Covering maps for locally path-connected spaces

arXiv:0801.4967v3 [math.GT] 14 Feb 2008 COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA Abstract. We define Peano covering maps and prove basic properties analogous to classical covers. Their domain is always locally path-connected but the range may be an arbitrary topological space. One of characterizations of Peano covering maps is via the uniqueness of homotopy lifting property for all locally path-connected spaces. Regular Peano covering maps over path-connected spaces are shown to be identical with generalized regular covering maps introduced by Fischer and Zastrow [15]. If X is path-connected, then every Peano covering map is equive alent to the projection X/H → X, where H is a subgroup of the fundamental e equipped with the topology used in [2], [15] and introduced group of X and X e in [23, p.82]. The projection X/H → X is a Peano covering map if and only e for if it has the unique path lifting property. We define a new topology on X e which one has a characterization of X/H → X having the unique path lifting property if H is a normal subgroup of π1 (X). Namely, H must be closed in π1 (X). Such groups include π(U , x0 ) (U being an open cover of X) and the kernel of the natural homomorphism π1 (X, x0 ) → π̌1 (X, x0 ). Contents 1. Introduction 2. Constructing Peano spaces 2.1. Universal Peano space e 2.2. Basic topology on X e 3. A new topology on X 4. Path lifting 5. Peano maps 6. Peano covering maps 7. Peano subgroups 8. Appendix: Pointed versus unpointed References 1 3 3 5 8 10 15 17 21 23 24 1. Introduction As locally complicated spaces naturally appear in mathematics (examples: boundaries of groups, limits under Gromov-Hausdorff convergence) there is an effort to extend homotopy-theoretical concepts to such spaces. This paper is devoted to Date: February 14, 2008. 2000 Mathematics Subject Classification. Primary 55Q52; Secondary 55M10, 54E15. Key words and phrases. covering maps, locally path-connected spaces. 1 2 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA a theory of coverings by locally path-connected spaces. Zeeman’s example [16, 6.6.14 on p.258] demonstrates difficulty in constructing a theory of coverings by non-locally path-connected spaces (that example amounts to two non-equivalent classical coverings with the same image of the fundamental groups). For coverings in the uniform category see [1] and [3]. To simplify exposition let us introduce the following concepts: Definition 1.1. A topological space X is an lpc-space if it is locally pathconnected. X is a Peano space if it is locally path-connected and connected. Fischer and Zastrow [15] defined generalized regular coverings of X as functions p : X̄ → X satisfying the following conditions for some normal subgroup H of π1 (X): R1. X̄ is a Peano space. R2. The map p : X̄ → X is a continuous surjection and π1 (p) : π1 (X̄) → π1 (X) is a monomorphism onto H. R3. For every Peano space Y , for every continuous function f : (Y, y) → (X, x0 ) with f∗ (π1 (Y, y)) ⊂ H, and for every x̄ ∈ X̄ with p(x̄) = x0 , there is a unique continuous g : (Y, y) → (X̄, x̄)) with p ◦ g = f . Our view of the above concept is that of being universal in a certain class of maps and we propose a different way of defining covering maps between Peano spaces in Section 6. Our first observation is that each path-connected space X has its universal Peano space P (X), the set X equipped with new topology, such that the identity function P (X) → X corresponds to a generalized regular covering for H = π1 (X). That way quite a few results in the literature can be formally deduced from earlier results for Peano spaces. The way the projection P (X) → X is characterized in 2.2 generalizes to the concept of Peano maps in Section 6 and our Peano covering maps combine Peano maps with two classical concepts: Serre fibrations and unique path lifting property. Peano covering maps possess several properties analogous to the classical covering maps [18] (example: local Peano covering maps are Peano covering maps). bH of the universal path space X e One of them is that they are all quotients X equipped with the topology defined in the proof of Theorem 13 on p.82 in [23] and used successfully by Bogley-Sieradski [2] and Fischer-Zastrow [15]. It turns out the bH → X is a Peano covering map if and only if it has the endpoint projection X uniqueness of path lifts property (see 6.4). In an effort to unify Peano covering maps with uniform covering maps of [1] and [3] (we will explain the connection in eH (see Section 3). Its main advantage is [4]) we were led to a new topology on X eH → X to have the unique that there is a necessary and sufficient condition for X path lifting property in case H is a normal subgroup of π1 (X). It is H being closed in π1 (X). That explains Theorem 6.9 of [15] as the basic groups there turn out to be closed in π1 (X). As an application of our approach we show existence of a universal Peano covering map over a given path-connected space. We thank Sasha Dranishnikov for bringing the work of Fischer-Zastrow [15] to our attention. We thank Greg Conner, Katsuya Eda, Aleš Vavpetić, and Ziga Virk for helpful comments. COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 3 2. Constructing Peano spaces The purpose of this section is to discuss various ways of constructing new Peano spaces. 2.1. Universal Peano space. In analogy to the universal covering spaces we introduce the following notion: Definition 2.1. Given a topological space X its universal lpc-space lpc(X) is an lpc-space together with a continuous map (called the universal Peano map) π : lpc(X) → X satisfying the following universality condition: For any map f : Y → X from an lpc-space Y there is a unique continuous lift g : Y → lpc(X) of f (that means π ◦ g = f ). Theorem 2.2. Every space X has a universal lpc-space. It is homeomorphic to the set X equipped with a new topology, the one generated by all path-components of all open subsets of the existing topology of X. Proof. Let U be an open set in X containing the point x and c(x, U ) be the path component of x in U . Since z ∈ c(x, U ) ∩ c(y, V ) implies c(z, U ∩ V ) ⊂ c(x, U ) ∩ c(y, V ), the family {c(x, U )}, where U ranges over all open subsets of X and x ranges over all elements of U , forms a basis. Given a map f : Y → X and given an open set U of X containing f (y) one has f (c(y, f −1 (U ))) ⊂ c(f (y), U ). That proves f : Y → lpc(X) is continuous if Y is an lpc-space. It also proves lpc(X) is locally path-connected as any path in X induces a path in lpc(X).  Remark 2.3. The topology above was mentioned in Remark 4.17 of [15]. After the first version of this paper was written we were informed by Greg Conner of his unpublished preprint [7] with David Fearnley, where that topology is discussed and its properties (compactness, metrizability) are investigated. If X is path-connected, then lpc(X) is a universal Peano space P (X) in the following sense: given a map f : Z → X from a Peano space Z to X there is a unique lift g : Z → P (X) of f . In the remainder of this section we give sufficient conditions for a function on an lpc-space to be continuous. Those conditions are in terms of maps from basic Peano spaces: the arc in the first-countable case and hedgehogs (see Definition 2.8) in the arbitrary case. Proposition 2.4. Suppose f : Y → X is a function from a first-countable lpc-space Y . f is continuous if f ◦ g is continuous for every path g : I → Y in Y . Proof. Suppose U is open in X. It suffices to show that for each y ∈ f −1 (U ) there is an open set V in Y containing y such that the path component of y in V is contained in f −1 (U ). Pick a basis of neighborhoods {Vn }n≥1 of y in Y and assume for each n ≥ 1 there is a path αn in Vn joining y to a point yn ∈ / f −1 (U ). Those paths can be spliced to one path α from y to y1 and going through all points yn , n ≥ 2. f ◦ α starts from f (y) and goes through all points f (yn ), n ≥ 1. However, as U is open, it must contain almost all of them, a contradiction.  The construction of the topology on lpc(X) in 2.2 can be done in the spirit of the finest topology on X that retains the same continuous maps from a class of spaces. 4 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA Proposition 2.5. Suppose X is a path-connected topological space and P is a class of Peano spaces. The family T of subsets U of X such that f −1 (U ) is open in Z ∈ P for any map f : Z → X in the original topology, is a topology and P(X) := (X, T ) is a Peano space. Proof. Since f −1 (U ∩ V ) = f −1 (U ) ∩ f −1 (V ), T is a topology on X. Suppose U ∈ T and C is a path component of U in the new-topology. Suppose f : Z → X is a map and f (z0 ) ∈ C. As f −1 (U ) is open, there is a connected neighborhood V of z0 in Z satisfying f (V ) ⊂ U . As f (V ) is path-connected, f (V ) ⊂ C and C ∈ T .  In case of first-countable spaces X we have a very simple characterization of the universal Peano map of X: Corollary 2.6. If X is a first-countable path-connected topological space, then a map f : Y → X is a universal Peano map if and only if Y is a Peano space, f is a bijection, and f has the path lifting property. Proof. Consider A(X) as in 2.5, where A consists of the unit interval. Notice the identity function P (X) → A(X) is continuous as P (X) is first-countable (use 2.4). Since the topology on A(X) is finer than that on P (X), P (X) = A(X). Since f induces a homeomorphism from A(Y ) to A(X) (due to the uniqueness of path lifting property of f ), the composition A(Y ) → A(X) → P (X) is a homeomorphism and f : Y → P (X) must be a homeomorphism (its inverse is P (X) → A(Y ) → Y ).  The construction in 2.5 can be used to create counter-examples to 2.6 in case X is not first-countable. Example 2.7. Let X be the cone over an uncountable discrete set B. Subsets of X that miss the vertex v are declared open if and only if they are open in the CW topology on X. A subset U of X that contains v is declared open if and only if U contains all but countably many edges of the cone and U \ {v} is open in the CW topology on X (that means X is a hedgehog if B is of cardinality ω1 see 2.8). Notice A(X) is X equipped with the CW topology, the identity function A(X) → X has the path lifting property but is not a homeomorphism. Proof. Notice every subset of X \ {v} that meets each edge in at most one point is discrete. Hence a path in X has to be contained in the union of finitely many edges. That means A(X) is X with the CW topology.  We generalize 2.7 as follows: DefinitionW2.8. A generalized Hawaiian Earring is the wedge (Zs , zs ) of pointed Peano spaces indexed by a directed set S and (Z, z0 ) = s∈S equipped with the following topology (all wedges in this paper are considered with that particular topology): (1) U ⊂ Z \ {z0 } is open if and only if U ∩ Zs is open for each s ∈ S, (2) U is an open neighborhood of z0 if and only if there is t ∈ S such that Zs ⊂ U for all s > t and U ∩ Zs is open for each s ∈ S. W (Zs , zs ) such that each A hedgehog is a generalized Hawaiian Earring (Z, z0 ) = s∈S (Zs , zs ) is homeomorphic to (I, 0). COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 5 Our definition of generalized Hawaiian Earrings is different from the definition of Cannon and Conner [5]. Also, the preferred terminology in [5] is that of a big Hawaiian Earring. Observe each generalized Hawaiian Earring is a Peano space. Lemma 2.9. Let S be a basis of neighborhoods of x0 in X ordered by inclusion (i.e., U ≤ V means V ⊂ U ). If, for each U ∈ S, αU : I → U is a path in U starting from x0 , then their wedge _ _ αU : (IU , 0U ) → (X, x0 ) U∈S U∈S is continuous, where (IU , 0U ) = (I, 0) for each U ∈ S. W αU at the base-point of the hedgehog Proof. Only the continuity of g = U∈S W (IU , 0U ) is not totally obvious. However, if V is a neighborhood of x0 in X, U∈S then g −1 (V ) contains all IU if U ⊂ V and g −1 (V ) ∩ IW is open in IW for all W ∈ S.  Proposition 2.10. Suppose f : Y → X is a function from an lpc-space Y . f is continuous if f ◦ g is continuous for every map g : Z → Y from a hedgehog Z to Y . Proof. Assume U is open in X and x0 = f (y0 ) ∈ U . Suppose for each pathconnected neighborhood V of y0 in Y there W is a path αV : (I, 0) → (V, y0 ) such that αV is a map g from a hedgehog to αV (1) ∈ / f −1 (U ). By 2.9 the wedge g = V ∈S Y (here S is the family of all path-connected neighborhoods of y0 in Y ). Hence h = f ◦ g is continuous and there is V ∈ S so that IV ⊂ h−1 (U ). That means f (αV (I)) ⊂ U , a contradiction.  e The philosophical meaning of this section is that many 2.2. Basic topology on X. results can be reduced to those dealing with Peano spaces via the universal Peano space construction. Let us illustrate this point of view by discussing a topology on e X. e of homoSuppose (X, x0 ) is a pointed topological space. Consider the space X topy classes of paths in X originating at x0 . It has an interesting topology (see the proof of Theorem 13 on p.82 in [23]) that has been put to use in [2] and [15]. Its basis consists of sets B([α], U ) (U is open in X, α joins x0 and α(1) ∈ U ) defined as follows: [β] ∈ B([α], U ) if and only if there is a path γ in U from α(1) to β(1) such that β is homotopic rel. endpoints to the concatenation α ∗ γ. e equipped with the above topology will be denoted by X b as in [2]. X b Both [2] and [15] consider quotient spaces X/H, where H is a subgroup of π1 (X, x0 ). We find it more convenient to follow [23, pp.82-3]: eH as the set of Definition 2.11. Suppose H is a subgroup of π1 (X, x0 ). Define X equivalence classes of paths in X under the relation α ∼H β defined via α(0) = β(0) = x0 , α(1) = β(1) and [α ∗ β −1 ] ∈ H (the equivalence class of α under the relation ∼H will be denoted by [α]H ). eH we define sets BH ([α]H , U ) (denoted by < α, U > To introduce a topology on X on p.82 in [23]), where U is open in X, α joins x0 and α(1) ∈ U , as follows: 6 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA [β]H ∈ BH ([α]H , U ) if and only if there is a path γ in U from α(1) to β(1) such that [β ∗ (α ∗ γ)−1 ] ∈ H (equivalently, β ∼H α ∗ γ). eH equipped with the topology (which we call the basic topology on X eH ) X whose basis consists of BH ([α]H , U ), where U is open in X, α joins x0 and α(1) ∈ U , bH in analogy to the notation X b in [2] that corresponds to H being is denoted by X trivial. Given a path α in X and a path β in X from x0 to α(0) one can define a bH originating at [β]H by the formula α̂(t) = [β ∗ αt ]H , standard lift α̂ of it to X where αt (s) = α(s · t) for s, t ∈ I (see [16, Proposition 6.6.3]). Let us extract the essence of the proof of [23, Theorem 13 on pp.82–83]: Lemma 2.12. Suppose X is a path-connected space and H is a subgroup of π1 (X, x0 ). bH → X if and only if U is locally An open set U ⊂ X is evenly covered by pH : X path-connected and the image of hα : π1 (U, x1 ) → π1 (X, x0 ) is contained in H for any path α in X from x0 to any x1 ∈ U . Proof. Recall that U is evenly covered by pH (see [23, p.62]) if p−1 H (U ) bH each of which is mapped is the disjoint union of open subsets {Us }s∈S of X homeomorphically onto U by pH . Also, recall hα : π1 (U, x1 ) → π1 (X, x0 ) is given by hα ([γ]) = [α ∗ γ ∗ α−1 ]. Suppose U is evenly covered, γ is a loop in (U, x1 ), and α is a path from x0 to x1 . If [α]H 6= [α ∗ γ]H , then they belong to two different sets Uu and Uv , u, v ∈ S. −1 However, there is a path from [α]H to [α ∗ γ]H in pH (U ) given by the standard lift of γ, a contradiction. Thus [α]H = [α ∗ γ]H and [α ∗ γ ∗ α−1 ] ∈ H. To show that U is locally path-connected, take a point x1 ∈ U , pick a path α from x0 to x1 and select the unique s ∈ S so that [α]H ∈ Us . There is an open subset V of U satisfying BH ([α]H , V ) ⊂ Us . As pH |Us maps Us homeomorphically onto U , pH (BH ([α]H , V )) is an open neighborhood of x1 in U and it is path-connected. Suppose U is locally path-connected and the image of hα : π1 (U, x1 ) → π1 (X, x0 ) is contained in H for any path α in X from x0 to any x1 ∈ U . Pick a path component V of U and notice sets BH ([β]H , V ), β ranging over paths from x0 to points of V , are either identical or disjoint. Observe pH |BH ([β]H , V ) maps BH ([β]H , V ) homeomorphically onto V . Thus each V is evenly covered and that is sufficient to conclude U is evenly covered.  As in [23, p.81], given an open cover U of X, π(U, x0 ) is the subgroup of π1 (X, x0 ) generated by elements of the form [α ∗ γ ∗ α−1 ], where γ is a loop in some U ∈ U and α is a path from x0 to γ(0). Here is our improvement of [23, Theorem 13 on p.82] and [15, Theorem 6.1]: Theorem 2.13. If X is a path-connected space and H is a subgroup of π1 (X, x0 ), bH → X is a classical covering map if and only then the endpoint projection pH : X if X is a Peano space and there is an open covering U of X so that π(U, x0 ) ⊂ H. Proof. Apply 2.12.  \ bH if X is path-connected. Proposition 2.14. P (X)H is naturally homeomorphic to X Proof. Since continuity of f : (Z, z0 ) → (P (X), x0 ), for any Peano space Z, is equivalent to the continuity of f : (Z, z0 ) → (X, x0 ), paths in (P (X), x0 ) correspond to paths in (X, x0 ). Also, π1 (P (X), x0 ) → π1 (X, x0 ) is an isomorphism so H is a subgroup of both π1 (P (X), x0 ) and π1 (X, x0 ), and the equivalence classes of COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 7 ^ e Notice that basis open sets relations ∼H are identical in both spaces P (X) and X. \ b are identical in P (X)H and XH .  Remark 2.15. In view of 2.14 some results in [15] dealing with maps f : Y → X, where Y is Peano, can be derived formally from corresponding results for f : Y → P (X). A good example is Lemma 2.8 in [15]: p : X̃ → X has the unique path lifting property if and only if X̃ is simply connected. It follows formally from Corollary 4.7 in [2]: The universal endpoint projection p : Ẑ → Z for a connected and locally pathconnected space Z has the unique path lifting property if and only if Ẑ is simply connected. bH is equipped When working in the pointed topological category the space X with the base-point x b0 equal to the equivalence class of the constant path at x0 . bH in the case of H = π1 (X, x0 ). Let us illustrate X Proposition 2.16. If H = π1 (X, x0 ), then bH , x a. The endpoint projection pH : (X b0 ) → (X, x0 ) is an injection and pH (B([α]H , U )) is the path component of α(1) in U , bH is a Peano space, b. X c. Given a map g : (Z, z0 ) → (X, x0 ) from a pointed Peano space to (X, x0 ), bH , x there is a unique lift h : (Z, z0 ) → (X b0 ) of g (pH ◦ h = g). Proof. a). Clearly, pH (BH ([α]H , U )) equals path component of α(1) in U . If [β1 ]H and [β2 ]H map to the same point x1 , then β1 (1) = β2 (1) and γ = β1 ∗ β2−1 is a loop. Hence [γ] ∈ H and [β2 ]H = [γ ∗ β2 ]H = [β1 ]H proving pH is an injection. b) is well-established in both [2] and [15]. Notice it follows from a). c). For each z ∈ Z pick a path αz from z0 to z in Z. Define h(z) as [αz ]H and notice h is continuous as h−1 (BH ([αz ]H , U )) equals the path component of g −1 (U ) containing z (use Part a)). As pH is injective, there is at most one lift of g.  In view of 2.16 we have a convenient definition of a universal Peano space in the pointed category: Definition 2.17. By the universal Peano space P (X, x0 ) of (X, x0 ) we mean bH , x the pointed space (X b0 ), H = π1 (X, x0 ), and the universal Peano map of (X, x0 ) is the endpoint projection P (X, x0 ) → (X, x0 ). Equivalently, P (X, x0 ) is (P (C), x0 ), where C is the path component of x0 in X. bH → (X, x0 ) always has the Due to standard lifts the endpoint projection pH : X path lifting property. Thus the issue of interest is the uniqueness of path lifting property of pH . Here is a necessary and sufficient condition for pH to have the unique path lifting property (compare it to [2, Theorem 4.5] for Peano spaces): Proposition 2.18. If X is a path-connected space and x0 ∈ X, then the following conditions are equivalent: bH , x a. pH : (X b0 ) → (X, x0 ) has the unique path lifting property, bH , x b. The image of π1 (pH ) : π1 (X b0 ) → π1 (X, x0 ) is contained in H. 8 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA bH it must equal the standard lift of Proof. a) =⇒ b). Given a loop α in X bH one must have [β] ∈ H. β = pH (α). For the standard lift of β to be a loop in X b) =⇒ a) Given a lift ᾱ of a path α in (X, x0 ) it suffices to show ᾱ(1) = [α]H as that implies ᾱ is the standard lift of α (use α|[0, t] instead of α). Pick a path β bH , satisfying α̂(1) = [β]H and let β̂ be its standard lift. As ᾱ ∗ (β̂)−1 is a loop in X −1 its image γ = pH (ᾱ ∗ (β̂) ) generates an element [γ] of H. Hence α ∼ γ ∗ β and ᾱ(1) = [β]H = [α]H .  e 3. A new topology on X bH → We do not know how to characterize subgroups H of π1 (X, x0 ) for which pH : X X has the unique path lifting property. Therefore we will create a new topology eH for which analogous question has a satisfactory answer in the case H being on X a normal subgroup. Given an open cover U of X, a subgroup H of π1 (X, x0 ), a path α in X originating eH as follows: at x0 , and V ∈ U containing x1 = α(1) define BH ([α]H , U, V ) ⊂ X [β]H ∈ BH ([α]H , U, V ) if and only if there is a path γ0 in V originating at x1 = α(1) and a loop λ at x1 such that [λ] ∈ π(U, x1 ) and β ∼H α ∗ λ ∗ γ0 . Observe [β]H ∈ BH ([α]H , U, V ) implies BH ([α]H , U, V ) = BH ([β]H , U, V ) and BH ([α]H , U ∩ V, V1 ∩ V2 ) ⊂ BH ([α]H , U, V1 ) ∩ BH ([α]H , V, V2 ), so the family of sets eH . When we consider X eH {BH ([α]H , U, V )} forms a basis of a new topology on X as a topological space, then we use precisely that topology. In the particular case eH to X. e Observe that, of H = {1}, the trivial subgroup of π1 (X, x0 ), we simplify X e → X, any subgroup G as π1 (X, x0 ) is the fiber of the endpoint projection p : X e of π1 (X, x0 ) can be considered as a subspace of X and we may consider it as a topological space that way. bH → X eH is continuous. Indeed, BH ([α]H , V ) ⊂ Notice the identity function X BH ([α]H , U, V ) for any V ∈ U containing α(1). eH is equipped When dealing with the pointed topological category the space X with the base-point x e0 equal to the equivalence class of the constant path at x0 . Let us prove a basic functorial property of our construction. Proposition 3.1. Suppose f : (X, x0 ) → (Y, y0 ) is a map of pointed topological spaces. If H and G are subgroups of π1 (X, x0 ) and π1 (Y, y0 ), respectively, such eH , x that π1 (f )(H) ⊂ G, then f induces a natural continuous function f˜: (X e0 ) → e (YG , ye0 ). Proof. Put f˜([α]H ) = [f ◦ α]G and notice f˜(BH ([α]H , f −1 (U), f −1 (V ))) ⊂ BG (f˜([α]H ), U, V ) for any open covering U of Y and any neighborhood V of α(1). In connection to 2.13 let us prove the following:  Proposition 3.2. If X is a path-connected space and H is a subgroup of π1 (X, x0 ), then the following conditions are equivalent: eH → X has an isolated point, a) A fiber of the endpoint projection pH : X e b) The endpoint projection pH : XH → X has discrete fibers, c) There is an open covering U of X so that π(U, x0 ) ⊂ H, eH is a Peano space and pH : X eH → P (X) is a classical covering map. d) X COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 9 Proof. a) =⇒ c). Suppose [α]H ∈ p−1 H (x1 ) is isolated. There is an open covering U of X and V ∈ U containing x1 such that BH ([α]H , U, V ) ∩ p−1 H (x1 ) = {[α]H }. Given γ in π(U, x0 ), the homotopy class [α−1 ∗ γ ∗ α]H belongs to π(U, x1 ), so −1 [α ∗ α−1 ∗ γ ∗ α]H = [γ ∗ α]H belongs to BH ([α]H , U, V ) ∩ pH (x1 ). Hence [γ ∗ α]H = [α]H and [γ] ∈ H. c) =⇒ d). Suppose there is an open covering U of X so that π(U, x0 ) ⊂ H and W is a path component of U ∈ U. Notice BH ([α]H , U, U ) is mapped by pH bijectively onto W and that is sufficient for d). d) =⇒ b) and b) =⇒ a) are obvious.  Applying 3.2 to H being trivial one gets the following (see [12] for analogous result in case of a different topology on the fundamental group): Corollary 3.3. If X is a path-connected space, then π1 (X, x0 ) is discrete if and only if X is semilocally simply connected. Proposition 3.4. If π(V, x0 ) ⊂ H for some open cover V of X, then the identity bH → X eH is a homeomorphism. function X Proof. Let us show BH ([α]H , U, W ) = BH ([α]H , W ) if U is an open cover of X refining V and W is an element of U containing α(1). Clearly, BH ([α]H , W ) ⊂ BH ([α]H , U, W ), so assume [β]H ∈ BH ([α]H , U, W ). There are h ∈ H, [λ] ∈ π(U, α(1)), and a path γ in W such that [β] = [h ∗ α ∗ λ ∗ γ]. Choose h1 ∈ H so that [h1 ∗ α] = [α ∗ λ] (h1 = [α ∗ λ ∗ α−1 ] ∈ π(U, x0 ) ⊂ H). Now [β] = [h ∗ α ∗ λ ∗ γ] = [h ∗ h1 ∗ α ∗ γ] and [β]H ∈ BH ([α]H , W ). bH → X eH is open: given an open cover Now we can show the identity function X W of X and given a path α from x0 to x1 pick an element W of U = W ∩ V containing x1 and notice BH ([α]H , U, W ) ⊂ BH ([α]H , W ).  eG → Lemma 3.5. If G ⊂ H are subgroups of π1 (X, x0 ), then the projection p : X e XH is open. Proof. It suffices to show p(BG ([α]G , U, V )) = BH ([α]H , U, V ). Clearly, p(BG ([α]G , U, V )) ⊂ BH ([α]H , U, V ), so suppose [β]H ∈ BH ([α]H , U, V ) and [β] = [h ∗ α ∗ λ ∗ γ], where [λ] ∈ π(U, α(1)) and γ is a path in V originating at β(1). Observe [β]H = [α ∗ λ ∗ γ]H = p([α ∗ λ ∗ γ]G ).  e We arrived at the fundamental result for the new topology on XH : Theorem 3.6. Suppose G ⊂ H are subgroups of π1 (X, x0 ). If G is normal in eG → π1 (X, x0 ), then H/G, identified with the fiber p−1 ([x̃0 ]H ) of the projection p : X e e XH , is a topological group and acts continuously on XG so that eG → X eG × X eG defined by ([α]G , [β]G ) 7→ a) The natural map (H/G) × X ([α ∗ β]G , [β]G ) is an embedding, eG to the orbit space corresponds to the projection b) The quotient map from X e e p : XG → XH . eG → X eH is the set of classes [α]G such Proof. The fiber F of the projection p : X eG → X eG as follows: given that [α] ∈ H, so it corresponds to H/G. Define µ : F × X e [α]G ∈ F and given [β]G ∈ XG put µ([α]G , [β]G ) = [α ∗ β]G . To see µ is well defined assume [γ1 ], [γ2 ] ∈ G. Now [γ1 ∗ α ∗ γ2 ∗ β]G [(α ∗ γ2 ∗ α−1 ) ∗ (α ∗ β)]G = [α ∗ β]G as [α ∗ γ2 ∗ α−1 ] ∈ G due to normality of G in H. Suppose U is an open cover of X, V, V1 ∈ U, and 10 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA eG , (1) [α]G ∈ F , [β]G ∈ X (2) [α1 ]G ∈ BG ([α]G , U, V1 ) ∩ F , and [β1 ]G ∈ BG ([β]G , U, V ). Thus [α1 ] = [g1 ∗ α ∗ λ1 ] for some [λ1 ] ∈ π(U, x0 ) and [g1 ] ∈ G. Similarly, [β1 ] = [g2 ∗ β ∗ λ2 ∗ γ], where [g2 ] ∈ G, [λ2 ] ∈ π(U, β(1)), and γ is a path in V . Now, −1 −1 [α−1 ∗ g1−1 ∗ g2 ∗ β ∗ λ2 ∗ γ]G = 1 ∗ β1 ]G = [λ1 ∗ α −1 [(λ1−1 ∗ α−1 ∗ g1−1 ∗ g2 ∗ α ∗ λ1 ) ∗ λ−1 ∗ β ∗ λ2 ∗ γ]G = 1 ∗α −1 −1 [λ−1 ∗β∗λ2 ∗γ]G = [(α−1 ∗β)∗(β −1 ∗α∗λ−1 ∗β)∗λ2 ∗γ]G ∈ BG ([α−1 ∗β]G , U, V ) 1 ∗α 1 ∗α −1 as [λ−1 ∗ g1−1 ∗ g2 ∗ α ∗ λ1 ] ∈ G and [β −1 ∗ α ∗ λ1−1 ∗ α−1 ∗ β] ∈ π(U, (α−1 ∗ β)(1)). 1 ∗α The above calculations amount to ρ((F ∩ BG (x, U, V1 )) × BG (y, U, V )) ⊂ BG (ρ(x, y), U, V ), where ρ(x, y) := µ(x−1 , y), which implies the following (1) F is a topological group, (2) µ is continuous, eG onto its image is open. (3) (x, y) → (µ(x−1 , y), y) from F × X As the map in (3) is injective, it is an embedding. Hence (x, y) → (µ(x, y), y) is an embedding. To see b) use 3.5 or check it directly.  4. Path lifting Definition 4.1. A pointed map f : (X, x0 ) → (Y, y0 ) has the path lifting property if any path α : (I, 0) → (Y, y0 ) has a lift β : (I, 0) → (X, x0 ). A surjective map f : X → Y has the path lifting property if for any path α : I → Y and any y0 ∈ f −1 (α(0)) there is a lift β : I → X of α such that β(0) = y0 . Definition 4.2. A pointed map f : (X, x0 ) → (Y, y0 ) has the uniqueness of path lifts property if any two paths α, β : (I, 0) → (X, x0 ) are equal if f ◦ α = f ◦ β. A pointed map f : (X, x0 ) → (Y, y0 ) has the unique path lifting property if it has both the path lifting property and the uniqueness of path lifts property. A map f : X → Y has the uniqueness of path lifts property if any two paths α, β : I → X are equal if f ◦ α = f ◦ β and α(0) = β(0). A surjective map f : X → Y has the unique path lifting property if it has both the path lifting property and the uniqueness of path lifts property. Corollary 4.3. Supppose G ⊂ H are subgroups of π1 (X, x0 ). If G is normal in π1 (X, x0 ), then the following conditions are equivalent: eG → X eH has the uniqueness of path lifts property, a) The natural map X b) π0 (H/G) = H/G, i.e. H/G has trivial path components. Proof. a) =⇒ b). If H/G has a non-trivial path component, then there is a eH . non-trivial lift of the constant path at the base-point of X eH and α(0) = b) =⇒ a). Suppose α and β are two lifts of the same path γ in X β(0). By 3.6 there is a path λ in H/G with the property λ(t) · α(t) = β(t) for each t ∈ I. As λ(0) = 1 ∈ H/G and H/G has trivial path components, λ(t) = 1 ∈ H/G for all t ∈ I and α = β.  COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 11 Proposition 4.4. Supppose G ⊂ H are subgroups of π1 (X, x0 ). If G is normal in π1 (X, x0 ), then the following conditions are equivalent: a) H/G is a T0 -space, b) H/G is Hausdorff, eG → X eH are T0 , c) Fibers of the projection p : X e e d) Fibers of the projection p : XG → XH are Hausdorff, e) For each h ∈ H − G there is a cover U such that (G · h) ∩ π(U, x0 ) = ∅, f) G is closed in H. Proof. In view of 3.6, a)≡c) and b)≡d). a) =⇒ e). Assume H/G is T0 and h ∈ H − G. Since [β]G ∈ BG ([α]G , U, V ) is equivalent to [α]G ∈ BG ([β]G , U, V ), there is an open cover U and V ∈ U containing x0 such that G · h ∈ / BG (G · 1, U, V ). That means precisely there is no λ ∈ π(U, x0 ) such that G · h = G · λ, hence (G · h) ∩ π(U, x0 ) = ∅. b)≡d) and a)≡c) follow from 3.6. e) =⇒ d). Suppose α, β are two paths in (X, x0 ) so that [α]H = [β]H but [α]G 6= [β]G . choose h ∈ H − G satisfying [h · α] = [β]. Pick an open cover U of X satisfying G · h ∩ π(U, x0 ) = ∅ and let V ∈ U contain α(1). Suppose [γ]G ∈ BG ([α]G , U, V ) ∩ BG ([β]G , U, V ) and [γ]H = [α]H . Let h0 ∈ H satisfy [h0 · α] = [γ]. Choose λ1 , λ2 ∈ π(U, α(1)) such that G · [h0 · α] = G · α · λ1 and −1 G · [h0 · α] = G · [h · α] · λ2 . As G is normal in H, G · h = h · G = G · (α · λ1 · λ−1 ), 2 α −1 −1 a contradiction as α · λ1 · λ2 · α ∈ π(U, x0 ). b) =⇒ a) is obvious. e)≡f). G being closed in H means existence, for each h ∈ H − G, of an open cover U such that G ∩ B(h, U, V ) = ∅ for some V ∈ U containing x0 . That, in turn, is equivalent to non-existence of λ ∈ π(U, x0 ) satisfying h · λ ∈ G, i.e. (G · h−1 ) ∩ π(U, x0 ) = ∅.  Corollary 4.5. Suppose G ⊂ H are subgroups of π1 (X, x0 ). If G is a normal subgroup of π1 (X, x0 ), then the following conditions are equivalent: a. H/G has trivial components, b. H/G has trivial path components, c. G is closed in H. Proof. b) =⇒ c). Suppose H/G has trivial path components. In view of 4.4 it suffices to show H/G is T0 to deduce G is closed in H. If there are two points u and v of H/G such that any open subset of H/G either contains both of them or contains none of them, then any function I → {u, v} ⊂ H/G is continuous. Hence u = v as H/G does not contain non-trivial paths. c) =⇒ a). Claim. If h1 , h2 ∈ H and G · f ∈ BH (G · h1 , U, V ) ∩ BH (G · h2 , U, V ) ∩ (H/G) for some open cover U of X and some V ∈ U containing x0 , then G·h−1 1 ·h2 ⊂ Gπ(U, x0 ). Proof of Claim: G·f = G·h1 ·λ1 and G·f = G·h2 ·λ2 for some λ1 , λ2 ∈ π(U, x0 ). −1 −1 Now h1 · G = h2 · G · (λ2 · λ−1  1 ) and (h1 · h2 ) · G ⊂ G · (λ1 · λ2 ) ⊂ Gπ(U, x0 ). Suppose G is closed in H and h ∈ H − G. By 4.4 there is a cover U such that (G·h)∩π(U, x0 ) = ∅. If there is a connected subset C of H/G containing G·h1 h and G · h1 for some h1 ∈ H, we consider the open cover {C ∩ BG (z, U, V )}z∈C of C and define the equivalence relation on C determined by that cover (z ∼ z ′ if there is a finite chain z = z1 , . . . , zk = z ′ in C such that BG (zi , U, V )∩BG (zi+1 , U, V )∩C 6= ∅ for all i < k). Equivalence classes of that relation are open, hence closed and 12 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA must equal C. Thus there is a finite chain h1 , . . . , hk = h1 · h in H such that BG ([hi ]G , U, V ) ∩ BG ([hi+1 ]G , U, V ) ∩ (H/G) 6= ∅ for all i < k. By Claim there are elements gi ∈ G (i < k) so that gi · h−1 i · hi+1 ∈ π(U, x0 ). By normality of G in H k−1 Q −1 hi · hi+1 = g · h ∈ π(U, x0 ), a contradiction.  there is g ∈ G satisfying g · i=1 Theorem 4.6. If G is a normal subgroup of π1 (X, x0 ), then the following conditions are equivalent: eG , x a. The endpoint projection pG : (X e0 ) → (X, x0 ) has the unique path lifting property, b. G is closed in π1 (X, x0 ), eG , x c. π1 (pG ) : π1 (X e0 ) → π1 (X, x0 ) is a monomorphism and its image equals G. eH is the Peanification of (X, x0 ) by Proof. Put H = π1 (X, x0 ) and observe X 2.16. a)≡b). By 4.3 the group H/G has trivial path components. Use 4.5. eG , x a) =⇒ c). Given a loop in (X e0 ) we may assume it is a canonical lift of a loop α in (X, x0 ). For that lift to be a loop we must have [α] ∈ G. Thus the eG , x image of π1 (pG ) : π1 (X e0 ) → π1 (X, x0 ) equals G (canonical lifts of elements of G show that the image contains G). If α is null-homotopic in (X, x0 ), then its eG , x canonical lift is null-homotopic as well. Thus π1 (pG ) : π1 (X e0 ) → π1 (X, x0 ) is a monomorphism. c) =⇒ a). If H/G has a non-trivial path component (we use 4.3), then there is a path from the base-point to a different point [α]G of H/G. Concatenating the eG , x canonical lift of α with the reverse of that path gives a loop in (X e0 ) whose image in π1 (X, x0 ) is [α] ∈ / G, a contradiction.  Proposition 4.7. Suppose (X, x0 ) is a pointed topological space and H is a subgroup of π1 (X, x0 ). The closure of H in π1 (X, x0 ) consists of all elements g ∈ π1 (X, x0 ) such that for each open cover U of X there is h ∈ H and λ ∈ π(U, x0 ) satisfying g = h · λ. If H is a normal subgroup of π1 (X, x0 ), then so is its closure. Proof. Suppose g ∈ π1 (X, x0 ) and for each open cover U of X there is h ∈ H and λ ∈ π(U, x0 ) satisfying g = h · λ. Notice B(g, U) contains h, so g belongs to the closure of H. If H is normal, then k · g · k −1 = (k · h · k −1 ) · (k · λ · k −1 ) also belongs to the closure of H.  Corollary 4.8. The closure of the trivial subgroup of π1 (X, x0 ) in π1 (X, x0 ) equals T π(U, x0 ), where COV stands for the family of all open covers of X. U ∈COV Example 4.9. The Harmonic ArchipelagoTHA of Bogley and Sieradski [2] is a π(U, x0 ). Hence π1 (X, x0 ) is the Peano space such that π1 (X, x0 ) equals U ∈COV only closed subgroup of π1 (X, x0 ). HA is built by stretching disks B(2−n , 2−n−2 ) to form cones over its boundary with the vertices at height 1 in the 3-space. Corollary 4.10. Suppose (X, x0 ) is a pointed topological space. The following subgroups of π1 (X, x0 ) are closed: a) Subgroups H containing π(U, x0 ) for some open cover U of X, T π(U, x0 ) for any family S of open covers of X, b) U ∈S COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 13 c) The kernel of π1 (f ) : π1 (X, x0 ) → π1 (Y, y0 ) for any map f : (X, x0 ) → (Y, y0 ) to a pointed semilocally simply connected space. d) The kernel of the natural homomorphism π1 (X, x0 ) → π̌1 (X, x0 ) from the fundamental group to the Čech fundamental group. Proof. a) Any subgroup containing π(U, x0 ) is open. Any open subgroup of a topological group is closed. b) easily follows from a). c) follows from 3.3 and 3.1 as π1 (f ) : π1 (X, x0 ) → π1 (Y, y0 ) is continuous and π1 (Y, y0 ) is discrete. d) follows from c). Indeed π̌1 (X, x0 ) is defined (see [9] or [19]) as the inverse limit of an inverse system {π1 (Ks , ks )}s∈S , where each Ks is a simplicial complex and there are maps fs : (X, x0 ) → (Ks , ks ) so that for t > s the map fs is homotopic to the composition of ft and the bonding map (Kt , kt ) → (Ks , ks ). That means the kernel of the natural homomorphism π1 (X, x0 ) → π̌1 (X, x0 ) is the intersection of kernels of all π1 (fs ), s ∈ S.  The concept of a space X being homotopically Hausdorff was introduced by Conner and Lamoreaux [8, Definition 1.1] to mean that for any point x0 in X and for any non-homotopically trivial loop γ at x0 there is a neighborhood U of x0 in X with the property that no loop in U is homotopic to γ rel.x0 in X. Subsequently, Fischer and Zastrow [15]) defined a space X to be homotopically Hausdorff relative to a subgroup H of π1 (X, x0 ) if for any g ∈ / H and for any path α originating at x0 there is an open neighborhood U of α(1) in X such that no element of H · g can be expressed as [α ∗ γ ∗ α−1 ] for some loop γ in (U, α(1)). We generalize this definition as follows: Definition 4.11. Suppose G ⊂ H are subgroups of π1 (X, x0 ). X is (H, G)homotopically Hausdorff if for any h ∈ H \ G and any path α originating at x0 there is an open neighborhood U of α(1) in X such that none of the elements of G · h can be expressed as [α ∗ γ ∗ α−1 ] for any loop γ in (U, α(1)). Notice X being homotopically Hausdorff relative to H corresponds to X being (π1 (X, x0 ), H)-homotopically Hausdorff. Let us characterize the concept of being (H, G)-homotopically Hausdorff in terms of the basic topology on the fundamental group. Proposition 4.12. If G ⊂ H are subgroups of π1 (X, x0 ), then X is (H, G)homotopically Hausdorff if and only if for every path α in X that terminates at x0 the group hα (G) is closed in hα (H) in the basic topology. Proof. hα (G) being closed in hα (H) means existence, for each h ∈ H \ G, of a neighborhood U of x1 = α(0) such that B([α∗h∗α−1], U )∩([α]·G·[α−1 ]) = ∅. Thus, for every loop γ in U at x1 , there is no g ∈ G satisfying [α∗h∗α−1 ∗γ −1 ] = [α∗g∗α−1 ]. The last equality is equivalent to [g ∗ h] = [α−1 ∗ γ ∗ α] which completes the proof.  Example 4.13. Proposition 4.12 allows for an easy construction of subgroups H of π1 (X, x0 ) such that X is L not homotopically Hausdorff relative to H. Namely, Q X = S 1 × S 1 × . . . and H = Z ⊂ Z = π1 (X). Let us show G being closed in H (in the new topology) is a stronger condition than X being (H, G)-homotopically Hausdorff. 14 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA Lemma 4.14. Suppose G ⊂ H are subgroups of π1 (X, x0 ). If G is closed in H, then X is (H, G)-homotopically Hausdorff. Proof. Given h ∈ H \ G pick an open cover U and W ∈ U containing x0 so that B(h, U, W ) does not intersect G. Given a path α in X from x0 to x1 choose V ∈ U containing x1 . Suppose there is a loop γ in (V, x1 ) so that [α∗γ∗α−1 ] = g·h for some g ∈ G. Now [α∗γ −1 ∗α−1 ] ∈ π(U, x0 ) and g −1 = h∗[α∗γ −1 ∗α−1 ] ∈ G∩B(h, U, W ), a contradiction.  Remark 4.15. The proof of 4.14 suggests that the trivial subgroup of π1 (X, x0 ) being closed is philosophically related to the concept of X being strongly homotopically Hausdorff (see [22]). Recall a metric space X is strongly homotopically Hausdorff if for any non-null-homotopic loop α in X there is an ǫ > 0 such that α is not freely homotopic to a loop of diameter less than ǫ. Lemma 4.16. Given subgroups G ⊂ H of π1 (X, x0 ) the following conditions are equivalent: bG → X bH are T0 , a) The fibers of the natural projection p : X bG → X bH are Hausdorff, b) The fibers of the natural projection p : X c) X is (H, G)-homotopically Hausdorff. Proof. a) =⇒ c). Suppose h ∈ H \ G and α is a path in X from x0 to x1 . As [h∗α]G 6= [α]G belong to the same fiber of p, there is a neighborhood U of x1 so that [h ∗ α]G ∈ / BG ([α]G , U ) or [α]G ∈ / BG ([h ∗ α]G , U ). Notice [h ∗ α]G ∈ / BG ([α]G , U ) is equivalent to [α]G ∈ / BG ([h ∗ α]G , U ). Suppose there is a loop γ in (U, x1 ) so that g·h = [α∗γ∗α−1 ] for some g ∈ G. Now [h∗α]G = [g·h∗α]G = [α∗γ]G ∈ BG ([α]G , U ), a contradiction. c) =⇒ b). Any two different elements of the same fiber of p can be represented as [h ∗ α]G 6= [α]G for some path α in X from x0 to x1 and some h ∈ H \ G. Choose a neighborhood U of x1 with the property that none of the elements of G · h can be expressed as [α ∗ γ ∗ α−1 ] for any loop γ in (U, x1 ). Suppose [β]G ∈ (H/G) ∩ BG ([α]G , U ) ∩ BG ([h ∗ α]G , U ). That means existence of loops γ1 , γ2 in (U, x1 ) so that [β]G = [h ∗ α ∗ γ1]G = [α ∗ γ2 ]G . Hence [h]G = [α ∗ (γ2 ∗ γ1−1 ) ∗ α−1 ]G , a contradiction.  Lemma 4.17. Supppose G ⊂ H are subgroups of π1 (X, x0 ), G is normal in bG , x π1 (X, x0 ), and X is (H, G)-homotopically Hausdorff. If α, β : (I, 0) → (X b0 ) b are two continuous lifts of the same path γ : (I, 0) → (XH , x b0 ), then for every h ∈ H the set S = {t ∈ I|α(t) = h · β(t)} is closed. Proof. Choose paths ut , vt in (X, x0 ) so that α(t) = [ut ]G and β(t) = [vt ]G for all t ∈ I. Assume [ut ]G 6= [h · vt ]G for some t ∈ I. Pick a neighborhood U of −1 x1 = ut (1) so that [vt ∗u−1 t ]·h·G 6= [vt ∗γ∗vt ]·G for any loop γ in (U, x1 ). There is a neighborhood V of t in I so that [us ]G ∈ BG ([ut ]G , U ) and [vs ]G ∈ BG ([vt ]G , U ) for all s ∈ V . That means [us ] = [g1 ∗ ut ∗ γ1 ] and [vs ] = [g2 ∗ vt ∗ γ2 ] for some g1 , g2 ∈ G and some paths γ1 , γ2 in U joining x1 and u1 (1) = vs (1). Put γ = γ1 ∗ γ2−1 and notice [us ∗ vs−1 ] = [g1 ∗ ut ∗ vt−1 ∗ (vt ∗ γ ∗ vt−1 ) ∗ g2−1 ]. As G is normal in π1 (X, x0 ), there is g3 ∈ G satisfying [g1 ∗ut ∗vt−1 ∗(vt ∗γ ∗vt−1 )∗g2−1 ] = [g3 ∗ut ∗vt−1 ∗(vt ∗γ ∗vt−1 )] and that element cannot belong to G · h by the choice of U .  COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 15 Corollary 4.18. Supppose G ⊂ H are subgroups of π1 (X, x0 ). If H/G is countable, G is normal in π1 (X, x0 ), and X is (H, G)-homotopically Hausdorff, then the bG → X bH has the uniqueness of path lifts property. natural map X Proof. Pick representatives hi ∈ H, i ≥ 1, of all right cosets of H/G so that bG of the same path in X bH , then h1 = 1. If α and β are two continuous lifts in X each set Si = {t ∈ I|α(t) = hi · β(t)} is closed, they are disjoint, and their union is the whole interval I. Hence only one of them is non-empty and it must be S1 . Thus α = β.  5. Peano maps This section is about one of the main ingredients of our theory of covering maps for lpc-spaces. It amounts to the following generalization of Peano spaces: Definition 5.1. A map f : X → Y is a Peano map if the family of path components of f −1 (U ), U open in Y , forms a basis of neighborhoods of X. Notice X is an lpc-space if f : X → Y is a Peano map. One may reword the above definition as follows: X is an lpc-space and lifts of short paths in Y are short in X. Indeed, given a neighborhood U of x0 ∈ X there is a neighborhood V of f (x0 ) in Y such that any path α in (f −1 (V ), x0 ) (i.e. f ◦ α is contained in V , hence short) must be contained in U . Proposition 5.2. Any product of Peano maps is a Peano map. Proof. Suppose fs : Xs → YS , s ∈ S, are Peano maps. Observe X = Q Xs is s∈S an lpc-space. Given a neighborhood U of x = {xs }s∈SQ ∈ X, we find a finite subset Us ⊂ U and Us = Xs for T of S and neighborhoods Us of xs in Xs such that s∈S s∈ / T . Choose neighborhoods Vs of fs (xs ) in Ys , s ∈ T , so that the path-component of xs in fs−1 (Vs ) is contained in U / T and observe the path s for s ∈ Qs . Put Vs = XQ Vs , is contained in U .  fs and V = component of x in f −1 (V ), f = s∈S s∈S Here is our basic class of Peano maps: Proposition 5.3. If H is a subgroup of π1 (X, x0 ), then the endpoint projection bH → X is a Peano map. pH : X Proof. It suffices to show that for any U open in X the path component of any [α]H in p−1 H (U ) is precisely BH ([α]H , U ). It’s straightforward that BH ([α]H , U ) is path-connected so suppose β is a path in p−1 H (U ) starting at [α]H . We wish to show that β([0, 1]) ⊂ BH ([α]H , U ). Let T = {t : β(t) ∈ BH ([α]H , U )}. Now T is nonempty since β(0) = [α]H and open as the inverse image of an open set. It suffices to prove [0, t) ⊂ T implies [0, t] ⊂ T . Set β(t) = [b]H . Now pH β([0, 1]) ⊂ U so in particular pH ([b]H ) ∈ U. Consider BH ([b]H , U ). There is an ε > 0 such that β(t − ε, t] ⊂ BH ([b]H , U ). Pick s ∈ (t − ε, t). Then β(s) = [c1 ]H and [b]H = [b1 ]H such that c1 ≃ b1 ∗ γ1 for some γ1 with γ1 [0, 1] ⊂ U. But β(s) ∈ BH ([α]H , U ) so β(s) = [c2 ]H and [α]H = [a1 ]H such that c2 ≃ a1 ∗γ2 for some γ2 with γ2 ([0, 1]) ⊂ U. Then b ≃H b1 ≃ c1 ∗ γ1−1 ≃H c2 ∗ γ1−1 ≃ a1 ∗ γ2 ∗ γ1−1 ≃H a ∗ γ2 ∗ γ1−1 and  (γ2 ∗ γ1−1 )([0, 1]) ⊂ U so [b]H ∈ BH ([α]H , U ) and t ∈ T. Therefore T = [0, 1]. In analogy to path lifting and unique path lifting properties (see 4.1 and 4.2) one can introduce the corresponding concepts for hedgehogs: 16 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA Definition 5.4. A W surjective map f : X → Y has the hedgehog lifting property Is → Y from a hedgehog and any y0 ∈ f −1 (α(0)) there is a if for any map α : s∈S W Is → X of α such that β(0) = y0 . continuous lift β : s∈S Definition 5.5. f : X → Y has the unique hedgehog lifting property if it has both the hedgehog lifting property and the uniqueness of path lifts property. Theorem 5.6. If f : X → Y has the unique hedgehog lifting property, then f : lpc(X) → Y is a Peano map. Proof. Assume U is open in X and x0 ∈ U . Suppose for each neighborhood V −1 of f (x0 ) in X there / U . By W is a path αV : (I, 0) → (f (V ), x0 ) such that αV (1) ∈ f ◦ αV is a map g from a hedgehog to Y (here S is the family of 2.9 the wedge V ∈S W αV . However all neighborhoods of f (x0 ) in Y ). Its lift must be the wedge h = V ∈S h−1 (U ) is not open in lpc(X), a contradiction.  Definition 5.7. Given a map f : X → Y of topological spaces its Peano map P (f ) : Pf (X) → Y is f on X equipped with the topology generated by path components of sets f −1 (U ), U open in Y . Notice that in the case of f = idX the range PidX (X) of P (idX ), where idX : X → X is the identity map, is identical to lpc(X) as defined in 2.2. Recall f : X → Y is a Hurewicz fibration if every commutative diagram α K × {0} −−−−→   y H X  f y K × I −−−−→ Y has a filler G : K × I → X (that means f ◦ G = H and G extends α). If the above condition is satisfied for K being any n-cell I n , n ≥ 0 (equivalently, for any finite polyhedron K), then f is called a Serre fibration. Notice for K being a point this is the classical path lifting property. If the above condition is satisfied for K being any hedgehog, then f is called a hedgehog fibration. If the above condition is satisfied for K being any Peano space, then f is called a Peano fibration. We will modify those concepts for maps between pointed spaces as follows: Definition 5.8. A map f : (X, x0 ) → (Y, y0 ) is a Serre 1-fibration if any commutative diagram α (I × {0}, ( 21 , 0)) −−−−→ (X, x0 )   f  y y H (I × I, ( 12 , 0)) −−−−→ (Y, y0 ) has a filler G : (I × I, ( 21 , 0)) → (X, x0 ) (that means f ◦ G = H and G extends α). Observe Serre 1-fibrations have the path lifting property in the sense that any path in Y starting at y0 lifts to a path in X originating at x0 . COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES Theorem 5.9. Suppose 17 g1 (T, z0 ) −−−−→ (X, x0 )    f iy y g (Z, z0 ) −−−−→ (Y, y0 ) is a commutative diagram in the topological category such that (Z, z0 ) is a Peano space and i is the inclusion from a path-connected subspace T of Z. If f is a Serre 1-fibration, then there is a continuous lift h : (Z, z0 ) → (Pf (X), x0 ) of g extending g1 if the image of π1 (g) : π1 (Z, z0 ) → π1 (Y, y0 ) is contained in the image of π1 (f ) : π1 (X, x0 ) → π1 (Y, y0 ). Proof. For each point z ∈ Z pick a path αz in Z from z0 to z and let βz be a lift of g : αz 7→ Y . In case of z = z0 we pick the constant paths αz and βz . In case z ∈ T the path αz is contained in T and βz = g1 ◦ αz . Define h : (Z, z0 ) → (Pf (X), x0 ) by h(z) = βz (1). Given a neighborhood U of g(z) in Y , let V be the path component of h(z) in f −1 (U ) and let W be the path component of g −1 (U ) containing z. Our goal is to show h(W ) ⊂ V as that is sufficient for h : (Z, z0 ) → (Pf (X), x0 ) to be continuous. For any t ∈ W choose a path µt in W from z to t. Let γ be a loop in X at x0 so that f (γ) is homotopic to g(αz ∗ µt ∗ α−1 t ). Notice f (βz ) is homotopic to f (γ ∗ βt ) via a homotopy H so that H({1} × I) ⊂ U . By lifting that homotopy to X we get a path in f −1 (U ) from h(z) to h(t), i.e., h(t) ∈ V .  Corollary 5.10. A Peano map f : X → Y is a Peano fibration if and only if it is a Serre 1-fibration. Proof. Assume f : X → Y is a Peano map and a Serre 1-fibration (in the other direction 5.10 is left as an exercise), g : Z × {0} → X is a map from a Peano space, and H : Z × I → Y is a homotopy starting from f ◦ g. Pick z0 ∈ Z and put x0 = g(z0 , 0), y0 = f (x0 ). Notice the image of π1 (g) : π1 (Z × {0}, (z0 , 0)) → π1 (Y, y0 ) is contained in the image of π1 (f ). Use 5.9 to produce an extension G : Z × I → X of g that is a lift of H.  6. Peano covering maps 5.9 suggests the following concept: Definition 6.1. A map f : X → Y is called a Peano covering map if the following conditions are satisfied: (1) f is a Peano map, (2) f is a Serre fibration, (3) The fibers of f have trivial path components. Notice 3) above can be replaced by f having the unique path lifting property (see 8.3). Also notice that, in case fibers of a Peano map f : X → Y are T0 spaces, path-components of fibers are trivial. Indeed, two points in a path-component of a fiber are always in any open set that contains one of them. Proposition 6.2. Any product of Peano covering maps is a Peano covering map. Q fs , Proof. Suppose fs : Xs → YS , s ∈ S, are Peano covering maps. Put f = s∈S Q Q Ys . By 5.2 f is a Peano map. It is obvious f is a Serre Xs , and Y = X= s∈S s∈S fibration and has the uniqueness of path lifting property.  18 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA Corollary 6.3. Suppose f : (X, x0 ) → (Y, y0 ) is a Peano covering map. If (Z, z0 ) is a Peano space, then any map g : (Z, z0 ) → (Y, y0 ) has a unique continuous lift h : (Z, z0 ) → (X, x0 ) if the image of π1 (g) is contained in the image of π1 (f ). Proof. By 5.9 a lift h exists and is unique by the uniqueness of path lifting property.  Our basic example of Peano covering maps is related to the basic topology: Theorem 6.4. If X is a path-connected space and x0 ∈ X, then the following conditions are equivalent: bH , x a. pH : (X b0 ) → (X, x0 ) has the unique path lifting property, b b. pH : XH → X is a Peano covering map. bH , x Proof. a) =⇒ b). In view of 5.3 and 8.4 it suffices to show pH : (X b0 ) → (X, x0 ) is a Serre fibration. Suppose f : (Z, z0 ) → (X, x0 ) is a map from a simply connected Peano space Z (the case of Z = I n is of interest here). There is a standard bH of f defined as g(z) = [αz ]H , where αz is a path in Z from z0 to lift g : (Z, z0 ) → X bH , x z. If T is a path-connected subspace of Z containing z0 and h : (T, z0 ) → (X b0 ) is any continuous lift of f |T , then h = g|T due to the uniqueness of the path lifting property of pH . That proves pH is a Serre fibration in view of 8.4. b) =⇒ a) is obvious.  Theorem 6.5. If f : X → Y is a map and X is an lpc-space, then the following conditions are equivalent: a) f is a Peano covering map, b) f is a Peano fibration and has the uniqueness of path lifting property, c) f is a hedgehog fibration and has the uniqueness of path lifting property, d) For any x0 ∈ X and any map g : (Z, z0 ) → (Y, f (x0 )) from a simplyconnected Peano space there is a lift h : (Z, z0 ) → (X, x0 ) of g and that lift is unique. Proof. a) =⇒ b). Suppose H : Z × I → Y is a homotopy, Z is a Peano space, and G : Z × {0} → X is a lift of H|Z × {0}. Pick z0 ∈ Z, put x0 = G(z0 , 0) and y0 = f (x0 ), and notice im(π1 (Z × I, (z0 , 0))) ⊂ im(π1 (f )). Using 5.9 there is a lift of H and that lift is unique, hence it agrees with G on Z × {0}. b) =⇒ c) is obvious. d) =⇒ c) is obvious. a) =⇒ b) follows from 5.9. c) =⇒ a). Notice f has the unique hedgehog lifting property and is a Serre 1-fibration. By 5.6 f is a Peano map.  Corollary 6.6. Suppose f : X → Y and g : Y → Z are maps of path-connected spaces and Y is a Peano space. If any two of f , g, h = g ◦ f are Peano covering maps, then so is the third provided its domain is an lpc-space. Proof. In view of 6.5 it amounts to verifying that the map has uniqueness of lifts of simply-connected Peano spaces, an easy exercise.  Proposition 6.7. Suppose f : X → Y is a map. a. If f : X → Y is a Peano covering map, then f : f −1 (U ) → U is a Peano covering map for every open subset U of Y . COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 19 b. If every point y ∈ Y has a neighborhood U such that f : f −1 (U ) → U is a Peano covering map, then f is a Peano covering map. Proof. a). f : f −1 (U ) → U is clearly a Peano map, is a fibration, and has the unique path lifting property. b). f is a Serre 1-fibration and path components of fibers are trivial. If V is an open subset of Y containing y we pick an open subset U of X containing f (y) such that f : f −1 (U ) → U is a Peano covering map. There is an open neighborhood W of f (y) in U so that the path component of y in f −1 (W ) is open and is contained in V ∩ f −1 (U ). That proves f : Y → X is a Peano map.  In analogy to regular classical covering maps let us introduce regular Peano covering maps: Definition 6.8. A Peano covering map f : X → Y is regular if lifts of loops in Y are either always loops of are always non-loops. Corollary 6.9. Given a map f : X → Y the following conditions are equivalent if X is path-connected: a) f is a regular Peano covering map, b) f is a Peano covering map and the image of π1 (f ) is a normal subgroup of π1 (Y, f (x0 )) for all x0 ∈ X, c) f : X → Y is a generalized covering map in the sense of Fischer-Zastrow. Proof. a) =⇒ b). If the image of π1 (f ) is not a normal subgroup of π1 (Y, f (x0 )) for some x0 ∈ X, then there is a loop α in Y at y0 = f (x0 ) that lifts to a loop in X at x0 and there is a loop β in Y at y0 such that β ∗ α ∗ β −1 does not lift to a loop in X at x0 . Let γ be a lift of α originating at x0 . Let x1 = β(1). Notice the lift of α originating at x1 cannot be a loop, a contradiction. b) =⇒ c). As im(π1 (f )) is a normal subgroup H of π1 (Y, y0 ), it does nor depend on the choice of the base-point of X in f −1 (y0 ). Using 5.9 one gets f is a generalized covering map. c) =⇒ a). Since each hedgehog is contractible, f has the unique hedgehog lifting property and is a Peano map by 5.6. It is also a Serre fibration, hence a Peano covering map. Also, as im(π1 (f )) is a normal subgroup H of π1 (Y, y0 ), it does nor depend on the choice of the base-point of X in f −1 (y0 ). Hence a loop in Y lifts to a loop in X if and only if it represents an element of H. Thus f is a regular Peano covering map.  In the remainder of this section we will discuss the relation of Peano covering maps to classical covering maps. Proposition 6.10. If f : Y → X is a Peano covering map and U is an open subset of X such that every loop in U is null-homotopic in X, then f −1 (V ) → P (V ) is a a trivial discrete bundle for every path component V of U . Proof. Consider a path component W of f −1 (U ) intersecting f −1 (V ). f maps W bijectively onto V and it is easy to see f |W : W → V is equivalent to P (V ) → V .  Corollary 6.11. If X is a semilocally simply connected Peano space, then f : Y → X is a Peano covering map if and only if it is a classical covering map and Y is connected. 20 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA Proof. If f is a classical covering map and Y is connected, then Y is locally path-connected, f has unique path lifting property and is a Serre 1-fibration. Thus it is a Peano covering map. Suppose f is a Peano covering map and x ∈ X. Choose a path-connected neighborhood U of x in X such that any loop in U is null-homotopic in X. By 6.10 U is evenly covered by f .  Corollary 6.12. If f : Y → P (X) is a classical covering map, then f : Y → X is a Peano covering map. Proof. By 6.7, f : Y → P (X) is a Peano covering map. As the identity function induces a Peano covering map P (X) → X, f : Y → X is a Peano covering map by 6.6.  Proposition 6.13. If f : Y → X is a Peano covering map and X is path-connected, then all fibers of f have the same cardinality. Proof. Given two points x1 , x2 ∈ X fix a path α from x1 to x2 and notice lifts of α establish bijectivity of fibers f −1 (x1 ) and f −1 (x2 ).  The following result has its origins in Lemma 2.3 of [8] and Proposition 6.6 of [15]. Proposition 6.14. Suppose f : Y → X is a regular Peano covering map. If f −1 (x0 ) is countable and x0 has a countable basis of neighborhoods in X, then there is a neighborhood U of x0 in X such that f −1 (V ) → P (V ) is a classical covering map, where V is the path component of x0 in U . Proof. Switch to X being Peano by considering f : Y → P (X). Notice x0 has a countable basis of neighborhoods and f is open. Suppose there is no open subset U of X containing x0 such that U is evenly covered. That means path components of f −1 (U ) are not mapped bijectively onto their images. First, we plan to show there is a neighborhood U of x0 in X such that the image of π1 (U, x0 ) → π1 (X, x0 ) is contained in the image of π1 (f ) : π1 (Y, y0 ) → π1 (X, x0 ). In particular, there is a lift of P (U, x0 ) → (Y, y0 ) of the inclusion induced map P (U, x0 ) → (X, x0 ). Suppose no such U exists. By induction we will find a basis of neighborhoods {Ui } of x0 in X and elements [αi ] ∈ π1 (Ui , x0 ) that are not contained in the image of π1 (Ui+1 , x0 ) → π1 (X, x0 ) and whose lifts are not loops and end at points yi such that yi 6= yj if i 6= j. Given a neighborhood Ui pick a loop αi in (Ui , x0 ) whose lift (as a path) in (Y, y0 ) is not a loop and ends at yi 6= y0 . There is a neighborhood Ui+1 of x0 in Ui such that the no path components of f −1 (Ui+1 ) contains both y0 and some yj , j ≤ i. Pick a loop αi+1 in (Ui+1 , x0 ) whose lift is not a loop. As in [21] one can create infinite concatenations αi(1) ∗ . . . ∗ αi(k) ∗ . . . for any increasing sequence {i(k)}k≥1 . By looking at lifts of those infinite concatenations, there are two different infinite concatenations αi(1) ∗ . . . ∗ αi(k) ∗ . . . and αj(1) ∗ . . . ∗ αj(k) ∗ . . . whose lifts end at the same point y ∈ f −1 (x0 ). Pick the smallest k0 so that i(k0 ) 6= j(k0 ). We may assume i(k0 ) < j(k0 ) and conclude there are loops β in (Uk0 +1 , x0 ) and γ in (Y, y0 ) so that αi(k0 ) ∼ f (γ) ∗ β i which case the lift of αi(k0 ) in (Y, y0 ) ends in the path component of f −1 (Ui(k0 )+1 ) containing y0 , a contradiction. As f is a regular Peano covering map, we can find lifts (U, x0 ) → (Y, y) of the inclusion map (U, x0 ) → (X, x0 ) for any y ∈ f −1 (x0 ).  COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 21 7. Peano subgroups Definition 7.1. Suppose (X, x0 ) is a pointed path-connected space. A subgroup H of π1 (X, x0 ) is a Peano subgroup of π1 (X, x0 ) if there is a Peano covering map f : Y → X such that H is the image of π1 (f ) : π1 (Y, y0 ) → π1 (X, x0 ) for some y0 ∈ f −1 (x0 ). Proposition 7.2. If H is a Peano subgroup of π1 (X, x0 ), then X is homotopically Hausdorff relative to H. In particular, H is closed in π1 (X, x0 ) equipped with the basic topology. Proof. Choose a Peano covering map f : Y → X so that im(π1 (f )) = H for some y0 ∈ f −1 (x0 ). If g ∈ π1 (X, x0 ) \ H and α is a path in X from x0 to x1 , then lifts of α and g · α end in two different points y1 and y2 of the fiber f −1 (x1 ) and there is a neighborhood U of x1 in X such that no path component of f −1 (U ) contains both y1 and y2 . Suppose there is a loop γ in (U, x1 ) with the property [α ∗ γ ∗ α−1 ] ∈ H · g. In that case the lifts of both α ∗ γ and g · α end at y2 . Since the lift of α ends in the same path component of f −1 (U ) as the lift of α ∗ γ, both y1 and y2 belong to the same component of f −1 (U ), a contradiction. Use 4.12 to conclude H is closed in π1 (X, x0 ) equipped with the basic topology.  Remark 7.3. In case of H being the trivial subgroup, Lemma 2.10 of [15] seems to imply that X is homotopically Hausdorff but the proof of it is valid only in a special case. Proposition 7.4. If H is a Peano subgroup of π1 (X, x0 ), then any conjugate of H is a Peano subgroup of π1 (X, x0 ). Proof. Choose a Peano covering map f : Y → X so that im(π1 (f )) = H for some y0 ∈ f −1 (x0 ). Suppose G = g · H · g −1 and choose a loop α in (X, x0 ) representing g −1 . Let β be a path in (Y, y0 ) that is the lift of α. Put y1 = β(1) and notice the image of π1 (f ) : π1 (Y, y1 ) → π1 (X, x0 ) is G.  Proposition 7.5. Suppose (X, x0 ) is a pointed path-connected topological space. If f : (Y, y0 ) → (X, x0 ) is a Peano covering map with image of π1 (f ) equal H, then f bH → X. is equivalent to the projection pH : X bH , x Proof. Define h : (X b0 ) → (Y, y0 ) by choosing a lift α b of every path α in X starting at x0 and declaring h([α]H ) = α b(1). Note h is a bijection. Given y1 = α b(1) and given a neighborhood U of y1 in Y choose a neighborhood V of f (y1 ) = α(1) in X so that the path component of f −1 (V ) containing y1 is a subset of U . Observe BH ([α]H , V ) ⊂ h−1 (U ) which proves h is continuous. Conversely, given a neighborhood W of α(1) in X the image h(BH ([α]H , W )) of BH ([α]H , W ) equals the path component of α b(1) in f −1 (W ) and is open in Y .  Theorem 7.6. If X is a path-connected space, x0 ∈ X, and H is a subgroup of π1 (X, x0 ), then the following conditions are equivalent: a. H is a Peano subgroup of π1 (X, x0 ), bH , x b. The endpoint projection pH : (X b0 ) → X is a Peano covering map, b c. The image of π1 (pH ) : π1 (XH , x b0 ) → π1 (X, x0 ) is contained in H, bH , x d. pH : (X b0 ) → (X, x0 ) has the unique path lifting property. 22 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA Proof. c)≡d) is done in 2.18. b)≡d) is contained in 6.4. a) =⇒ b) follows from 7.5. b) =⇒ a) holds as c) implies the image of π1 (pH ) is H. Let us state a straightforward consequence of 7.6:  Corollary 7.7. If X is a path-connected space and x0 ∈ X, then the following conditions are equivalent: b → X is a Peano covering map, a. The endpoint projection p : X b b. π1 (p) : π1 (X, x b0 ) → π1 (X, x0 ) is trivial, b is simply connected, c. X b x d. p : (X, b0 ) → (X, x0 ) has the unique path lifting property. Corollary 7.8. Closed and normal subgroups of π1 (X, x0 ) are Peano subgroups of π1 (X, x0 ). eH , x Proof. By 4.6 the endpoint projection pH : (X e0 ) → X has unique path b lifting property. Since pH : (XH , x b0 ) → X has path lifting property, this implies bH , x pH : (X b0 ) → X has the unique path lifting property.  Corollary 7.9. If H(s) is a Peano subgroup of π1 (X, x0 ) for each s ∈ S, then T H(s) is a Peano subgroup of π1 (X, x0 ). G= s∈S bG , x Proof. The projection pG : (X b0 ) → (X, x0 ) factors through T bH(s) , x H(s) = G pH(s) : (X b0 ) → (X, x0 ) for each s ∈ S. Therefore im(π1 (pG )) ⊂ s∈S and 6.4 (in conjunction with 2.18) says G is a Peano subgroup of π1 (X, x0 ).  Corollary 7.10. For each path-connected space X there is a universal Peano covering map p : Y → X. Thus, for each Peano covering map q : Z → X and any points z0 ∈ Z and y0 ∈ Y satisfying q(z0 ) = p(y0 ), there is a Peano covering map r : Y → Z so that r(y0 ) = z0 . Moreover, the image of π1 (Y ) is normal in π1 (X). Proof. Let H be the intersection of all Peano subgroups of π1 (X, x0 ) by 7.9 bH and use 6.3.  and 7.4 it is a normal Peano subgroup of π1 (X, x0 ). Put Y = X It would be of interest to characterize path-connected spaces X admitting a b being simply universal Peano covering that is simply connected (that amounts to X connected). Here is an equivalent problem: Problem 7.11. Characterize path-connected spaces X so that the trivial group is a Peano subgroup of π1 (X, x0 ). So far the following classes of spaces belong to that category: (1) Any product of spaces admitting simply connected Peano cover (see 6.2). (2) Subsets of closed surfaces: it is proved in [14] that if X is any subset of a closed surface, then π1 (X, x0 ) → π̌1 (X, x0 ) is injective. (3) 1-dimensional, compact and Hausdorff, or 1-dimensional, separable and metrizable: π1 (X, x0 ) → π̌1 (X, x0 ) is injective by [11, Corollary 1.2 and Final Remark]. It is shown in [10] (see proof of Theorem 1.4) that the b → X has the uniqueness of path-lifting property if X is 1projection X dimensional and metrizable. See [6] for results on the fundamental group of 1-dimensional spaces. COVERING MAPS FOR LOCALLY PATH-CONNECTED SPACES 23 (4) Trees of manifolds: If X is the limit of an inverse system of closed PLmanifolds of some fixed dimension, whose consecutive terms are obtained by connect summing with closed PL-manifolds, which in turn are trivialized by the bonding maps, then X is called a tree of manifolds. Every tree of manifolds is path-connected and locally path-connected, but it need not be semilocally simplyconnected at any one of its points. Trees of manifolds arise as boundaries of certain Coxeter groups and as boundaries of certain negatively curved geodesic spaces [13]. It is shown in [13] that if X is a tree of manifolds (with a certain denseness of the attachments in the case of surfaces), then π1 (X, x0 ) → π̌1 (X, x0 ) is injective. b → X does not have the unique Notice Example 2.7 in [15] gives X so that p : X path lifting property (one can construct a simpler example with X being the Harmonic Archipelago). However, X is not homotopically Hausdorff. b →X Problem 7.12. Is there a homotopically Hausdorff space X such that p : X does not have the uniqueness of path lifting property? Corollary 7.13. Suppose H is a normal subgroup of π1 (X, x0 ). If there is a Peano subgroup G of π1 (X, x0 ) containing H such that G/H is countable, then H is a Peano subgroup of π1 (X, x0 ) if and only if X is homotopically Hausdorff relative to H. Proof. By 7.2, X is homotopically Hausdorff relative to H if H is a Peano subgroup of π1 (X, x0 ). bH of Suppose X is homotopically Hausdorff relative to H. Given two lifts in X b b the same path in X, their composition with XH → XG are the same by 7.6. By 4.18 the two lifts are identical and 7.6 says H is a Peano subgroup of π1 (X, x0 ).  Corollary 7.14. Suppose H is a normal subgroup of π1 (X, x0 ). If π1 (X, x0 )/H is countable, then H is a Peano subgroup of π1 (X, x0 ) if and only if X is homotopically Hausdorff relative to H. 8. Appendix: Pointed versus unpointed In this section we discuss relations between pointed and unpointed lifting properties. Lemma 8.1. If f : (X, x0 ) → (Y, y0 ) has the uniqueness of path lifts property and X is path-connected, then f : X → Y has the uniqueness of path lifts property. Proof. Given two paths α and β in X originating at the same point and satisfying f ◦ α = f ◦ β, choose a path γ in X from x0 to α(0). Now f ◦ (γ ∗ α) = f ◦ (γ ∗ β), so γ ∗ α = γ ∗ β and α = β.  Lemma 8.2. If f : (X, x0 ) → (Y, y0 ) has the unique path lifting property and X is path-connected, then f : X → Y has the unique path lifting property. Proof. In view of 8.2 it suffices to show f : X → Y is surjective and has the path lifting property. If y1 ∈ Y , we pick a path α from y0 to y1 and lift it to (X, x0 ). The endpoint of the lift maps to y1 , hence f is surjective. Suppose α is a path in Y and f (x1 ) = α(0). Choose a path β in X from x0 to x1 and lift (f ◦ β) ∗ α to a path γ in (X, x0 ). Due to the uniqueness of path lifts property of f : (X, x0 ) → (Y, y0 ) 24 N. BRODSKIY, J. DYDAK, B. LABUZ, AND A. MITRA one has γ(t) = β(2t) for t ≤ 21 . Hence γ( 12 ) = x1 and λ defined as λ(t) = γ( 21 + 2t ) for t ∈ I is a lift of α originating from x1 .  Lemma 8.3 (Lemma 15.1 in [17]). If f : X → Y is a Serre 1-fibration, then f has the unique path lifting property if and only if path components of fibers of f are trivial. Proof. Suppose the fibers of f have trivial path components and α, β are two lifts of the same path in Y that originate at x1 ∈ X. Let H : I × I → Y be the standard homotopy from f ◦ (α−1 ∗ β) to the constant path at f (x1 ). There is a lift G : I × I → X of H starting from α−1 ∗ β. As path components of f are trivial, α = β due to the way the standard homotopy H is defined.  Lemma 8.4. Suppose n ≥ 1. If f : (X, x0 ) → (Y, y0 ) is a Serre n-fibration, both X and Y are path-connected, and f has the uniqueness of path lifts property, then f : X → Y is a Serre n-fibration. Proof. Suppose H : I n × I → Y is a homotopy and G : I n × {0} → X is its partial lift. 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