aa r X i v : . [ m a t h . A T ] A ug GENERAL THEORY OF LIFTING SPACES
GREGORY R. CONNER ∗ AND PETAR PAVEˇSI ´C ∗∗ Abstract.
In his classical textbook on algebraic topology Edwin Spa-nier developed the theory of covering spaces within a more general frame-work of lifting spaces (i.e., Hurewicz fibrations with unique path-liftingproperty). Among other, Spanier proved that for every space X thereexists a universal lifting space, which however need not be simply con-nected, unless the base space X is semi-locally simply connected. Thequestion on what exactly is the fundamental group of the universal spacewas left unanswered.The main source of lifting spaces are inverse limits of covering spacesover X , or more generally, over some inverse system of spaces convergingto X . Every metric space X can be obtained as a limit of an inversesystem of polyhedra, and so inverse limits of covering spaces over thesystem yield lifting spaces over X . They are related to the geometry (inparticular the fundamental group) of X in a similar way as the coveringspaces over polyhedra are related to the fundamental group of their base.Thus lifting spaces appear as a natural replacement for the concept ofcovering spaces over base spaces with bad local properties.In this paper we develop a general theory of lifting spaces and provethat they are preserved by products, inverse limits and other impor-tant constructions. We show that maps from X to polyhedra give riseto coverings over X and use that to prove that for a connected, lo-cally path connected and paracompact X , the fundamental group of theabove-mentioned Spanier’s universal space is precisely the intersectionof all Spanier groups associated to open covers of X , and that the latercoincides with the shape kernel of X .We examine in more detail lifting spaces over X that arise as inverselimits of coverings over some approximations of X . We construct an ex-act sequence relating the fundamental group of X with the fundamentalgroup and the set of path-components of the lifting space, study rela-tion between the group of deck transformations of the lifting space withthe fundamental group of X and prove an existence theorem for lifts ofmaps into inverse limits of covering spaces.In the final section we consider lifting spaces over non-locally pathconnected base and relate them to the fibration properties of the socalled hat space (or ’Peanification’) construction. Keywords : covering space, lifting space, inverse system, deck transfor-mation, shape fundamental group, shape kernel
AMS classification:
Primary 57M10; Secondary 55R05, 55Q05, 54D05. ∗ Supported by Simons Foundation collaboration grant 246221. ∗∗ Supported by the Slovenian Research Agency program P1-0292 and grants N1-0083,N1-0064. Introduction
The theory of covering spaces is best suited for spaces with nice local prop-erties. In particular, if X is a semi-locally simply-connected space, then thereis a complete correspondence between coverings of X and the subgroups ofits fundamental group π ( X ) (see [17, Section II, 5]). On the other hand,Shelah [16] showed that if the fundamental group of a Peano continuum X is not countable, then it is has at least one ‘bad’ point, around which itis not semi-locally simply-connected. It then follows that most subgroupsof π ( X ) do not correspond to a covering space. Thus, the nice relationbetween covering spaces and π ( X ) breaks apart as soon as X is not locallynice. Note that the situation considered by Shelah is by no means exotic:Peano continua with uncountable fundamental group appear naturally asattractors of dynamical systems [9], as fractal spaces [11, 13], as boundariesof non-positively curved groups [10], and in many other important contexts.Attempts to extend covering space theory to more general spaces includeFox overlays [8], and more recently, universal path-spaces by Fisher andZastrow [6] (see also [7]), and Peano covering maps by Brodskiy, Dydak,Labuz and Mitra [1]. Presently there is still an animated debate concerningthe correct way to define generalized covering spaces, fundamental groupsand other related concepts in order to retain as much of the original theoryas possible.In [17, Chapter II] Spanier introduced coverings spaces and maps as aspecial case of a more general concept of fibrations with unique path-liftingproperty. The latter turn out to be much more flexible than coverings. Inparticular, one can always construct the universal fibration over a given basespace X , where universality is interpreted as being the initial object in thecorresponding category. This universal object however need not be simplyconnected, and the basic question of what is its fundamental group is leftopen in Spanier’s book.As a part of his approach Spanier characterized subgroups of π ( X ) thatgive rise to covering spaces by showing that a covering subgroup of π ( X )must contain all U -small loops with respect to some cover U of X . Con-sequently elements of π ( X ) that are small with respect to all covers of X cannot be ’unravelled’ in any covering space and even in any fibration withunique path-lifting property. This result was one of the motivations for ourwork.In this paper we give a systematic treatment of lifting spaces and theirproperties, with particular emphasis to inverse limits of covering spaces,universal lifting spaces and their fundamental groups. In Section 2 we givean alternative definition of lifting spaces and derive their basic properties,which include stability under arbitrary products, compositions and inverselimits. Then we study the structure of the category of lifting spaces overa given base X and prove the existence of the universal lifting space over X . In Section 3 we study subgroups of the fundamental group π ( X ) that ENERAL THEORY OF LIFTING SPACES 3 correspond to covering spaces over X (without the assumption that X issemi-locally simply-connected). The main result is Theorem 3.5 in whichwe prove that fundamental group of the universal lifting space over X isprecisely the shape kernel of X (i.e. the kernel of the homomorphism fromthe fundamental group of X to its shape fundamental group). In Section 4we restrict our attention to a special class of lifting spaces over X that canbe constructed as limits of inverse systems of covering spaces over polyhedralapproximations of X . We show that much of the theory of covering spacescan be extended to this more general class. We construct an exact sequencerelating the fundamental group of the base with the fundamental group andthe set of path-components of the lifting space. Furthermore, we relatethe group of deck transformations of a lifting space with the fundamentalgroup of the base and the density of the path-components of the liftingspace. At this point one may expect that every lifting space over a givenbase space X can be obtained as a limit of an inverse system of coveringspaces over X or its approximations. However, a closer look reveals thatthis a subtle question, in [3] we constructed several classes of lifting spacesthat are not inverse limits of coverings, and in [4] we proved a detectionand classification theorem for lifting spaces that are inverse limits of finitecoverings. On the other hand, we can construct a universal lifting spaces forthe class of lifting spaces studied in Section 4. This is achieved in Section5 by taking a polyhedral expansion of the base space and considering thecorresponding inverse system of universal coverings. It turns out that the soobtained universal lifting space is in many aspects analogous to the universalcovering space, especially when the base space is locally path-connected.The first main result is Theorem 5.10 which summarizes the basic propertiesof lifting spaces whose base is a Peano continuum. In addition we give inTheorem 5.12 a criterion for maps between inverse limits of covering spaces.Finally, in Section 6 we partially extend our results to spaces that are notlocally path-connected. To this end we consider the hat construction (or‘peanification’), which to a non-locally path-connected space assigns the’closest’ locally path-connected one. The main result is that by applying thehat construction on a locally compact metric space we obtain a fibration (infact, a lifting projection) if and only if the hat space is locally compact.2. Fundamental groups of lifting spaces
Much of the exposition of covering spaces in [17, Chapter II] is done in amore general setting of (Hurewicz) fibrations with unique path lifting prop-erty. To work with Hurewicz fibrations we will use the following standardcharacterization in terms of lifting functions. Every map p : L → X inducesa map p : L I → X I × L , p : γ ( p ◦ γ, γ (0)). In general p is not surjective, GREGORY R. CONNER AND PETAR PAVEˇSI´C in fact its image is the subspace X I ⊓ L := { ( γ, l ) ∈ X I × L | p ( l ) = γ (0) } ⊂ X I × L. A lifting function for p is a section of p , that is, a map Γ : X I ⊓ L → L I suchthat p ◦ Γ is the identity map on X I ⊓ L . Then we have the following basicresult: Theorem 2.1. (cf. [14, Theorem 1.1] ) A map p : L → X is a Hurewiczfibration if and only if it admits a continuous lifting function Γ . Moreover, unique path-lifting property for p means that for ( γ, l ) , ( γ ′ , l ′ ) ∈ X I ⊓ L the equality Γ( γ, l ) = Γ( γ ′ , l ′ ) implies that ( γ, l ) = ( γ ′ , l ′ ). Thiscondition is clearly is equivalent to the injectivity of p , which leads to thefollowing definition.A map p : L → X is a lifting projection if p : L I → X I ⊓ L is a homeo-morphism, or equivalently, if the following diagram L Ip ◦− (cid:15) (cid:15) ev / / L p (cid:15) (cid:15) X I ev / / X is a pull-back in the category of topological spaces. Given a path γ : I → X and an element l ∈ L with p ( l ) = γ (0) we will denote by h γ, l i the uniquepath in L which starts at l and covers γ (i.e. p ( h γ, l i ) = ( γ, l )). The liftingspace is the triple ( L, p, X ) where p : L → X is a lifting projection. Wewill occasionally abuse the notation and refer to the space L or the map p : L → X itself as a lifting space over X .Clearly, every covering map is a lifting projection. Before giving furtherexamples we list some basic properties of lifting spaces (cf. [17], Section 2.2.,short proofs are included here to illustrate the efficiency of the alternativedefinition). Proposition 2.2. (1)
Arbitrary pull-backs, compositions, products, fi-bred products and inverse limits of lifting spaces are lifting spaces. (2)
In a lifting space p : L → X , given a path γ : I → X the formula f γ : l
7→ h γ, l i (1) determines a homeomorphism f γ : p − ( γ (0)) → p − ( γ (1)) between the fibres. In particular, if X is path-connectedthen any two fibres of p are homeomorphic. (3) A fibration p : L → X is a lifting space if, and only if its fibres aretotally path-disconnnected (i.e. admit only constant paths).Proof. (1) All claims follow from general facts about pull-backs as wenow show. ENERAL THEORY OF LIFTING SPACES 5
Pull-backs:
Let p : L → X be a lifting projection and let f : B → X be any map. Then in the following commutative cube B ⊓ L / / (cid:15) (cid:15) L (cid:15) (cid:15) ( B ⊓ L ) I / / (cid:15) (cid:15) ♣♣♣♣♣♣ L I (cid:15) (cid:15) > > ⑤⑤⑤⑤ B / / XB I / / ♦♦♦♦♦♦♦♦ X I = = ④④④④ the front, back and right vertical face are pull-backs, which by ab-stract nonsense implies that the left vertical square is also a pull-back. Therefore the pull-back projection B ⊓ L → B is a liftingprojection. Compositions: If p : L → X and q : K → L are lifting projec-tions then in the following diagram K I / / (cid:15) (cid:15) L I / / (cid:15) (cid:15) X I (cid:15) (cid:15) K q / / L q / / X the two inner squares are pull-backs, which implies that the outersquare is a pull-back, hence p ◦ q is also lifting projection. Products: If { p i : L i → X i } is a family of lifting projections,then the diagram Q i L Ii / / (cid:15) (cid:15) Q i L i Q p i (cid:15) (cid:15) Q i X Ii / / Q i X i is a product of pull-back diagrams, hence a pull-back diagram itself.It follows that Q p i is a lifting projection. Fibred products:
The fibred product of a family { p i : L i → X } i ∈I of lifting spaces over X is obtained by pulling back theirproduct along the diagonal map X → X I , hence is a lifting spaceby the above. Inverse limits:
An inverse system of lifting spaces is given by adirected set I , two I -indexed inverse systems L = ( L i , u ij : L j → L i )and X = ( X i , v ij : X j → X i ), and a morphism of systems p : L → X ,such that p i : L i → X i is a lifting projection for all i ∈ I . In orderto prove that the limit maplim ←− p i : lim ←− L i → lim ←− X i GREGORY R. CONNER AND PETAR PAVEˇSI´C is a lifting projection it is sufficient to observe that we have thenatural identifications (cid:16) lim ←− L i (cid:17) I = lim ←− L Ii , (cid:16) lim ←− X i (cid:17) I ⊓ (cid:16) lim ←− L i (cid:17) = lim ←− (cid:0) X Ii ⊓ L i (cid:1) , and that the projection (cid:16) lim ←− L i (cid:17) I −→ (cid:16) lim ←− X i (cid:17) I ⊓ (cid:16) lim ←− L i (cid:17) is ahomeomorphism because it is the inverse limit of homeomorphisms p i : L Ii ≈ −→ X Ii ⊓ L i .(2) Let γ denote the inverse path of the path γ . We claim that the map f γ : p − ( y ) → p − ( x ) , l
7→ h γ, l i (1) is the inverse of f γ . Indeed, since p (cid:0) h γ, l i (cid:1) = (cid:0) γ, h γ, l i (1) (cid:1) we get the equality h γ, l i = h γ, h γ, l i (1) i andso f γ ( f γ ( l )) = h γ, h γ, l i (1) i (1) = h γ, h γ, l i (1) i (1) = h γ, l i (1) = h γ, l i (0) = l. That f γ f γ is also the identity map is proved analogously.(3) Assume p : L → X is a lifting space and let γ be a path in p − ( x ) ⊂ L . Then p ( γ ) = (const x , γ (0)) = p (const γ (0) ), hence γ = const γ (0) .Conversely, assume that all fibres of p admit only constant paths.Since p is a fibration there is a map Γ : X I ⊓ L → L I such that p ◦ Γ = Id, and we only need to prove that γ = Γ( p ◦ γ, γ (0)) for all γ : I → L . For s ∈ I let γ s denote the path γ s ( t ) := γ ( st ), and let H be the standard homotopy starting at ( pγ ) s · ( pγ ) s and ending atconst pγ (0) . Let moreover e H : I × I → L be a lifting of H starting at e H | × I = γ s · Γ( pγ, γ (0)) s . It is easy to check that the restriction of e H to I × ∪ × I ∪ I × p − ( pγ ( t )) from γ ( s ) to Γ( pγ, γ (0))( s ), so by the assumption γ ( s ) = Γ( pγ, γ (0))( s ). (cid:3) We have recently proved in [15, Theorem 3.2] that, under very generalassumptions, lifting spaces are preserved by the mapping space construction,which yields a host of examples of lifting spaces that are very far from beingcoverings. The following examples illustrate typical ways how a lifting spacecan fail to be a covering space.
Example 2.3.
Let p : R → S be the usual covering of the circle. Then thecountable product p N : R N → ( S ) N is a lifting space by Proposition 2.2, butis not a covering space. In fact, the fibre of p is not a discrete space, beingan infinite product of Z . Even more drastically, one can easily verify that theinfinite product of circles is not semi-locally simply connected at any point,which means that it cannot have at all a simply connected covering space. ENERAL THEORY OF LIFTING SPACES 7
Example 2.4.
Another basic example is given by the following inverse limitof n -fold coverings S (cid:15) (cid:15) S (cid:15) (cid:15) o o S (cid:15) (cid:15) o o · · · o o S o o p (cid:15) (cid:15) S S S · · · S which presents the dyadic solenoid S as a lifting space over the circle. Byvarying the choice of coverings we obtain an entire family of non-equivalentlifting spaces over the circle leading to the following interesting problem: isit possible to classify all lifting spaces over the circle? One should keep inmind that this necessarily require the study of non-locally path-connected to-tal spaces. In fact, Spanier [17, Proposition 2.4.10] proved that a lifting space p : L → X over a locally path-connected and semi-locally simply-connectedbase X is a covering space if, and only if L is locally path-connected. Example 2.5.
Let us describe a simple but useful construction that some-times allow to extend results to spaces that are not locally path-connected.Given any space X let b X denote the set X endowed with the minimal topol-ogy that contains all path-components of open sets in X . Clearly, if X islocally path-connected then b X = X , but for non locally path-connected spaceswe obtain a strictly stronger topology, so the identity map ι : b X → X is acontinuous bijection but not a homeomorphism. For example, if W is thestandard Warsaw circle, then one can easily check that c W is homeomorphicto the interval [0 , . Note that this construction was called Peanification in [1] , but some care is needed, because in general b X is not a Peano space.The hat-construction is clearly functorial (in fact, together with the pro-jection to the original space it forms an idempotent augmented functor), sothat for every map f : Y → X we obtain a commutative diagram b Y b f / / (cid:15) (cid:15) b X (cid:15) (cid:15) Y f / / X It follows that every map from a locally path-connected space to X liftsuniquely to a map to b X , so in particular, the projection from the hat spaceadmits unique path liftings. Even more, it is always a Serre fibration, but itis not in general a Hurewicz fibration (and hence not a lifting space). Weare going to study this question in detail in the last section of the paper. Every covering space and every locally trivial fibration is an open map.In view of the above examples it would be interesting to know whether alllifting projections over a locally path-connected base are open maps.In the solenoid example above the total space is not path-connected. Clearly,
GREGORY R. CONNER AND PETAR PAVEˇSI´C if p : L → X is a lifting space then the restriction of p to any path-componentof L is a lifting space, too. In order to study the fundamental groups oflifting spaces, we now restrict our attention to based path-connected spaces.Let Lift X denote the category whose objects are path-connected open liftingspaces over X , and morphisms are fibre-preserving maps between them.All spaces have base-points and all maps are base-point preserving, but wesystematically omit the base-points from the notation. The category Lift X shares many properties with its full subcategory of covering spaces Cov X but is in some aspects more flexible. Proposition 2.6.
Morphisms in
Lift X are lifting projections and Lift X isan ordered category (i.e. there is at most one morphism between any twoobjects).Proof. Let f : L → K be a morphism between lifting spaces p : L → X and q : K → X . Then we have the natural identification K I ⊓ L = ( X I ⊓ K ) ⊓ L = X I ⊓ L = L I induced by f , therefore f is a lifting projection.If f, g : L → K are morphisms in Lift X then the unique path-lifting prop-erty imply that f and g coincide on path-components. Since f and g coincideon the base-point, and since L is path-connected, we have f = g . (cid:3) Clearly the category
Lift X has equalizers as there are no parallel pairs ofdistinct maps. It also has products: one can easily check that the categoricalproduct of a set of lifting spaces { L i → X } i ∈I is obtained by taking the path-component of the fibred product of { L i → X } i ∈I containing the base-point.Since categorical products and equalizers suffice for the construction of anyset-indexed categorical limit we obtain the following fact. Proposition 2.7.
Category
Lift X has arbitrary small (i.e. set-indexed) lim-its. Corollary 2.8.
For every path-connected space X the category Lift X hasthe universal (initial) object e X . The correspondence X e X determinesan idempotent augmented functor, as f : X → Y induce the commutativediagram e X ˜ f / / (cid:15) (cid:15) e Y (cid:15) (cid:15) X f / / Y Proof.
We first observe that the isomorphism classes of objects in
Lift X forma set. In fact by [17, Theorem 2.3.9] the points on any fibre of a lifting space p : L → X are in bijection with the set cosets of the subgroup p ♯ ( π ( L )) in π ( X ). This means that every lifting space over X corresponds to a choice ofa subgroup of π ( X ), together with a choice of a topology on the cartesianproduct of the set X with the set of cosets of p ♯ ( π ( L )) in π ( X ). Weconclude that the class of possible lifting spaces over X whose total spaces ENERAL THEORY OF LIFTING SPACES 9 is path-connected, forms a set. By Proposition 2.7 the categorical productof a set of representatives of all objects in
Lift X exists and is clearly theinitial object of the category. The other properties of the universal liftingspace follow from general properties of initial objects. (cid:3) As for covering spaces, it is of crucial importance to determine the fun-damental group of the universal lifting space. The following result is a stepin that direction.
Proposition 2.9.
Fundamental group of a categorical product in
Lift X isthe intersection of the fundamental groups of its factors.Proof. Let p : L → X be the categorical product of the family of liftingspaces { p i : L i → X } . By Proposition 2.6 the projection maps q i : L → L i are lifting projections, so by [17, Theorem 2.3.4] they induce monomor-phisms ( q i ) ♯ : π ( L ) → π ( L i ). It follows that π ( L ) ∼ = Im p ♯ ≤ π ( X ) is con-tained in T i Im( p i ) ♯ . For the converse implication, let the loop α : S → X represent an element of T i π ( L i ) ∼ = T i Im( p i ) ♯ ≤ π ( X ). By the uniquepath lifting property there are unique lifts α i : S → L i for the loop α , sothey define an element e α ∈ π ( L ). We therefore conclude that π ( L ) ∼ = T i π ( L i ). (cid:3) In order to achieve a more precise identification of the fundamental groupof e X we need a better understanding of subgroups of the π ( X ) that corre-spond to covering spaces of X .3. Covering subgroups
In this section we consider the question which subgroups of the funda-mental group of X correspond to coverings of X and relate them to theshape kernel of X .Let ( X, x ) be a based space. A subgroup G ≤ π ( X, x ) is a coveringsubgroup if there is a covering space p : ( e X, ˜ x ) → ( X, x ), such that Im p ♯ = G . Spanier gave a simple characterization of covering subgroups in terms of U -small loops. Given a covering U of X a based loop in ( X, x ) is said tobe U -small if it is of the form γ · α · γ where α is a (non-based) loop whoseimage is contained in some element of U and γ is a path in X connecting x and α (0). We denote by π ( X, x ; U ) the subgroup of π ( X, x ) generatedby classes of U -small loops. It is clear that π ( X, x ; U ) is always a normalsubgroup of π ( X, x ), and that π ( X, x ; V ) is contained in π ( X, x ; U )whenever V is a covering of X that refines U .The covering subgroups can be characterized as follows: Theorem 3.1 ([17] Lemma 2.5.11 and Theorem 2.5.13) . Let X be connectedand locally path-connected. Then G ≤ π ( X, x ) is a covering subgroup if, and only if G contains a subgroup of the form π ( X, x ; U ) for some cover U of X . A natural source of covering subgroups are continuous maps into polyhe-dra (or more generally, into semi-locally simply connected spaces).
Corollary 3.2.
Let f : X → K be a map from X to a semi-locally sim-ply connected space K . Then the kernel of the induced homomorphism f ♯ : π ( X, x ) → π ( K, f ( x )) is a covering subgroup of π ( X, x ) .Proof. Let U be a cover of K , such that for all U ∈ U the inclusion U ֒ → K induces a trivial homomorphism on the fundamental group. Then the group π ( X, x ; f − U ) is contained in the kernel of f ♯ because f maps every f − U -small loop in X to a homotopically trivial loop in K . Theorem 3.1 thenimplies that Ker f ♯ is a covering subgroup of π ( X, x ). (cid:3) For a partial converse to the above result assume that X has a numerablecover U = { U } with a subordinated locally finite partition of unity { ρ U } ,and let |U | denote the geometric realization of the nerve of U . Then theformula f ( x ) := P U ∈U ρ U ( x ) · U defines a map f : X → |U | . It is wellknown that the choice of the partition of unity does not affect the homotopyclass of f , so the induced homomorphism f ♯ depends only on the cover U . Lemma 3.3.
Let X be a path-connected space with a numerable cover U consisting of path-connected open sets. Then there is a short exact sequence −→ π ( X, x ; 2 U ) −→ π ( X, x ) f ♯ −→ π ( |U | , f ( x )) −→ , where U is the cover of X consisting of all unions of pairs of intersectingsets in U .Proof. By a suitable modification of the partition of unity we may assumewithout loss of generality that every U ∈ U contains some point x U ∈ U such that ρ U ( x U ) = 1.For every intersecting pair of sets U, V ∈ U choose a path in U ∪ V between x U and x V . These paths determine a map g : |U (1) | → X . For every1-simplex σ in |U (1) | the image f ( g ( σ )) is contained in the open star of σ ,which implies that the map f ◦ g is contiguous to the inclusion i : |U (1) | ֒ → |U | so the following diagram homotopy commutes: X f / / |U ||U (1) | g a a ❈❈❈❈❈❈❈❈ .(cid:14) i < < ③③③③③③③③ ENERAL THEORY OF LIFTING SPACES 11
By applying the fundamental group functor we obtain the diagram π ( X, x ) f ♯ / / π ( |U | , f ( x )) π ( |U (1) | , f ( x )) g ♯ h h PPPPPPPPPPP ( (cid:8) i ♯ ❧❧❧❧❧❧❧❧❧❧❧❧ Since the homomorphism i ♯ is surjective, so must be f ♯ .If U, V ∈ U intersect then every loop, whose image is contained in U ∪ V ismapped by f to the star of the simplex in |U | spanned by the vertices U and V . It follows that f ♯ is trivial on 2 U -small loops, therefore π ( X, x ; 2 U ) ⊆ Ker f ♯ .For the converse we extend the above diagram to obtain the followingone: Ker i ♯ / / g ♯ (cid:15) (cid:15) ✤✤✤ π ( |U (1) | , u ) i ♯ / / g ♯ (cid:15) (cid:15) π ( |U | , u ) (cid:15) (cid:15) ✤✤✤ ∼ = π ( X, x ) / Ker f ♯ π ( X, x ; 2 U ) / / π ( X, x ) / / f ♯ ❦❦❦❦❦❦❦❦❦❦❦❦❦❦❦ π ( X, x ) /π ( X, x ; 2 U )Since Ker i ♯ is generated by loops given by the boundaries of 2-simplexes inthe nerve of U we may use the same method as in the proof of Theorem7.3(2) in [2] to show that every such loop is mapped by g ♯ to a sum of2 U -small loops, so g ♯ (Ker i ♯ ) ⊆ π ( X, x ; 2 U ). Thus we obtain the inducednatural map π ( X, x ) / Ker f ♯ → π ( X, x ) /π ( X, x ; 2 U ) , and so Ker f ♯ ⊆ π ( X, x ; 2 U ). (cid:3) Lemma 3.4.
For every cover U of a paracompact space X there is a nu-merable cover V such that its double V is a refinement of U . Moreover, if X is locally path-connected, then V can be chosen so that its elements arepath connected.Proof. Let U ′ be a locally finite refinement of U with a subordinated parti-tion of unity. Then we have a map f : X → |U ′ | defined as before. Let V be the cover of X obtained by taking preimages of open stars of vertexes inthe barycentric subdivision of the nerve of U ′ . Clearly, two elements of V can have a non-empty intersection only if they are both contained in someelement of U ′ , which means that 2 V refines U ′ , and hence U .If X is locally path-connected, then we can further refine V by taking thecover formed by the path-components of elements of V . (cid:3) Assume that a pointed space (
X, x ) can be represented as a limit of aninverse system of pointed polyhedra ( X, x ) = lim ←− (cid:0) ( K i , k i ) , p ij , I (cid:1) (or moregenerally, that X has a polyhedral resolution in the sense of [12]). Thenthe homomorphisms ( p i ) ♯ : π ( X, x ) → π ( K i , k i ) induce a homomorphism ∂ : π ( X, x ) → ˇ π ( X, x ), where ˇ π ( X, x ) is the so called first shape group of X , and is defined as the limit of the inverse system of fundamental groups (cid:0) π ( K i , k i ) , ( p ij ) ♯ , I (cid:1) . Although the definitions are based on a specific reso-lution of X , it is a standard fact (see [12]) that both the first shape groupof X and the homomorphism ∂ : π ( X, x ) → ˇ π ( X, x ) are independent ofthe chosen resolution for X .We are now ready to prove the main theorem of this section which relatesvarious subgroups of the fundamental group. Theorem 3.5.
Let X be a connected, locally path-connected and paracom-pact space. Then the the following subgroups of π ( X, x ) coincide: (1) Intersection of all groups of U -small loops π ( X, x ; U ) for U a coverof X . (2) Intersection of all covering subgroups of X . (3) Intersection of all kernels
Ker f ♯ for maps f from X to a polyhedron. (4) The shape kernel of X , defined as ShKer(X) := Ker( ∂ : π ( X ) → ˇ π ( X )) .Proof. The inclusion (1) ⊆ (2) follows from Theorem 3.1 because every cover-ing subgroup contains some subgroup of U -small loops. Corollary 3.2 impliesthat (2) ⊆ (3).To prove the inclusion (3) ⊆ (1) observe that by Lemma 3.3 (3) is containedin the intersection of all groups of the form π ( X, x ; 2 U ), while by Lemma3.4 the latter is contained in (4).Finally the equality (3)=(4) amounts to the standard description of theshape kernel. (cid:3) As a consequence we obtain the following description of the fundamentalgroup of the universal lifting space:
Theorem 3.6. If X is a connected, locally path-connected and paracompactthen π ( e X ) coincides with the shape kernel of X .Proof. By Corollary 2.8 and Proposition 2.9 the fundamental group of theuniversal lifting space is contained in the intersection of all covering sub-groups of π ( X ), which by Theorem 3.5 coincides with the shape kernel of X . For the converse, take a loop α representing an element of the intersec-tion T π ( X ; U ), and approximate α by a sequence of homotopically trivialloops α i , such that α ≃ α i (mod U i ). As each α i lift to a loop in the universalspace, we may apply the fibration property to show that α also lift to a loop,hence α ∈ π ( e X ). (cid:3) Inverse limits of coverings
Inverse limits of coverings are in many aspects the most tractable classof lifting spaces (with the exception of coverings, of course). To simplifythe notation we agree that all spaces have base-points which are omitted
ENERAL THEORY OF LIFTING SPACES 13 from the notation whenever they are not explicitly used, and all maps arebase-point preserving.Let I be a directed set and X = ( X i , u ij : X j → X i , i, j ∈ I ) an I -indexedinverse system of path-connected and semi-locally simply-connected spaces.For each i ∈ I let q i : e X i → X i be the universal cover of X i . By the standardlifting criterion for maps between covering spaces (see [17, Theorem 2.4.5])there are unique maps ˜ u i,j : e X j → e X i such that the diagram e X j ˜ u ij / / q j (cid:15) (cid:15) e X iq i (cid:15) (cid:15) X j u ij / / X i commutes. Furthermore, e X := (cid:0) e X i , ˜ u i,j , I (cid:1) is an inverse system of spacesand q := ( q i ) : e X → X is a level-preserving mapping of inverse systems.More generally, let us for every i ∈ I choose a subgroup G i ≤ π ( X i )and consider maps q i : e X i → e X i /G i and p i : e X i /G i → X i , where p i is thecovering projection corresponding to the group G i . If ( u ij ) ♯ ( G j ) ⊆ G i , thenagain by the lifting theorem for covering spaces there exist unique maps¯ u ij : e X j /G j → e X i /G i such that the following diagram commutes e X j ˜ u ij / / q j (cid:15) (cid:15) e X iq i (cid:15) (cid:15) e X j /G j ¯ u ij / / p j (cid:15) (cid:15) e X i /G ip i (cid:15) (cid:15) X j u ij / / X i We will say that the an I -indexed family of groups G i ≤ π ( X i ) form a coherent thread with respect to the inverse system X if ( u ij ) ♯ ( G j ) ⊆ G i forall i ≤ j or, in other words, if G = (cid:0) G i , ( u ij ) ♯ , I (cid:1) is an inverse system ofgroups.Clearly, every coherent thread G for X induces an inverse system of spaces e X / G := ( e X i /G i , ¯ u ij , I ). Moreover, the inverse systems X , e X and e X / G arerelated by level preserving morphisms of inverse systems p := ( p i ) : e X / G → X and q := ( q i ) : e X → X / G . Observe that the systems e X and X maybe viewed as special instances of e X / G with respect to coherent threadsconsisting of trivial groups or of groups G i = π ( X i ) respectively.Since the bonding maps are base-point preserving, the inverse limits X :=lim ←− X , e X := lim ←− e X and e X G := lim ←− e X / G are non-empty. We will denote by u i : X → X i , ˜ u i : e X → e X i and ¯ u i : e X G → e X i /G i the projections from thelimit spaces to the system. Proposition 4.1.
The limit map p G := lim ←− p : e X G → X is a lifting projec-tion. Moreover, if X is locally path-connected then p G is an open map.Proof. The first claim follows directly from the fact that inverse limits oflifting projections is a lifting projection, as proved in Proposition 2.2.Toward the proof of the second claim observe that X and e X G may beviewed as subspaces of the topological products Q i ∈I X i and Q i ∈I X i /G i ,respectively. Therefore, it is sufficient to show that p G ( e U ) is open in X for every sub-basic open set of the form e U = X G ∩ ( e U i × Q j = i X j /G j ),where e U i is an open subset of X i /G i that is homeomorphically projectedto an elementary open subset U i = p i ( e U i ) ⊂ X i . Let x = ( x i ) ∈ p G ( e U )be the projection of the point ˜ x = (˜ x i ) ∈ e U . Since X is locally path-connected, there exists a path-connected open set V ⊂ X such that x ∈ V ⊂ X ∩ ( U i × Q j = i X j ). For every y ∈ V we can find a path α : ( I, , → ( V, x, y ). Then there is a unique lifting e α : ( I, → ( e U , ˜ x ) of α along thelifting projection p . As p i : e U i ≈ U i the construction of the lifting functionimplies that e α (1) i ∈ e U i , therefore y = p ( e α (1)) ∈ p ( e U ). We may thereforeconclude that V ⊂ p ( e U ), and consequently, that p ( e U ) is open in X . (cid:3) The same argument can be used to prove a more general statement thatif G and G ′ are coherent threads such that G i ≤ G ′ i ≤ π ( X i ) for every i ∈ I , then we obtain a lifting projection e X G → e X G ′ , which is an openmap, whenever e X G ′ is locally path-connected.In the following proposition we give an explicit description of the fibre of p G in terms of the fundamental groups of the spaces in the inverse system X and the coherent thread G : Proposition 4.2.
The fibre of p G is naturally homeomorphic to the inverselimit of the system of cosets (cid:0) π ( X i ) /G i , ( u ij ) ♯ , I (cid:1) .Proof. For each i ∈ I let x i and ¯ x i denote respectively the base-points of X i and e X i /G i , and let x = ( x i ) be the base-point of X . The correspondingfibres over x i and x are F i := p − i ( x i ) ⊂ e X i /G i and F := p − G ( x i ). If i ≤ j then the restriction of ¯ u ij maps F j to F i so we obtain the inverse system (cid:0) F i , ¯ u ij , I (cid:1) whose limit is precisely F , and the projection maps F → F i maybe identified with the restrictions of the projections ¯ u i : e X G → e X i /G i .It is well-known that the function ∂ : π ( X i ) → F i , that to every loop α ∈ π ( X i ) assigns the end point of the lifting of α to e X i , ∂ ( α ) := h e α, ¯ x i i (1) ∈ F i ,induces a bijection l i : π ( X i ) /G i → F i . The bijections l i are compatible withthe bonding homomorphisms in the inverse system, as for each i ≤ j we have ENERAL THEORY OF LIFTING SPACES 15 a commutative diagram π ( X j ) /G j (¯ u ij ) ♯ / / l j (cid:15) (cid:15) π ( X i ) /G il i (cid:15) (cid:15) F j ¯ u ij / / F i Observe that the bijections l i are actually homeomorphisms, as the fibres ofcovering spaces are discrete topological spaces. We conclude that the mor-phisms of inverse systems ( l i ) induces a natural homeomorphism between F = lim ←− F i and lim ←− π ( X i ) /G i . (cid:3) We may be lead to expect that p G is never a covering projection but thatis not the case. Indeed, to determine if p G is a covering projection it issufficient to consider the topology on its fibre. The product topology on thelimit of an inverse system of discrete spaces is not discrete, unless almostall bonding maps in the system are injective. Thus we have the followingcorollary (where we assume, for simplicity, that I = N , so that the inversesystem is in fact an inverse sequence). Corollary 4.3. p G is a covering projection if and only if the connectingmorphisms in the inverse system (cid:0) π ( X i ) /G i , ( u ij ) ♯ , I (cid:1) are eventually injec-tive.Proof. If there exists N such that ( u i i − ) ♯ are injective for i > N then p G is the pullback of the covering projection p N : e X N /G N → X N and so it isitself a covering projection. Conversely, if infinitely many bonding maps inthe sequence are non-injective then the limit space is not discrete, hence p G is a lifting projection but not a covering. (cid:3) Observe that Examples 2.3 (infinite product of coverings) and 2.4 (dyadicsolenoid) are inverse limits of coverings whose fibres are not discrete, so theyare not covering spaces over the circle. On the other hand, the fibre of thehat construction described in Example 2.5 is discrete but the total space isusually disconnected, so it is not a covering space in the usual sense, butrather a disjoint union of coverings. Here is another interesting lifting space:
Example 4.4.
Let W be the Warsaw circle, and let ( W i , u ij , N ) be the usualsequence of approximations of W by topological annuli. Then π ( W i ) ∼ = Z and ( u ij ) ♯ are isomorphisms. By choosing a coherent thread G with G i :=2 i Z we obtain an inverse sequence which is at group level analogous to that ofexample 2.4. The limit p G : f W G → W is an interesting lifting space that re-sembles a dyadic solenoid over W , so we may call it a dyadic Warsawonoid . Homotopy exact sequence.
Next we consider the long homotopyexact sequence of the lifting space p G . As the fibres of a lifting spaceare totally path-disconnected (cf. [17, Theorem 2.2.5]), we conclude that π n ( e X G ) ∼ = π n ( X ) for n ≥
2, and that π ( F ) may be identified with F . Furthermore, we have the following exact sequence of groups and pointedsets1 / / π ( e X G ) p G ♯ / / π ( X ) ∂ / / π ( F ) / / π ( e X G ) / / π ( X ) / / ∗ where the function ∂ is determined by the action of π ( X ) on the fibre F .We will normally assume that X is path-connected in which case the aboveexact sequence ends at the term π ( e X G ). The following theorem identifies π ( e X G ) and π ( e X G ). Observe that the fibre F is a closed subspace of e X G ,which by Proposition 4.2 may be identified with the inverse limit of thesystem of cosets F = lim ←− π ( X i ) /G i . Theorem 4.5.
Assume X is path-connected. Then there is an exact se-quence of groups and pointed sets / / π ( e X G ) p G ♯ / / π ( X ) ∂ / / lim ←− π ( X i ) /G i / / π ( e X G ) / / ∗ where the connecting function ∂ is given by the composition π ( X ) lim ←− ( u i ) ♯ / / lim ←− π ( X i ) / / lim ←− π ( X i ) /G i Consequently, p G induces an isomorphism π ( e X G ) ∼ = \ i ∈I ( u i ) − ♯ ( G i ) . Moreover, if G i is a normal subgroup of π ( X i ) for each i , then lim ←− π ( X i ) /G i is a group, ∂ is a homomorphism and π ( e X G ) may be identified with the setof cosets (cid:0) lim ←− π ( X i ) /G i (cid:1) /∂ ( π ( X )) . Proof.
By its definition, ∂ maps every α ∈ π ( X ) to the end point of thelifting to e X G of any representative of α , ∂ ( α ) = e α (1) ∈ F . The uniqueness ofliftings in covering spaces imply the commutativity of the following diagram π ( X ) ∂ / / ( u i ) ♯ (cid:15) (cid:15) F u i (cid:15) (cid:15) π ( X i ) ∂ / / (cid:15) (cid:15) (cid:15) (cid:15) F i π ( X i ) /G i l i / / F i which, together with Proposition 4.2, leads to the above description of theconnecting map ∂ .The description of ∂ implies that ( u i ) ♯ ( p G ( π ( e X G ))) ⊆ G i for all i ∈ I ,therefore Im p G ⊆ T i ( u i ) − ♯ ( G i ) . Conversely, let α be a loop representingan element of T i ( u i ) − ♯ ( G i ). Then for every i the loop u i ◦ α represents an ENERAL THEORY OF LIFTING SPACES 17 element of G i ≤ π ( X i ), so it lifts to a loop e α i in e X i /G i . The uniquenessof liftings imply that ¯ u ij ◦ e α j = e α i whenever i ≤ j , hence we obtain a loop e α := lim ←− e α i in e X G . Clearly, p G ◦ e α = α , therefore α ∈ Im p G .The proof of the last claim is straightforward. (cid:3) There are two special cases of the Theorem worth mentioning:1. For a subgroup G ≤ π ( X ) one may define a coherent thread G by letting G i := ( u i ) ♯ ( G ). Then ( u i ) − ♯ ( G i ) = G · Ker( u i ) ♯ , and so π ( e X G ) ∼ = G · \ I Ker( u i ) ♯ . Moreover if G is a normal subgroup of π ( X ), and if all homomorphisms( u i ) ♯ are surjective, then lim ←− π ( X i ) /G i is a group and π ( e X ) is a set ofcosets.2. If G is a thread of trivial groups then clearly e X G = e X . In this case π ( e X ) ∼ = \ I Ker( u i ) ♯ , while π ( e X ) may be identified with the set of cosets lim ←− π ( X i ) /∂ ( π ( X )).4.2. Deck transformations.
In the theory of covering spaces deck trans-formations provide a crucial connection between the algebra of the funda-mental group and the geometry of the covering space. Their role is evenmore important in the theory of lifting spaces because they enclose the in-formation about the interleaving of the path-components of the total spaceand induce a topology on the fundamental group of the base.A deck transformation of a lifting space p : L → X is a homeomorphism f : L → L , such that p ◦ f = p : L f / / p ❅❅❅❅❅❅❅ L p (cid:127) (cid:127) ⑦⑦⑦⑦⑦⑦⑦ X The set of all deck transformations of p clearly form a group, which wedenote by A ( p ). We will consider two basic questions: is the action of A ( p )free and is it transitive on the fibres of p . Our first result is valid for generallifting spaces. Proposition 4.6.
Let p : L → X be a lifting space such that each path-component of L is dense in L . Then the action of A ( p ) on L is free.Proof. Assume that f ( x ) = x for some f ∈ A ( p ) and x ∈ L . For every path α : ( I, → ( L, x ) the equality p ◦ f ◦ α = p ◦ α implies that f ◦ α and α are lifts of the path p ◦ α : I → X . Since f ( α (0)) = f ( x ) = x = α (0) and since p has unique path liftings, we conclude that f ( α (1)) = α (1) and so f fixes all elements of the path-component of L that contains x . But eachpath-component of L is dense in L and so the deck transformation f mustbe the identity. (cid:3) We must therefore determine when are the path-components of L dense.For inverse limits of coverings it is sufficient to consider the function ∂ thatwe introduced earlier. Proposition 4.7.
Let X be an inverse system of spaces, G a coherent threadof groups and and let F be the fibre of p G .If the image of the function ∂ : π ( X ) → π ( F ) = lim ←− π ( X i ) /G i isdense in π ( F ) (with respect to the inverse limit topology) then every path-component of e X G is dense in e X G .Conversely, if X is locally path-connected and every path-component of e X G is dense, then the image of ∂ is dense in F .Proof. Recall that we have identified F with π ( F ) and that the function ∂ is determined by the liftings of representatives of π ( X ) in e X G . This inparticular means that all elements of the image of ∂ belong to the samepath-component L of e X G .For a given ˜ x ∈ e X G let α be a path in X from x to x := p G (˜ x ). Thenthe formula g (˜ y ) := h α, ˜ y i (1) determines a homeomorphism between F andthe fibre p − G ( x ). This implies that ˜ x is in the closure of g (Im ∂ ), and hencein the closure of the path-component L , because clearly g (Im ∂ ) ⊂ L .To show that some other path-component L of e X G is also dense, it issufficient to choose a base-point for e X G in L and repeat the above argu-ment. Observe that a different choice of a base-point simply conjugates thefunction ∂ , so that its image is still dense in F .To show the converse statement, assume that some ˜ x ∈ F = p − G ( x ) is inthe closure of the path component L and let e U := Q i ∈F e U i × Q i/ ∈F e X i /G i be a product neighbourhood of ˜ x . Here F is some finite subset of I and each e U i is homeomorphically projected by p i to some elementary neighbourhoodin X i . We must show that e U intersects Im ∂ = L ∩ F . By Proposition 4.1the projection p G is an open map, so we may find a path-connected openset V such that x ∈ V ⊆ p G ( e U ). Since ˜ x is in the closure of L there exists˜ y ∈ e U ∩ p − G ( V ) ∩ L . Let α be a path in V from p G (˜ y ) to x . Then h α, ˜ y i (1)is by construction an element of Im ∂ ∩ e U , which proves that Im ∂ is densein F (cid:3) For arbitrary lifting spaces it can be sometimes hard to determine whetherthe image of ∂ is dense in the fibre, but for inverse limits of coverings wemay rely on the following algebraic criterion. Observe that if in an inversesystem of sets ( A i , u ij , I ) we replace each A i by the corresponding stableimage A Stab i := T j ≥ i u ij ( A j ) (and the bonding maps by their restrictions) ENERAL THEORY OF LIFTING SPACES 19 then lim ←− ( A i , u ij , I ) = lim ←− ( A Stab i , u ij , I ), i.e. the inverse limit depends onlyon stable images of bonding maps. In fact, the converse is also true, as thestable image is precisely the image of the projection from the inverse limit, A Stab i = ϕ i (lim ←− A i ). It is clear that for every morphism from a set A intothe system, ( ϕ i : A → A i , i ∈ I ), we have ϕ i ( A ) ⊆ A Stab i for all i ∈ I . Wewill say that such a morphism is stably surjective if ϕ i ( A ) = A Stab i for all i ∈ I . Obviously, every morphism that consists of surjective maps is stablysurjective. Lemma 4.8.
Let ( ϕ i : A → A i , I ) be a stably surjective morphism from A to an inverse system of sets ( A i , u ij , I ) . Then the image of the limit map ϕ := lim ←− ϕ i : A → lim ←− A i is dense in lim ←− A i (with respect to the producttopology of discrete spaces).Proof. The statement is trivial when the index set I is finite, so we willassume that I is infinite. Let ( a i ) be an element of lim ←− A i . By definition ofa product topology, a local basis of neighbourhoods at ( a i ) is given by thesets of the form U F := Q i ∈F { a i } × Q i/ ∈F A i , where F is a finite subset of I . Given any such F , let j ∈ I be bigger then all elements of F . By theassumptions, there exists a ∈ A such that ϕ j ( a ) = a j , but then ϕ ( a ) i = a i for every i ≤ j , and so ϕ ( a ) ∈ U F . We conclude that ϕ ( A ) is dense inlim ←− A i . (cid:3) We may finally formulate our main result about the free action of thegroup of deck transformations of an inverse limit of coverings.
Theorem 4.9.
Let X be an inverse system of spaces and let G be a coherentthread of groups. If the morphism (( u i ) ♯ : π ( X ) → π ( X i ) /G i , i ) is stablysurjective, then A ( p G ) acts freely on e X G .Proof. By Theorem 4.5 the function ∂ can be identified with the inverselimit lim ←− ( u i ) ♯ . By Lemma 4.8 the image of ∂ is dense in the lim ←− π ( X i ) /G i ,which is by Proposition 4.2 homeomorphic to the fibre of p G . Proposition 4.7implies that the path-components of e X G are dense, and finally Proposition4.6 implies that the action of A ( p G ) on e X G is free. (cid:3) Our next objective is to examine the transitivity of the action of A ( p ) onthe fibre of p . It is well-known that for a covering space p : e X/G → X theaction of A ( p ) on the fibre is transitive if and only if G is a normal subgroupof π ( X ). It is therefore reasonable to restrict our attention to coherentthreads of normal subgroups. Let ( X i , u ij , I ) be an inverse system of spacesand G = ( G i , I ) a coherent thread of normal subgroups, i.e. G i ⊳ π ( X i )for every i ∈ I . Then we have group isomorphisms l i : A ( p i ) → π ( X i ) /G i ,explicitly given as l i ( f ) := [ p ◦ ˜ α ], where ˜ α is any path in e X i /G i from the base-point ˜ x i to its image f (˜ x i ), and [ p ◦ ˜ α ] is the coset in π ( X i ) /G i determined bythe loop p ◦ ˜ α (cf. the description in [17, Section 2.6]). Moreover, whenever j ≥ i there is a homomorphism ˆ u ij : A ( p j ) → A ( p i ), where ˆ u ij ( f ) is defined to be the unique deck transformation of p i : e X i /G i → X i that maps thebase-point ˜ x i to ˆ u ij ( f (˜ x j )). It is easy to check that whenever i ≤ j we havea commutative diagram A ( p j ) ˆ u ij / / l j ∼ = (cid:15) (cid:15) A ( p i ) l i ∼ = (cid:15) (cid:15) π ( X j ) /G j ( u ij ) ♯ / / π ( X i ) /G i Thus the homomorphisms l i determine an isomorphism of inverse systems( l i ) : ( A ( p i ) , ˆ u ij , I ) → ( π ( X i ) /G i , ( u ij ) ♯ , I ) . Theorem 4.10.
There is an isomorphism of groups lim ←− l i : lim ←− A ( p i ) ∼ = −→ lim ←− π ( X i ) /G i . The group lim ←− A ( p i ) acts freely and transitively on the fibre of p G . Since lim ←− A ( p i ) is a subgroup of A ( p G ) it follows that A ( p G ) also acts transitivelyon the fibre of p G .Proof. That lim ←− l i is an isomorphism follows from the above discussion. To-ward the proof of transitivity, let (˜ x i ) , (˜ x ′ i ) be elements of the fibre of p G :then for every i ∈ I there exists a unique deck transformation f i ∈ A ( p i )such that f i (˜ x i ) = ˜ x ′ i . In order to prove that transformations f i representan element of the inverse limit, consider an element ˜ y ∈ e X j /G j and a path e α : ( I, , → ( e X j /G j , ˜ x j , ˜ y ). Then p i f i ¯ u ij e α = p i ¯ u ij e α = u ij p j e α = u ij p j f j e α = p i ¯ u ij f j e α, which means that f i ¯ u ij e α and ¯ u ij f j e α are paths with the same initial point andthe same projection in e X i /G i , so by the monodromy theorem they coincide.In particular f i (¯ u ij (˜ y )) = ¯ u ij ( f j (˜ y )) for every ˜ y ∈ e X j /G j , therefore ( f i ) isan element of lim ←− A ( p i ) that maps (˜ x i ) to (˜ x ′ i ).On the other side, if lim ←− f i (˜ x i ) = (˜ x i ) for some ( f i ) ∈ lim ←− A ( p i ) and( x i ) ∈ e X G then f i ( e x i ) = x i for each i ∈ I . It follows that all f i are identitydeck transformations on their respective domains, therefore the action oflim ←− A ( p i ) is free. (cid:3) One could expect that A ( p ) coincides with lim ←− A ( p i ) but we don’t know ifthat is true in general. In fact, there are inverse limits of coverings where thepath-components are not dense (like the Warsawonoid from Example 4.4),and so it is conceivable that the action of A ( p ) may not be free. In view ofthe above Theorem, this would imply that lim ←− A ( p i ) is a proper subgroup of A ( p ). This problem disappears, if the fundamental group of the limit mapssurjectively to the fundamental groups of its approximations, so we have thefollowing result. ENERAL THEORY OF LIFTING SPACES 21
Corollary 4.11.
Let X be an inverse system of spaces and let G be acoherent thread of normal subgroups. If the morphism (( u i ) ♯ : π ( X ) → π ( X i ) /G i , I ) is stably surjective, then A ( p G ) acts freely and transitivelyon the fibre of p G and is therefore isomorphic to lim ←− A ( p i ) .Furthermore, the map p = lim ←− p i : e X G → X induces a continuous bijection p : e X G /A ( p G ) → X . In addition, if X is locally path-connected then p is ahomeomorphism.Proof. The action of A ( p G ) on the fibre F is free and transitive by theassumptions and Theorems 4.9 and 4.10. Thus we may conclude that thereis a bijection A ( p G ) → F ∼ = lim ←− A ( p i ), which is clearly compatible with thegroup structures.Map p is induced by p : e X G → X as in the following diagram e X G p / / (cid:15) (cid:15) X e X G /A ( p G ) p : : ✈✈✈✈✈ and is clearly a continuous bijection. If X is locally connected then Propo-sition 4.1 implies that the map p is open. Then by the definition of quotienttopology p is also an open map and hence a homeomorphism. (cid:3) Universal lifting spaces
We are now going to apply methods developed in previous sections toconstruct certain lifting spaces that in many aspect behave as the universalcovering spaces. We will use freely terminology and constructions that arestandard in shape theory and are described, for example, in [12]. Recall thatall spaces are based (even if the base-points are normally omitted from thenotation) and the maps are base-point preserving.Throughout this section X will be a path-connected metric compactum.We can choose an embedding X ֒ → M into an absolute retract (for metricspaces) M , and consider the inverse system X of open neighbourhoods of X in M , ordered by the inclusion. Then X is an ANR expansion of X inthe sense of [12], and moreover X = lim ←− X . Observe that each space in theexpansion is semi-locally simply-connected and therefore admit all coveringspaces, including the universal one. As in the previous section, we maydefine e X to be the associated inverse system of universal coverings, and let e X := lim ←− e X . We are going to prove that the definition of e X is independentof the embedding X ֒ → M .To this end we will first show that the above construction is functorial. Let Y ֒ → N be another metric compactum embedded into an absolute retract N , and let f : X → Y be any map. By the standard properties of the ANRexpansions, f induces a morphism of systems f : X → Y . Note that themorphism f is in general not level-preserving, and commutes with bondingmaps only up to homotopy. Nevertheless, when composed with projectionsfrom X , it commutes exactly, so we may write f = lim ←− f . Every mapbetween base-spaces admits a unique lifting to a map between respectiveuniversal coverings, so we have a morphism of systems ˜f : e X → e Y that is theunique lifting of f . Here again, the morphism ˜f commutes with the bondingmaps only up to homotopy, but it commutes exactly when composed withprojections from e X , so we are justified to define ˜ f := lim ←− ˜f . Clearly, e X = 1 e X and ] g ◦ f = e g ◦ e f . Proposition 5.1.
The definition of e X is independent of the choice of em-bedding for X . The correspondence X e X and f e f determines a functorfrom compact metric spaces to metric spaces. This functor is augmented inthe sense that the following diagram is commutative e X ˜ f / / p X (cid:15) (cid:15) e Y p Y (cid:15) (cid:15) X f / / Y Proof.
Let i : X ֒ → M and j : X ֒ → N be embeddings of X into absoluteretracts M and N . Then we have corresponding ANR expansions X M and X N . Let f : X M → X N and g : X N → X M morphisms of systems inducedby the identity map on X . By the above discussion f and g induce maps˜ f , ˜ g : e X → e X which are inverse one to the other, which implies that e X isuniquely defined up to a natural homeomorphism. The functoriality and therelation between f and ˜ f follow directly from the definitions. (cid:3) It is a standard fact of shape theory (see for example [12, I, 5.1]) thatfor metric compacta every inverse system of polyhedra whose limit is X isan expansion and that for every such expansion we can choose a cofinalsequence. Thus we obtain a more manageable description of e X : Corollary 5.2. If X is a metric compactum then e X = lim ←− e X i for anyinverse sequence of polyhedra, converging to X . Clearly, if X is a compact polyhedron, then e X is just the universal cover-ing space of X . For a less trivial example, if W denotes the Warsaw circle,then it is easy to see that f W is the ’Warsaw line’ , which we may describe asthe real line in which every segment [ n, n + 1) is replaced by the topologist’ssine curve. As another illustration, if H is the Hawaiian earring, then e H isthe inverse limit of trees that are universal covering spaces for finite wedgesof circles.In order to explain in what sense is e X universal with respect to the liftingspaces that are inverse limits of coverings, let us consider an expansion ENERAL THEORY OF LIFTING SPACES 23 X = ( X i , u ij , I ) of a metric compactum X . Then any inverse system ofcoverings over X uniquely corresponds to a choice of a coherent thread G = ( G i , I ), i.e. is of the form ( e X i /G i , ¯ u ij , I ). By the lifting properties ofcovering spaces, for every i ≤ j in I there are unique maps q i , q j for whichthe following diagram commutes e X j ˜ u ij / / q j | | ③③③③③③③③ ˜ q j (cid:5) (cid:5) ☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛☛ e X iq i } } ③③③③③③③③ ˜ q i (cid:6) (cid:6) ☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞☞ e X j /G j ¯ u ij / / p j (cid:15) (cid:15) e X i /G ip i (cid:15) (cid:15) X j u ij / / X i Thus we obtain a commutative diagram of inverse systems and the corre-sponding limits e X Gp (cid:15) (cid:15) e X q o o ˜q ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ e X G p (cid:15) (cid:15) e X q o o ˜ q ~ ~ ⑤⑤⑤⑤⑤⑤⑤⑤ X X Theorem 5.3.
Let ˜ q : e X → X be the universal lifting space for X and let p : X → X be a lifting space obtained as a limit of an inverse system ofcoverings over some expansion of X . Then X = e X G for some (essentiallyunique) coherent thread G , and there is a unique (base-point preserving)lifting space projection q : e X → e X G for which p ◦ q = e q .Furthermore, let π ( X i ) Stab and ( π ( X i ) /G i ) Stab be the stable images ofthe systems of groups and cosets induced by the inverse systems for X and X . Then q ( e X ) is dense in e X G if and only if the induced map π ( X i ) Stab −→ ( π ( X i ) /G i ) Stab is surjective for every i ∈ I .Proof. The first assertion follows from the above discussion, so it remains toprove the characterization of density. Assume that the map between stableimages is surjective and consider an element ¯ x = (˜ x i G i ) ∈ X (with ˜ x i ∈ e X i as representatives of the orbits). By choosing a path α in X connecting thebase-point to p (¯ x ) we obtain a coherent sequence of paths ( α i : I → X i ),where α i connects the base point of X i to u i ( p (¯ x )). The unique path-lifting along α i yields a commutative diagram π ( X i ) (cid:31) (cid:127) / / (cid:15) (cid:15) e X iq i (cid:15) (cid:15) π ( X i ) /G i (cid:31) (cid:127) / / (cid:15) (cid:15) e X i /G ip i (cid:15) (cid:15) ∗ (cid:31) (cid:127) / / X i where the horizontal maps are π ( X i )-equivariant bijections onto the fibresover u i ( p (¯ x )) of the lifting spaces p i and ˜ q i . By construction, these diagramscommute with the bonding maps of the respective inverse systems so theyform a commutative diagram of inverse systems( π ( X i ) , (˜ u ij ) ♯ , I ) / / (cid:15) (cid:15) ( e X i , ˜ u ij , I ) q i (cid:15) (cid:15) ( π ( X i ) /G i , (¯ u ij ) ♯ , I ) / / (cid:15) (cid:15) ( e X i /G i , ¯ u ij , I ) p i (cid:15) (cid:15) ∗ / / ( X i , u ij , I )As a consequence, we obtain bijections π ( X i ) Stab → e X Stab i ∩ ˜ q − i ( u i ( p (¯ x )))and ( π ( X i ) /G i ) Stab → ( e X i /G i ) Stab ∩ ˜ p − i ( u i ( p (¯ x ))).A local basis of neighbourhoods at (˜ x i G i ) is given by the sets of the form U F := Q i ∈F { x i G i } × Q i/ ∈F e X i /G i , where F is any finite subset of I . Let j ∈ I be bigger than all elements of F . Since x j G j ∈ ( e X j /G j ) Stab , itcorresponds to to some g j G j , where by our assumption g j ∈ π ( X j ) Stab . Itfollows that there is an ˜ x ∈ e X such that ˜ u j (˜ x ) G j = ˜ x j G j , and consequently˜ u i (˜ x ) G i = ˜ x i G i for all i ∈ F . We have thus proved that q ( e X ) intersectsevery open set in e X G , hence the image of q is dense in e X G .Conversely, if for some i ∈ I the map π ( X i ) Stab −→ ( π ( X i ) /G i ) Stab is not surjective, then we may use the previously described correspondencebetween π ( X i ) and π ( X i ) /G i with the fibres of q i and p i to find an element¯ x ∈ e X G , such that q ( e X ) does not intersect its open neighbourhood { ¯ u i (¯ x ) }× Q j = i e X j /G j . (cid:3) In various practical situations it may be hard to verify whether the imageof e X is dense in X directly from the theorem. The following corollaryprovides several sufficient conditions. Corollary 5.4.
Any of the following conditions imply that q ( e X ) is dense in e X G . ENERAL THEORY OF LIFTING SPACES 25 (1)
The inverse system ( π ( X i ) , ( u ij ) ♯ , I ) has the Mittag-Leffler property(this holds in particular, if all bonding homomorphisms in the systemare surjective) and G is arbitrary. (2) G i is a subgroup of π ( X i ) Stab for every i ∈ I . (3) G i · π ( X i ) Stab = π ( X i ) for every i ∈ I . (4) There exists a cofinal subsequence
C ⊆ I , and all groups in the thread G are finite.Proof. (1) Recall that ( π ( X i ) , ( u ij ) ♯ , I ) satisfy the Mittag-Leffler con-dition if the stable images are achieved at some finite stage, i.e. forevery i ∈ I there is a j ≥ i such that π ( X i ) Stab = ( u ik ) ♯ ( π ( X k ))for every k ≥ j . Given an element g i G i ∈ π ( X i ) Stab there exists g j ∈ π ( X j ) such that g i G i = ( u ij ) ♯ ( g j G j ) = ( u ij ) ♯ ( g j ) G i , so g i G i isthe image of ( u ij ) ♯ ( g j ), which is by the Mittag-Leffler condition anelement of π ( X i ) Stab .(2) Assume that g i G i ∈ ( π ( X i ) /G i ) Stab , so that for every j ≥ i there ex-ists some g j ∈ π ( X j ) satisfying g i G i = ( u ij ) ♯ ( g j G j ) = ( u ij ) ♯ ( g j ) G i .It follows that ( u ij ) ♯ ( g j ) = g i g for some g ∈ G i . Since G i ⊆ π ( X i ) Stab there exists some h ∈ π ( X j ) such that ( u ij ) ♯ ( h ) = g ,and so g i = ( u ij ) ♯ ( g j h − ), which shows that g i is in the image of( u ij ) ♯ for every j ≥ i , therefore it is in the stable image.(3) The condition implies that every element of π ( X i ) /G i , and henceevery element of ( π ( X i ) /G i ) Stab , is in the image of π ( X i ) Stab .(4) If g i G i ∈ ( π ( X i ) /G i ) Stab , then for every j ∈ C , j ≥ i there exists g j ∈ π ( X j ) such that ( u ij ) ♯ ( g j G j ) = g i G i . This means that ( u ij )( g j )is one of the finitely many elements of the coset x i G i and so at leastone of them must appear infinitely many times as value of ( u ij ) ♯ for j ∈ C , j ≥ i . By the cofinality of C this element must be in π ( X i ) Stab . (cid:3) Example 5.5. If X is a polyhedron, then we may always take the trivialexpansion so the bonding maps on the associated system of fundamentalgroups are identity homomorphisms. If p : X → X is a limit of an inversesystem coverings over X then by Theorem 5.3 there exists a unique liftingprojection q : e X → X such that p ◦ q = ˜ q : e X → X , the standard universalcovering space projection. Moreover, by Corollary 5.4 the image q ( e X ) isdense in X . Observe that in principle the density of q ( e X ) in X depends on the prop-erties of the expansion used in the construction of X . However, we havethe following result that allows to avoid this objection. Our argument isbased on the fact that the limit of the inverse system of fundamental groupsassociated to an expansion of X actually depends only on the space itself:the shape fundamental group is defined as ˇ π ( X ) := lim ←− ( π ( X i ) , ( u ij ) ♯ , I ),where ( X i , u ij , I ) is any expansion of X (cf. [12, II, 3.3]). Lemma 5.6.
Let ( X i , u ij , I ) and ( X ′ i , u ′ ij , I ′ ) be two polyhedral expansionsfor X , and assume that the bonding homomorphisms in the associated systemof fundamental groups ( π ( X i ) , ( u ij ) ♯ , I ) are surjective. Then the system ( π ( X ′ i ) , ( u ′ ij ) ♯ , I ′ ) satisfies the Mittag-Leffler condition.Proof. By the properties of expansions, for a given i ′ ∈ I ′ we may find an i ∈ I and a map v : X i → X i ′ such that v ◦ u i = u ′ i ′ . Similarly, we may findsome j ′ ∈ I ′ and a map w : X j ′ → X i satisfying w ◦ u ′ j ′ = u i . X u i / / X i v ❇❇❇❇❇❇❇❇ X u ′ j ′ / / X j ′ u ′ i ′ j ′ / / w = = ⑤⑤⑤⑤⑤⑤⑤⑤ X i ′ Then, by applying fundamental groups we obtain the following diagramˇ π ( X ) = lim ←− π ( X i ) ( u i ) ♯ / / / / π ( X i ) v ♯ $ $ ■■■■■■■■■■ ˇ π ( X ) = lim ←− π ( X ′ i ) ( u ′ j ′ ) ♯ / / π ( X j ′ ) ( u ′ i ′ j ′ ) ♯ / / w ♯ : : ✉✉✉✉✉✉✉✉✉✉ π ( X i ′ )Since ( u i ) ♯ is surjective, so must be w ♯ , thus we have π ( X ′ i ′ ) Stab = ( u i ′ ) ♯ (ˇ π ( X )) = v ♯ ( π ( X i )) = ( u ′ i ′ j ′ ) ♯ ( π ( X j ′ ))which proves that the stable image coincides with the image of the bondingmap ( u ′ i ′ j ′ ) ♯ , as required by the Mittag-Leffler condition. (cid:3) Example 5.7.
We have already mentioned that the universal lifting spaceof the Warsaw circle W is the Warsaw line f W . Since we may obtain W as a limit of shrinking annuli, where the induced homomorphisms on thefundamental group are identities, the above results imply that f W is mappeddensely into every inverse limit of coverings over any expansion of W . Thisapplies in particular to all Warsawonoids that we described in Example 4.4. In general one cannot expect to find an expansion of X for which thebonding homomorphisms in the induced system of fundamental groups aresurjective. Nevertheless, this important property can be always achieved forexpansions of locally path-connected spaces. Lemma 5.8.
Every locally path-connected space X admits an expansion ( X i , u ij , I ) such that the induced homomorphisms ( u i ) ♯ : π ( X ) → π ( X i ) are surjective for all i ∈ I .Proof. We are going to exploit the technique used in the proof of Lemma 3.3.In fact, let U be a finite, non-redundant open cover of X and let f : X → |U | the map induced by some choice of a partition of unity subordinated to U . Then by Lemma 3.3 the induced homomorphism f ♯ : π ( X ) → π ( |U | ENERAL THEORY OF LIFTING SPACES 27 is surjective. Therefore, if we take the standard ˇCech expansion of X bypolyhedra X i that are nerves of covers of X by metric balls of radius 1 /i for i = 1 , , , . . . , then the resulting inverse sequence satisfy the desiredsurjectivity property. (cid:3) As a consequence we obtain strong surjectivity properties of inverse sys-tems of fundamental groups associated to expansions of locally path-connectedspaces.
Corollary 5.9.
Let ( X i , u ij , I ) be any expansion of a connected and lo-cally path-connected compact metric space X . Then the induced morphism ( π ( X ) → π ( X i ) , i ∈ I ) is stably surjective and the associated inverse sys-tem ( π ( X i ) , ( u ij ) ♯ , I ) satisfies the Mittag-Leffler condition.Proof. By Lemma 5.8 and the properties of an expansion we may find foreach i ∈ I a polyhedron P and maps v : X → P and v i : P → X i so that v i ◦ v = u i and v ♯ : π ( X ) → π ( P ) is a surjection. On the other side, we mayalso find a j ≥ i and a map v j : X j → P so that v j ◦ u j = v and v ◦ v j ≃ u ij .Thus we obtain the following diagram X u j / / v ❆❆❆❆❆❆❆❆ X j u ij / / v j (cid:15) (cid:15) X i P v i > > ⑤⑤⑤⑤⑤⑤⑤⑤ which induces a commutative diagram of fundamental groups π ( X ) ( u j ) ♯ / / v ♯ $ $ $ $ ■■■■■■■■■ π ( X j ) ( u ij ) ♯ / / ( v j ) ♯ (cid:15) (cid:15) π ( X i ) π ( P ) ( v i ) ♯ : : ttttttttt Since ( u ij ) ♯ factors through ( v i ) ♯ , it follows that ( v i ) ♯ ( π ( P ) contains thestable image π ( X i ) Stab . But then the surjectivity of v ♯ implies that thehomomorphism ( u i ) ♯ : π ( X ) → π ( X i ) Stab is also surjective.The second claim follows immediately by Lemma 5.6. (cid:3)
Connected and locally path-connected compact metric spaced form a largeclass of spaces that include all finite polyhedra, compact manifolds and manyother important spaces. They are in fact more commonly known as
Peanocontinua . This name is a distant echo of the Peano space-filling curves,consolidated by the famous Hahn-Mazurkiewicz Theorem that characterizesPeano continua as Hausdorff spaces that can be obtained as a continuousimage of an arc. The following theorem summarizes the main properties oflifting spaces over Peano continua.
Theorem 5.10.
Let X be a Peano continuum, ˜ q : e X → X its universallifting space and p : X → X any lifting space that can be obtained as a limit of an inverse system of coverings p i : X i → X i over some expansion of X .Then: (1) There exists a unique lifting projection q : e X → X such that p ◦ q = ˜ q .Moreover, the image q ( e X ) is dense in X . (2) The group of deck transformations A ( p ) acts freely on X . (3) If X is an inverse limit of normal coverings then A ( p ) acts freely andtransitively on the fibres of p , and there is an isomorphism A ( p ) ∼ =lim ←− A ( p i ) . In particular, A (˜ q ) is naturally isomorphic with the shapefundamental group ˇ π ( X ) . (4) If X is an inverse limit of normal coverings then p induces a home-omorphism ¯ p : X/A ( p ) → X . (5) There is an exact sequence of groups and sets / / π ( e X ) (˜ q ) ♯ / / π ( X ) ∂ / / ˇ π ( X ) / / π ( e X ) / / ∗ In particular, π ( e X ) may be identified with the kernel of the naturalhomomorphism ∂ : π ( X ) → ˇ π ( X ) , also known as shape kernel of X . Similarly, π ( e X ) may be identified with the shape cokernel of X ,namely the set of cosets ˇ π ( X ) /∂ ( π ( X )) . (6) Let G = ( G i ) be the coherent thread of groups, determined by X .Then the map q : e X → X induces a commutative ladder: / / π ( e X ) q ♯ (cid:15) (cid:15) ˜ q ♯ / / π ( X ) ∂ / / ˇ π ( X ) / / (cid:15) (cid:15) π ( e X ) / / q ♯ (cid:15) (cid:15) ∗ / / π ( X ) p ♯ / / π ( X ) ∂ / / lim ←− π ( X i ) /G i / / π ( X ) / / ∗ Proof.
The existence of the map q : e X → X was proved in Theorem 5.3.As for the second claim, observe that by Lemma 5.8 X admits an expan-sion such that the induced homomorphisms between fundamental groups aresurjective. By Corollary 5.9 the induced system satisfies the Mittag-Lefflercondition, and then by Corollary 5.4 it follows that q ( e X ) is dense in X .The statements 2., 3. and 4. also follow from Corollary 5.9, combinedwith Theorem 4.9 and Corollary 4.11.Finally, 5. and 6. follow from Theorem 4.5, in particular from the natu-rality of the exact sequence of a fibration. (cid:3) Observe that the shape kernel and shape cokernel are not shape invariants:in fact they are more like ’anti-invariants’ as they measure the variation ofthe structure of the universal lifting space within shape-equivalent spaces.We conclude the section with a lifting theorem for inverse limits of cov-ering spaces. Given a map f : X → Y and lifting spaces p : e X → X and q : e Y → Y we would like to know if there exists a map ˜ f for which the ENERAL THEORY OF LIFTING SPACES 29 following diagram commutes. e X ˜ f / / ❴❴❴ p (cid:15) (cid:15) e Y q (cid:15) (cid:15) X f / / Y If e X is connected and locally path connected then the answer is given bythe classical lifting theorem [17, Theorem II.4.5]: e f exists if, and only if f ♯ ( p ♯ ( π ( e X )) ⊆ q ♯ ( π ( e Y )). We are going to extend this result to more generallifting spaces. Observe that a very special case was already considered as apart of Proposition 5.1.Let us first show that the property of being the limit of a sequence ofcovering spaces over a polyhedral expansion for X is independent on thechoice of the expansion. Proposition 5.11.
Let X be the limit of an polyhedral expansion X =( X i , u ij : X j → X i , i, j ∈ N ) and let Let p G : e X G → X be the lifting spacedetermined by a coherent thread G of subgroups of π ( X i ) . Then for everypolyhedral expansion P = ( P i , v ij : P j → P i , i, j ∈ N ) for X there exists acoherent thread H , such that e X H = e X G and p H = p G .Proof. We will use properties of polyhedral expansions to obtain approxima-tions of the identity map of X with respect to the expansions X and P (cf.[12]). For every i there exists j and a map f i : P j → X i such that f i v j = u i .Furthermore, there exists a k and a map g j : X k → P j such that g j u k = v j .Finally, since f i and g j are both approximations of the identity on X thereexists some index l such that u il ≃ f i g j u kl . X u l / / X l u il / / X i P j f i > > ⑦⑦⑦⑦⑦⑦⑦⑦ X u l / / X l u kl / / X k g k > > ⑥⑥⑥⑥⑥⑥⑥⑥ Let H j := ( f i♯ ) − ( G i ) ≤ π ( P j ). Then ( g k u kl ) ♯ ( G l ) ⊆ H j and we obtain asequence of covering projections e X l /G l / / p l (cid:15) (cid:15) e P j /H jq j (cid:15) (cid:15) / / e X i /G ip i (cid:15) (cid:15) X l g k u kl / / P j f i / / X i which shows that we may refine the system of coverings over X i by coveringsover P i . It follows that e X G can be represented as an inverse limit of coveringsover the system P with respect to some coherent thread of groups H . (cid:3) Theorem 5.12.
Let X be a polyhedral expansion for X and p G : e X G → X the lifting space determined by a coherent thread G . Similarly, let Y be apolyhedral expansion for Y and q H : e Y H → Y the lifting space determined bya coherent thread H .Then a map f : X → Y can be lifted to a map ˜ f : e X G → e Y H if, and onlyif for a morphism f : X → Y induced by f we have that for every index i there exists some index j such that ( f i u ij ) ♯ ( G j ) ≤ H i .The condition for the existence of a lifting is independent from the choicesof expansions for X and Y .Proof. The assumption that for every i there is some j , such that ( f i v ij ) ♯ ( H j ) ≤ G i implies that there exist maps ˜ f i so that the following diagram commutes e X j /G j ˜ f i / / p j (cid:15) (cid:15) e Y i /H iq i (cid:15) (cid:15) X j f i u ij / / Y i After suitable reindexing we see that maps ˜ f i determine a morphism ofsystems ˜ f : e X / G → e Y / H , and hence define the lifting ˜ f : e X G → e Y H for f .That the existence of the lifting is independent from the choice of theexpansions for X and Y follows from Proposition 5.11. We prefer to omittechnical details. The converse implication is obvious. (cid:3) When is the hat space a lifting space?
In the preceding sections we have mostly assumed that the spaces underconsideration are locally path-connected. For more general spaces we mayfirst construct the hat space mentioned in Example 2.5. Since the uniquepath-lifting property of the map ι : b X → X is obvious, ι is a lifting space ifand only if it is a fibration. When this happens, every lifting space over b X is automatically a lifting space over X . Moreover, every lifting projection p : Y → X with Y locally path-connected, factors through b X as a composi-tion of two lifting projections.In this section we discuss in detail the fibration properties of the hatconstruction. In particular we prove that the hat construction over a metricspace yields a fibration for the class of all metric (in fact 1-countable) spacesif and only if the hat space is locally compact. ENERAL THEORY OF LIFTING SPACES 31
Recall that the hat space b X is obtained from a space X generated bytaking the path components of open sets in the topology of X as a sub-basis.The identity map ι : b X → X is continuous and bijective. If f : Y → X isa continuous map and C a path-component of an open set U ⊆ X , then f − ( C ) is a union of components of the open set f − ( U ). Therefore, if Y is locally path-connected, then f : Y → b X is also continuous. In particularthe paths in X correspond precisely to paths in b X and the same holds forpath-components of subsets as well. This implies that b X is locally path-connected, so d ( b X ) = b X . In addition, if f : Y → X is continuous, thenˆ f : b Y → b X (where ˆ f = f as a function between sets) is also continuous, andthe following diagram commutes b Y b f / / ι (cid:15) (cid:15) b X ι (cid:15) (cid:15) Y f / / X We may summarize these facts in categorical terms by saying that the hatconstruction is an idempotent augmented functor.What can be said about the fibration properties of the hat construction?Since maps and homotopies from cubes have unique liftings, the projection ι : b X → X is a Serre fibration with the unique path-lifting property. Inparticular, this shows that X and b X have isomorphic homotopy groups.However, the following example show that it is not in general a Hurewiczfibration. Example 6.1.
Let us denote by S := { /n | n ∈ N } ∪ { } ⊆ R the ’model’convergent sequence, and let C S be the cone over S , which we view as asubspace of the plane, e.g. C S := { ( t, tu ) ∈ R | t ∈ [0 , , u ∈ S } . It is easyto see that c C S can be identified with a countable one-point union of intervals.Consider the homotopy H : C S × I → C S , given by H (cid:0) ( x, y ) , t (cid:1) := ( tx, ty ) .Then the constant map H lifts to c C S but the entire homotopy H cannot belifted, because its final stage would give a continuous inverse to ι : c C S → C S ,which would imply that C S is locally path-connected, a contradiction. Is there some natural condition that would imply that ι : b X → X isa fibration for a sufficiently large class of spaces, like the metric spaces?Toward an answer to this question we prove that in order to check thefibration property for metric spaces it is sufficient to check that ι has thecovering homotopy property for maps from the model sequence S . Lemma 6.2.
The canonical map ι : b X → X is a fibration for the class offirst countable spaces if and only if it has the homotopy lifting property formaps from S to X . Proof.
One direction is immediate. For the other implication assume that Y is a 1-countable space and that we are given a homotopy H : Y × I → X such that the lifting b H : Y × → b X is continuous. In order to prove that b H : Y × I → b X is continuous we take a sequence { ( y i , t i ) } in Y × I convergingto ( y, t ). The sequence { y i } converges to y and hence determines a map g : S → Y . The initial stage of the homotopy F := H ◦ ( g ×
1) : S × I → X lifts to the continuous map b F = b H ◦ g : Y × → b X , so by the assumption b F : S × I → b X is continuous as well. In particular the sequence b F ( y i , t i )converges to b F ( y, t ). (cid:3) Recall that a space is sequentially compact if every sequence in it hasa convergent subsequence. In general the compactness and the sequentialcompactness are not directly related as none of them implies the other.However, they coincide for the class of metric spaces and for 1-countablespaces (or more generally, for sequential spaces ) the compactness impliessequential compactness. Moreover, a space is locally sequentially compact ifevery point has a neigborhood whose closure is sequentially compact.
Theorem 6.3. If X is a Hausdorff space and if b X is locally sequentiallycompact, then the canonical map ι : b X → X has the homotopy lifting prop-erty for maps from S .Proof. Let F : S × I → X be a map, such that the restriction of the uniquelift b F : S × I → b X is continuous when restricted to S ×
0. We are going toshow that b F is also continuous.First observe that the local path-connectedness of ( S − × I imply that b F is continuous on ( S − × I , so it only remains to prove the continuity of b F on 0 × I . Since b F is continuous on S × t := sup { s ∈ I | b F | S × [0 ,s ] is continuous } . By the local sequential compactness of b X and the definition of the hat-topology we may choose a neighborhood U ⊂ b X of b F (0 , t ) that is a path-component of an open set V ⊂ X , and whose closure is sequentially compact.By the continuity of F there exist ε, δ > B := ( S ∩ [0 , δ )) × ( t − ε, t + ε ) is contained in F − ( V ). Moreover, sincethe restriction b F | × I is continuous, we may assume (by decreasing ε , ifnecessary) that b F (0 × ( t − ε, t ]) ⊂ U . Furthermore, by the definition of t ,there is an s ∈ ( t − ε, t ], such that b F | S × s is continuous, so we may assume(by adjusting δ if necessary) that b F (cid:0) ( S ∩ [0 , δ )) × s (cid:1) ⊂ U . Since U is path-connected, it follows that b F ( B ) ⊂ U . We claim that b F is continuous on0 × ( t − ε, t + ε ).Take an x ∈ ( t − ε, t + ε ) and let { x i } be a sequence in B converging to(0 , x ). To show that { b F ( x i ) } converges to b F (0 , x ) we must show that everyopen neighborhood W of b F (0 , x ) contains all but finitely many elements ofthe sequence. Indeed, otherwise the elements of { b F ( x i ) } outside W would ENERAL THEORY OF LIFTING SPACES 33 have an accumulation point u in the compact set U − W . This would implythat F (0 , x ) and ι ( u ) are distinct accumulation points of the convergentsequence { F ( x i ) } , a contradiction.Our argument shows that the assumption t < t = 1, and therefore e F : S × I → b X iscontinuous. (cid:3) Example 6.4.
The obvious map from [0 , to the Warsaw circle is a fi-bration for the class of 1-countable spaces, while the projection from thecountable one-point union of intervals to C S is not. If X is first countable and Hausdorff then so is b X : the Hausdorff propertyis obvious, for the other simply observe that if we take a countable localbasis around the point x in X , then the path-components of the basic setscontaining the point x constitute a countable local basis for x in b X . If inaddition b X is locally compact (and therefore locally sequentially compact),then we have just proved that ι : b X → X is a fibration for the class ofall first countable spaces. But a map between first countable spaces thathas the covering homotopy property for maps between from first countablehas automatically the covering homotopy property for maps from arbitraryspaces, so we have the following Theorem 6.5.
Let X be a first countable, Hausdorff space. If b X is locallycompact then ι : b X → X is a lifting space.Proof. By 6.2 and 6.3 we already know that ι has the covering homotopyproperty for 1-countable spaces. Since we assumed that X is 1-countablethat b X and X I are also 1-countable, and therefore the space ( b X ⊓ X I ) × I is1-countable, because it is a subspace of b X × X I . It follows that there existsthe unique map H that makes commutative the following diagram( b X ⊓ X I ) × (ˆ x,α, ˆ x / / (cid:127) _ (cid:15) (cid:15) b X ι (cid:15) (cid:15) ( b X ⊓ X I ) × I H ♠♠♠♠♠♠♠♠ (ˆ x,α,t ) α ( t ) / / X Its adjoint map e H : b X ⊓ X I → b X I is the continuous inverse for the continuousbijection ¯ ι : b X I → b X ⊓ X I , therefore ι is a lifting space. (cid:3) Remark 6.6.
The above results can be easily extended to more generalspaces. Indeed, let ℵ be any cardinal number, and let ω = ω ( ℵ ) be thecorresponding initial ordinal. Then we may consider spaces for which everypoint has local bases of cardinality at most ℵ and spaces that are ω -compact,i.e. each ’sequence’ indexed by ω has an accumulation point. Then we canrepeat the above proofs word-by-word to show that if X has local basis ofcardinality at most ℵ and if b X is locally ω -compact then ι b X → X is a liftingspace. To prove the converse of Theorem 6.3 we need to assume that X is ametric space. Let us first show that if ( X, d ) is a path-connected metricspace then b X is also metrizable. Indeed, a suitable metric on b X can bedefined by taking into account the path structure on X . Let the breadth ofa path α : I → X be defined asbr( α ) := sup (cid:8) d (cid:0) α (0) , α ( t ) (cid:1) + d (cid:0) α ( t ) , α (1) (cid:1) | t ∈ [0 , (cid:9) . Then we get a metric ρ on b X by ρ ( x, x ′ ) = inf (cid:8) br( α ) | α : ( I, , → ( X, x, x ′ ) (cid:9) . Note that always d ( x, x ′ ) ≤ ρ ( x, x ′ ), and that for a path-connected set A ⊆ X we have diam ρ ( A ) ≤ · diam d ( A ) . We can easily verify that the topology of b X is induced by ρ . In fact, let C be a component of an open set U ⊆ X . For every x ∈ C there is an ε -ball B d ( x, ε ) ⊆ U , and since d -distance does not exceed ρ -distance, we also havethat B ρ ( x, ε ) ⊆ U . It follows that all points of B ρ ( x, ε ) can be connected bya path in U , therefore B ρ ( x, ε ) ⊆ C , and so C is open with respect to themetric ρ . On the other hand, the ball B ρ ( x, ǫ ) is clearly open with respectto the metric d , so the path-component of B ρ ( x, ǫ ) containing x is an openset in b X contained in that ball. Theorem 6.7.
Suppose X is a locally compact, path-connected metric space.If ι : b X → X has the homotopy lifting property for maps from S then b X islocally compact.Proof. Assume by contradiction that b X is not locally compact, so that thereis a point x ∈ X that does not possess any compact neighborhood in b X .Let ε > ε > ε > . . . be a strictly decreasing sequence converging to zero,such that all B d ( x, ε i ) are relatively compact. For every i the correspondingball B ρ ( x, ε i ) ⊂ b X is path-connected, contained in B d ( x, ε i ) and, by theassumption, not relatively compact. Therefore, we can choose for each i asequence x i , x i , . . . of points in B ρ ( x, ε i ) that converges in X to a point inthe closure of B d ( x, ε i ) but does not have any accumulation points in b X . Forevery j , we have ρ ( x ij , x i +1 j ) < ε i + ε i +1 , so we can find a path of breadth lessthen ε i + ε i +1 connecting x ij and x i +1 j . We can concatenate these paths toobtain a path α j : I → X , running from x j to x j on [0 , / x j to x j on [1 / , / α j (taken in reverse direction) togetherwith the constant path in x define a homotopy H : S × I → X , given by theformula H ( u, t ) = ( x if u = 0 α u (1 − t ) otherwise . Since H is a constant map it can be lifted to b X , but we cannot lift theentire homotopy H , because then b H would send the convergent sequence S to the sequence x , x , x , . . . that does not converge in b X . (cid:3) ENERAL THEORY OF LIFTING SPACES 35
We may now combine Theorem 6.7 with Theorems 6.3 and 6.5 to obtainthe following result.
Theorem 6.8.
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Math Department, Brigham Young University, Provo, UT 84602, USA
E-mail address : [email protected] (Petar Paveˇsi´c) Faculty of Mathematics and Physics, University of Ljubljana, Slovenija
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