). It follows that e Y n covers e Z n for every n and we obtain the following diagram: e Y (cid:15) (cid:15) e Y (cid:15) (cid:15) o o · · · o o e Y n − (cid:15) (cid:15) o o e Y n (cid:15) (cid:15) o o e Y n +1 (cid:15) (cid:15) o o · · · o o e Y (cid:15) (cid:15) o o e Z (cid:15) (cid:15) e Z (cid:15) (cid:15) o o · · · o o e Z n − (cid:15) (cid:15) o o e Z n (cid:15) (cid:15) o o e Z n +1 (cid:15) (cid:15) o o · · · o o e Z (cid:15) (cid:15) o o e X e X o o · · · o o e X n − o o e X n o o e X n +1 o o · · · o o e X o o By construction, the fibre of the lifting projection e Y → e X maps surjectivelyto the fibre of e Z → e X . Since every point in e Z (respectively e Y ) is connectedby a path to a point in the fibre, it follows, that the projection e Y → e Z issurjective, therefore e Y is not path-connected. (cid:3) References [1] A. K. Bousfield and D. M. Kan.
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