Covering maps for locally path-connected spaces

CO VERING MAPS F OR LOCALL Y P A TH-CONN ECTED SP A CES N. BRODSKIY, J . DYD AK, B. LABUZ, AND A. MITRA Abstract. W e define P eano cov ering maps and pro v e basic prop erties anal- ogous to c lassical co v ers. Their do main is alw a ys locally path-connect ed but the range may be an ar bitrary top ological space. One of c haracterizations of Pe ano co v ering maps is via th e uniqueness of homotop y lif ting propert y for all lo call y path-connec ted space s. Regular Pea no cov ering maps ov er path-connect ed spaces ar e sho wn to b e iden tical with generalized regular cov ering maps i n troduced by Fisc her and Zastro w [15]. If X i s path-conne cted, then ev ery P eano co v ering map is equiv- alen t to the pro jection e X /H → X , where H is a subgroup of the fundamenta l group of X and e X eq uipp ed with the topology used in [2], [15] and int ro duced in [23 , p. 82]. The pro j ection e X /H → X is a Peano c ov ering map i f and only if it has the unique path lifting prop erty . W e define a ne w topol ogy on e X for which one has a characte rization of e X /H → X having the unique path li fting property if H is a normal subgroup of π 1 ( X ). Namely , H must be closed in π 1 ( X ). Such groups include π ( U , x 0 ) ( U being an op en cov er of X ) and the k ernel of the natural homomorphism π 1 ( X, x 0 ) → ˇ π 1 ( X, x 0 ). Contents 1. Int ro duction 1 2. Constructing Peano spaces 3 2.1. Univ ersal Peano space 3 2.2. Basic topo logy on e X 5 3. A new topolo gy on e X 8 4. Path lif ting 10 5. Peano ma ps 15 6. Peano c overing maps 17 7. Peano s ubgroups 21 8. Appendix: P oin ted v ersus unpointed 23 References 24 1. Introduction As lo cally complicated spaces naturally appear in mathematics (examples: b ound- aries of g roups, limits under Gromov-Hausdorff conv ergence) there is a n effort to extend homotopy-theoretical concepts to s uch spaces. This pap er is d evoted to Date : F ebruary 14, 2008. 2000 Math ematics Subje ct Classific ation. Primary 55Q52; Secondary 55M10, 54E15. Key w or ds and phr ases. co v ering maps, lo cally path-connecte d spaces. 1 2 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA a theory of c overings by lo cally path-co nnected spaces. Zeeman’s example [16, 6.6.14 on p.25 8] demonstrates difficult y in co nstructing a theor y of coverings by non-lo cally path-connected spaces (that example amount s to t w o non-equiv alent classical cov erings with the same image of the fundamen tal gro ups). F or coverings in the uniform category see [1] and [3]. T o simplify expo sition let us in tro duce the following concepts: Definition 1.1. A top ological space X is an lpc-space if it is locally path- connected. X is a Pea no s pace if it is lo cally path-connected and connected. Fischer and Zastrow [15] defined generalized regular co verings of X as func- tions p : ¯ X → X satis fying the following co nditions for some normal subgro up H of π 1 ( X ): R1. ¯ X is a Peano space. R2. The map p : ¯ X → X is a contin uous surjection and π 1 ( p ) : π 1 ( ¯ X ) → π 1 ( X ) is a monomorphism onto H . R3. F or every Peano spac e Y , for every co ntin uo us function f : ( Y , y ) → ( X , x 0 ) with f ∗ ( π 1 ( Y , y )) ⊂ H , and for ev ery ¯ x ∈ ¯ X with p ( ¯ x ) = x 0 , there is a unique contin uous g : ( Y , y ) → ( ¯ X , ¯ x )) with p ◦ g = f . Our view o f the ab ov e concept is tha t of b eing universal in a cer tain class o f ma ps and w e propos e a different way of defining cov ering maps b etw e e n P eano spa ces in Section 6. Our fir st obser v atio n is that each path-connected space X has its universal Peano space P ( X ), the set X equipp ed with new top olo g y , such that the identit y function P ( X ) → X corr esp onds to a generalized r egular cov ering for H = π 1 ( X ). That wa y quite a few results in the literatur e c a n b e forma lly deduced from ea rlier results for Peano s paces. The wa y the pro jection P ( X ) → X is characterized in 2.2 g e neralizes to the concept of P eano maps in Section 6 and our P eano cov ering maps com bine Peano maps with tw o class ical co ncepts: Serre fibrations and unique path lifting prop erty . Peano cov ering maps p o ssess s e veral prop erties analo gous to the classic a l cov ering maps [18] (example: lo cal Peano cov ering maps are Peano cov ering maps). One of them is that they are a ll quo tients b X H of the universal path spa c e e X equipp e d wit h the topolog y defined in t he pro of of Theor em 13 on p.82 in [23] and used successfully b y Bog ley-Sieradsk i [2] and Fisc her-Zastr ow [15]. It tur ns out the endpo int pr o jection b X H → X is a Peano covering map if and only if it has the uniqueness of path lifts property (se e 6.4). In an effor t to unify P eano co v ering maps with unifor m covering maps of [1] and [3] (we will explain the connection in [4]) we were led to a new top ology on e X H (see Section 3). Its main adv an tage is that there is a necessary and sufficient condition for e X H → X to hav e the unique path lifting prop erty in case H is a normal subgroup of π 1 ( X ). It is H b eing closed in π 1 ( X ). That explains Theorem 6.9 of [15] as the basic groups ther e turn out to b e clos ed in π 1 ( X ). As a n application of o ur approa ch w e show existence of a universal Peano cov ering map o ver a given path-connected space. W e thank Sasha Dranishniko v for bringing the work o f Fischer-Zas tr ow [15] to our attention. W e thank Greg Co nner, K atsuya Eda, Ale ˇ s V a vp eti´ c, and Zig a Virk for helpful comments. CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 3 2. Constructing P eano sp aces The purp os e of this section is t o discuss v arious wa ys of constr ucting new P eano spaces. 2.1. Universal Peano space. In a nalogy to the univ ersal cov ering spaces we in- tro duce the following notio n: Definition 2.1. Given a top ologica l space X its universal l p c-space l pc ( X ) is an lp c-space together with a contin uous map (called the univ ersal P eano map ) π : lpc ( X ) → X s atisfying the following universality co ndition: F or a ny map f : Y → X from an lp c-space Y there is a unique co nt inuous lift g : Y → l pc ( X ) of f (t hat mea ns π ◦ g = f ). Theorem 2. 2. Every sp ac e X has a un iversal lp c-sp ac e. It is home omorph ic to the set X e quipp e d with a new top olo gy, the one gener ate d by al l p ath-c omp onents of al l op en subsets of t he exist ing top olo gy of X . Pro of. Let U b e an o pen set in X co ntaining the p oint x and c ( x, U ) b e the path compo nent 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 r anges ov er all open subsets o f X and x ranges ov er a ll elemen ts of U , forms a basis . Given a map f : Y → X and given an op e n set U of X containing f ( y ) one ha s f ( c ( y , f − 1 ( U ))) ⊂ c ( f ( y ) , U ). That proves f : Y → l pc ( X ) is contin uous if Y is an lpc- space. It also p rov es l pc ( X ) is locally path-connected as any path in X induces a path in lpc ( X ).  R emark 2.3 . The topolo gy a b ove was mentioned in Remark 4.17 of [15]. After t he first version of this pa per was written w e were informed by Greg Conner of his unpublished preprint [7] with David F earnley , where that top olog y is discussed and its prop erties (compactness, metrizability) are in v estigated. If X is path-connec ted, then l pc ( X ) is a univ ersal Peano space P ( X ) in the following sense: giv en a map f : Z → X fro m a Peano space Z to X there is a unique lift g : Z → P ( X ) o f f . In the remainder of this section we give sufficient conditions for a function o n an lp c-space to be contin uous. Those co nditions are in ter ms of maps from ba sic Peano spaces: the arc in the firs t-countable case and hedge hogs (s e e Definition 2.8) in the arbitrary case. Prop ositio n 2. 4. Supp ose f : Y → X is a fun ction fr om a first-c ountable lp c-sp ac e Y . f is c ontinuous if f ◦ g is c ontinu ous for every p ath g : I → Y in Y . Pro of. Suppose U is op en in X . It suffices to show that for each y ∈ f − 1 ( U ) there is an op en set V in Y containing y suc h that the path comp onent of y in V is contained in f − 1 ( U ). Pick a basis o f neighbor ho o ds { V n } n ≥ 1 of y in Y and assume for each n ≥ 1 there is a path α n in V n joining y to a p oint y n / ∈ f − 1 ( U ). Those paths can be spliced to one path α fro m y to y 1 and going through all po in ts y n , n ≥ 2. f ◦ α starts fro m f ( y ) and go es through all points f ( y n ), n ≥ 1. How ev er, as U is open, it must contain almost all o f them, a contradiction.  The cons tr uction of the top o lo gy on l pc ( X ) in 2.2 can b e done in the spirit of the fines t top ology on X tha t retains the same con tin uous maps from a class of spaces. 4 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA Prop ositio n 2.5. Supp ose X is a p ath-c onne cte d top olo gic al s p ac e and P is a class of Pe ano sp ac es. The family T of su bsets U of X such that f − 1 ( U ) is op en in Z ∈ P for any map f : Z → X in the original top olo gy, is a top olo gy and P ( X ) := ( X , T ) is a Pe ano sp ac e. Pro of. Since f − 1 ( U ∩ V ) = f − 1 ( U ) ∩ f − 1 ( V ), T is a topolo gy on X . Supp ose U ∈ T a nd C is a path comp onent of U in the new-top olo gy . Suppo se f : Z → X is a map and f ( z 0 ) ∈ C . As f − 1 ( U ) is op en, ther e is a connected neig hbo rho o d V of z 0 in Z sa tisfying f ( V ) ⊂ U . As f ( V ) is path-connected, f ( V ) ⊂ C and C ∈ T .  In case o f first-c ountable space s X we have a v ery simple c haracterization of the universal Peano map of X : Corollary 2.6. If X is a first-c ount able p ath-c onne cte d top olo gic al sp ac e, then a map f : Y → X is a universal Pe ano map if and only if Y is a Pe ano sp ac e, f is a bije ction, and f has the p ath lifting pr op erty. Pro of. Co nsider A ( X ) a s in 2.5, where A consists of the unit in terv al. Notice the iden tit y function P ( X ) → A ( X ) is contin uous as P ( X ) is first-countable (use 2.4). Since the top olog y on A ( X ) is finer than that on P ( X ), P ( X ) = A ( X ). Since f induces a homeomo r phism from A ( Y ) to A ( X ) (due to the uniqueness of path lifting prop er ty of f ), the co mp os ition A ( Y ) → A ( X ) → P ( X ) is a homeo mo rphism and f : Y → P ( X ) must be a homeomorphism (its inv erse is P ( X ) → A ( Y ) → Y ).  The construction in 2.5 can be used to create co un ter-examples to 2.6 in case X is not first-countable. Example 2. 7 . Let X b e the cone over an uncoun table discrete set B . Subsets of X that miss the vertex v are declare d o p e n if and only if they ar e open in the CW top ology on X . A subset U o f X that con tains v is declared op en if a nd only if U contains all but countably ma n y edges of the cone and U \ { v } is op en in the CW topolog y on X (that means X is a he dg ehog if B is of cardina lit y ω 1 - see 2.8). Notice A ( X ) is X equipped with the CW top o lo gy , the iden tit y function A ( X ) → X ha s the path lifting prop erty but is no t a homeomorphism. Pro of. Notice every subset of X \ { v } that meets each edg e in at mo st o ne p oint is discrete. Hence a path in X has to b e co nt ained in the unio n of finitely many edges. That means A ( X ) is X with the CW top ology .  W e g eneralize 2.7 as follows: Definition 2.8. A generalized Ha w aiian Earring is the wedge ( Z, z 0 ) = W s ∈ S ( Z s , z s ) of po int ed Peano spa ces indexed by a directed set S and equipp e d wit h the follo wing to p o lo gy (all wedges in this pape r are considered wit h that particular top ology): (1) U ⊂ Z \ { z 0 } is op en if and only if U ∩ Z s is op en for each s ∈ S , (2) U is an o pe n neighborho o d of z 0 if and only if there is t ∈ S such that Z s ⊂ U for a ll s > t and U ∩ Z s is op en for each s ∈ S . A hedgehog is a generalized Haw aiian Ear ring ( Z, z 0 ) = W s ∈ S ( Z s , z s ) such that each ( Z s , z s ) is homeomorphic to ( I , 0). CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 5 Our definition of genera lized Ha waiian Ea rrings is different fro m the definition of Cannon and Conner [5]. Also , the preferr ed termino logy in [5] is that of a big Ha w aiian Earring . Observe eac h generalized Ha w aiian Earring is a P eano space. Lemma 2.9. L et S b e a b asis of neighb orho o ds of x 0 in X or der e d by inclu s ion (i.e., U ≤ V me ans V ⊂ U ). If, fo r e ach U ∈ S , α U : I → U is a p ath in U starting fr om x 0 , t hen their we dge _ U ∈ S α U : _ U ∈ S ( I U , 0 U ) → ( X , x 0 ) is c ontinuous, wher e ( I U , 0 U ) = ( I , 0) for e ach U ∈ S . Pro of. Only the co ntin uity of g = W U ∈ S α U at the base-p oint of the hedgeho g W U ∈ S ( I U , 0 U ) is no t totally obvious. Ho wev er, if V is a neig hborho o d of x 0 in X , then g − 1 ( V ) contains all I U if U ⊂ V and g − 1 ( V ) ∩ I W is o pe n in I W for all W ∈ S .  Prop ositio n 2.10. Supp ose f : Y → X is a funct ion fr om an lp c-sp ac e Y . f is c ontinu ous if f ◦ g is c ontinuous for every map g : Z → Y fr om a he dgeho g Z t o Y . Pro of. Assume U is op en in X and x 0 = f ( y 0 ) ∈ U . Supp ose for each path- connected neighborho o d V of y 0 in Y there is a pa th α V : ( I , 0) → ( V , y 0 ) such that α V (1) / ∈ f − 1 ( U ). By 2 .9 the wedge g = W V ∈ S α V is a map g fro m a hedgehog to Y (here S is the family of all path-c onnected neighborho o ds of y 0 in Y ). Hence h = f ◦ g is contin uous and there is V ∈ S so that I V ⊂ h − 1 ( U ). That mea ns f ( α V ( I )) ⊂ U , a cont radiction.  2.2. Basic top olog y on e X . The philoso phical mea ning of this section is that ma ny results can be reduced to those dealing with Peano spaces via the universal P eano space construction. Let us illustrate this point o f view b y discussing a top olog y o n e X . Suppo se ( X , x 0 ) is a pointed top ologic a l space. Consider the space e X of homo- topy class es of paths in X o riginating a t x 0 . It has an int eresting t op ology (see the pro of o f Theorem 13 on p.82 in [23]) that has b een put to use in [2] and [15]. Its basis consis ts of sets B ([ α ] , U ) ( U is op en in X , α joins x 0 and α (1) ∈ U ) defined as fo llows: [ 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. endp oints to the concatenation α ∗ γ . e X equipp ed with the ab ove to p o logy will be deno ted b y b X as in [2]. Both [2 ] a nd [15] consider quo tien t spa ces b X /H , where H is a subg roup of π 1 ( X, x 0 ). W e find it mo re conv enien t to follow [23, pp.82-3 ]: Definition 2. 11. Supp ose H is a subgroup of π 1 ( X, x 0 ). Define e X H as the set of equiv ale nce classes of paths in X under the relation α ∼ H β defined via α (0) = β (0) = x 0 , α (1) = β (1) and [ α ∗ β − 1 ] ∈ H (the equiv alence cla ss o f α under the relation ∼ H will b e denoted by [ α ] H ). T o introduce a to po logy on e X H we define sets B H ([ α ] H , U ) (denoted by < α, U > on p.82 in [23]), where U is op en in X , α jo ins x 0 and α (1) ∈ U , as follows: 6 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA [ β ] H ∈ B H ([ α ] H , U ) if and o nly if there is a path γ in U fro m α (1) to β (1) such that [ β ∗ ( α ∗ γ ) − 1 ] ∈ H (equiv alently , β ∼ H α ∗ γ ). e X H equipp e d with the top olo gy (which we c a ll the basic top ology on e X H ) whose basis consists o f B H ([ α ] H , U ), where U is o pen in X , α joins x 0 and α (1) ∈ U , is denoted b y b X H in analogy to the notation b X in [2] that corresp onds to H b eing trivial. Given a path α in X and a pa th β in X fro m x 0 to α (0) one can define a standard lift ˆ α of it to b X H originating at [ β ] H by the formula ˆ α ( t ) = [ β ∗ α t ] H , where α t ( s ) = α ( s · t ) for s, t ∈ I (see [16, Prop osition 6.6.3]). Let us extrac t the essence of the pro of of [23, Theorem 13 on pp.82– 83]: Lemma 2.12. S upp ose X is a p ath-c onne cte d sp ac e and H is a su b gr oup of π 1 ( X, x 0 ) . An op en set U ⊂ X is evenly c over e d by p H : b X H → X if and only if U is lo c al ly p ath-c onne cte d and the image of h α : π 1 ( U, x 1 ) → π 1 ( X, x 0 ) is c ontaine d in H for any p ath α in X fr om x 0 to any x 1 ∈ U . Pro of. Recall that U is evenly cov ered by p H (see [23, p.62]) if p − 1 H ( U ) is the disjoint union of o p e n subsets { U s } s ∈ S of b X H each o f which is mapp ed homeomorphica lly onto U by p H . Also, rec all h α : π 1 ( U, x 1 ) → π 1 ( X, x 0 ) is given by h α ([ γ ]) = [ α ∗ γ ∗ α − 1 ]. Suppo se U is evenly cov ered, γ is a loo p in ( U, x 1 ), and α is a pa th fro m x 0 to x 1 . If [ α ] H 6 = [ α ∗ γ ] H , then they b elong to t wo differen t sets U u and U v , u, v ∈ S . How ev er, there is a path fro m [ α ] H to [ α ∗ γ ] H in p − 1 H ( U ) giv en by the standard lift of γ , a contradiction. Thus [ α ] H = [ α ∗ γ ] H and [ α ∗ γ ∗ α − 1 ] ∈ H . T o show that U is lo cally path-c o nnected, take a p oint x 1 ∈ U , pick a path α from x 0 to x 1 and select the unique s ∈ S so that [ α ] H ∈ U s . There is a n op en subset V of U sa tisfying B H ([ α ] H , V ) ⊂ U s . As p H | U s maps U s homeomorphica lly onto U , p H ( B H ([ α ] H , V )) is an open neig hborho o d of x 1 in U and it is path-connected. Suppo se U is lo cally pa th-connected and the image of h α : π 1 ( U, x 1 ) → π 1 ( X, x 0 ) is contained in H for any path α in X from x 0 to any x 1 ∈ U . Pick a path comp onent V of U and notice sets B H ([ β ] H , V ), β ranging o ver paths from x 0 to points of V , a re e ither iden tical or disjoint. Observe p H | B H ([ β ] H , V ) maps B H ([ β ] H , V ) homeomorphica lly on to V . Thus eac h V is evenly co vered and that is sufficient to conclude U is evenly co v ered.  As in [23, p.81], given an op en cov er U of X , π ( U , x 0 ) is the subg roup of π 1 ( X, x 0 ) generated by elemen ts of the form [ α ∗ γ ∗ α − 1 ], where γ is a lo op in some U ∈ U and α is a path from x 0 to γ (0). Here is our improv ement of [23, Theor e m 13 on p.8 2] and [15, Theorem 6.1]: Theorem 2.13. If X is a p ath-c onne cte d sp ac e and H is a sub gr oup of π 1 ( X, x 0 ) , then the endp oint pr oje ction p H : b X H → X is a classic al c overing map if and only if X is a Pe ano sp ac e and ther e is an op en c overi ng U of X so t hat π ( U , x 0 ) ⊂ H . Pro of. Apply 2.1 2.  Prop ositio n 2.14. \ P ( X ) H is natu r al ly home omorphic t o b X H if X is p ath-c onn e cte d. Pro of. Since contin uit y of f : ( Z , z 0 ) → ( P ( X ) , x 0 ), for any Peano spa ce Z , is equiv ale nt to the contin uity of f : ( Z, z 0 ) → ( X , x 0 ), paths in ( P ( X ) , x 0 ) corr e sp o nd to paths in ( X , x 0 ). Also, π 1 ( P ( X ) , x 0 ) → π 1 ( X, x 0 ) is an iso morphism s o H is a subgroup of b oth π 1 ( P ( X ) , x 0 ) and π 1 ( X, x 0 ), and the equiv alence classes of CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 7 relations ∼ H are identical in both spaces ^ P ( X ) and e X . Notice that basis ope n sets are iden tical in \ P ( X ) H and b X H .  R emark 2.15 . In view o f 2.14 some results in [15] dealing with ma ps f : Y → X , where Y is Peano, can be derived formally from corr esp onding results for f : Y → P ( X ). A go o d example is Lemma 2.8 in [15]: p : ˜ X → X has the unique pa th lifting pro pe r ty if and only if ˜ X is simply connected. It follows formally fr om Corolla ry 4.7 in [2 ]: The universal endp oint pro jection p : ˆ Z → Z for a connected a nd lo cally path- connected space Z has the unique path lifting prop erty if and only if ˆ Z is s imply connected. When working in the pointed topolo gical category the space b X H is equipp ed with the base-p oint b x 0 equal to the equiv alence class of the constant path at x 0 . Let us illustrate b X H in the case of H = π 1 ( X, x 0 ). Prop ositio n 2.16. If H = π 1 ( X, x 0 ) , then a. The endp oint pr oje ction p H : ( b X H , b x 0 ) → ( X , x 0 ) is an inje ction and p H ( B ([ α ] H , U )) is the p ath c omp onent of α (1) in U , b. b X H is a Pe ano sp ac e, c. Given a map g : ( Z, z 0 ) → ( X, x 0 ) fr om a p ointe d Pe ano sp ac e to ( X , x 0 ) , ther e is a un ique lift h : ( Z , z 0 ) → ( b X H , b x 0 ) of g ( p H ◦ h = g ). Pro of. a). Clearly , p H ( B H ([ α ] H , U )) equals pa th comp onent of α (1) in U . If [ β 1 ] H and [ β 2 ] H map to t he same p oint x 1 , then β 1 (1) = β 2 (1) and γ = β 1 ∗ β − 1 2 is a lo op. Hence [ γ ] ∈ H and [ β 2 ] H = [ γ ∗ β 2 ] H = [ β 1 ] H proving p H is an injection. b) is well-established in both [2] and [15]. Notice it follows from a). c). F o r ea ch z ∈ Z pick a path α z from z 0 to z in Z . Define h ( z ) a s [ α z ] H and notice h is contin uous a s h − 1 ( B H ([ α z ] H , U )) equals the path comp onent of g − 1 ( U ) containing z (use Part a)). As p H is injective, there is at most one lif t of g .  In view of 2 .1 6 we hav e a con v enient definition of a univ ersal Peano s pace in the po int ed category: Definition 2.17. By the univ ersal P eano space P ( X , x 0 ) of ( X , x 0 ) we mean the p ointed s pa ce ( b X H , b x 0 ), H = π 1 ( X, x 0 ), and the univ ersal Peano map of ( X, x 0 ) is the endp oint pro jection P ( X , x 0 ) → ( X , x 0 ). Eq uiv alently , P ( X , x 0 ) is ( P ( C ) , x 0 ), where C is the path comp onent of x 0 in X . Due t o standard lifts the e ndpo int pro jection p H : b X H → ( X , x 0 ) alw a ys has the path lifting pr op erty . Thus the issue of in terest is the uniqueness of path lifting prop erty o f p H . Here is a necessa ry a nd sufficient condition for p H to have the unique path lifting prop erty (compa re it to [2, Theorem 4.5] for Peano spa ces): Prop ositio n 2.18. If X is a p ath-c onne ct e d sp ac e and x 0 ∈ X , then t he fol lowing c onditions ar e e quivale nt: a. p H : ( b X H , b x 0 ) → ( X , x 0 ) has the unique p ath lifting pr op erty, b. The image of π 1 ( p H ) : π 1 ( b X H , b x 0 ) → π 1 ( X, x 0 ) is c ontaine d in H . 8 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA Pro of. a) = ⇒ b). Given a loop α in b X H it m ust equal the standard lift of β = p H ( α ). F or the standard lift of β to b e a lo op in b X H one m ust hav e [ β ] ∈ H . b) = ⇒ a) Given a lift ¯ α of a path α in ( X , x 0 ) it suffices to show ¯ α (1) = [ α ] H as that implies ¯ α is the standard lift of α (use α | [0 , t ] instead of α ). P ick a path β satisfying ˆ α (1) = [ β ] H and let ˆ β b e its standard lift. As ¯ α ∗ ( ˆ β ) − 1 is a loop in b X H , its image γ = p H ( ¯ α ∗ ( ˆ β ) − 1 ) g enerates an elemen t [ γ ] of H . Hence α ∼ γ ∗ β and ¯ α (1 ) = [ β ] H = [ α ] H .  3. A new topolo gy on e X W e do not know how to characterize subgro ups H of π 1 ( X, x 0 ) for which p H : b X H → X has the unique path lifting prop erty . Therefor e we will create a new top o lo gy on e X H for whic h analogous ques tion has a satisfactor y a ns wer in the ca s e H b eing a normal subgroup. Given an o pen cover U of X , a subgro up H of π 1 ( X, x 0 ), a path α in X originating at x 0 , a nd V ∈ U containing x 1 = α (1) define B H ([ α ] H , U , V ) ⊂ e X H as fo llows: [ β ] H ∈ B H ([ α ] H , U , V ) if and only if there is a path γ 0 in V orig inating at x 1 = α (1) and a loo p λ at x 1 such that [ λ ] ∈ π ( U , x 1 ) and β ∼ H α ∗ λ ∗ γ 0 . Observe [ β ] H ∈ B H ([ α ] H , U , V ) implies B H ([ α ] H , U , V ) = B H ([ β ] H , U , V ) and B H ([ α ] H , U ∩ V , V 1 ∩ V 2 ) ⊂ B H ([ α ] H , U , V 1 ) ∩ B H ([ α ] H , V , V 2 ), so the family of sets { B H ([ α ] H , U , V ) } for ms a bas is of a new top olog y on e X H . When w e cons ider e X H as a top ologic a l space, then w e use precisely that to p o lo gy . In the par ticular cas e of H = { 1 } , the trivial subgr o up of π 1 ( X, x 0 ), w e simplify e X H to e X . Observ e that, as π 1 ( X, x 0 ) is the fib er of the endp oint pro jection p : e X → X , an y subg roup G of π 1 ( X, x 0 ) can b e considered as a subspace of e X and w e may consider it as a top ological space that w ay . Notice the identit y function b X H → e X H is contin uous. Indeed, B H ([ α ] H , V ) ⊂ B H ([ α ] H , U , V ) for any V ∈ U containing α (1). When dealing with the p ointed top o logical catego ry the space e X H is equipp ed with the base-p oint e x 0 equal to the equiv alence class of the constant path at x 0 . Let us prove a basic functorial prop erty of o ur construction. Prop ositio n 3.1. Supp ose f : ( X, x 0 ) → ( Y , y 0 ) is a map of p ointe d top olo gic al sp ac es. If H and G ar e sub gr oups of π 1 ( X, x 0 ) and π 1 ( Y , y 0 ) , r esp e ctively, s uch that π 1 ( f )( H ) ⊂ G , then f induc es a natur al c ontinuous function ˜ f : ( e X H , e x 0 ) → ( e Y G , e y 0 ) . Pro of. Put ˜ f ([ α ] H ) = [ f ◦ α ] G and notice ˜ f ( B H ([ α ] H , f − 1 ( U ) , f − 1 ( V ))) ⊂ B G ( ˜ f ([ α ] H ) , U , V ) for any op en covering U of Y and any neigh bo rho o d V o f α (1).  In connection to 2.13 let us prov e the follo wing: Prop ositio n 3. 2. If X is a p ath-c onne cte d sp ac e and H is a su b gr oup of π 1 ( X, x 0 ) , then the fol lowing c onditions ar e e quivalent: a) A fib er of the endp oint pr oje ct ion p H : e X H → X has an isolate d p oint, b) The endp oint pr oje ct ion p H : e X H → X has discr ete fib ers, c) Ther e is an op en c overi ng U of X so that π ( U , x 0 ) ⊂ H , d) e X H is a Pe ano sp ac e and p H : e X H → P ( X ) is a classic al c overing map. CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 9 Pro of. a) = ⇒ c). Suppose [ α ] H ∈ p − 1 H ( x 1 ) is isolated. There is an op en cov ering U o f X and V ∈ U containing x 1 such that B H ([ α ] H , U , V ) ∩ p − 1 H ( x 1 ) = { [ α ] H } . Given γ in π ( U , x 0 ), the homotopy class [ α − 1 ∗ γ ∗ α ] H belo ngs to π ( U , x 1 ), so [ α ∗ α − 1 ∗ γ ∗ α ] H = [ γ ∗ α ] H belo ngs to B H ([ α ] H , U , V ) ∩ p − 1 H ( x 1 ). Hence [ γ ∗ α ] H = [ α ] H and [ γ ] ∈ H . c) = ⇒ d). Supp ose there is an op en cov ering U o f X so that π ( U , x 0 ) ⊂ H and W is a path comp onent of U ∈ U . Notice B H ([ α ] H , U , U ) is mapped by p H bijectiv ely onto W and that is sufficient for d). d) = ⇒ b) and b) = ⇒ a) ar e ob vious.  Applying 3.2 to H b e ing trivial one gets the follo wing (see [12] for analog ous result in case of a differen t top ology on the fundamental group): Corollary 3. 3. If X is a p ath-c onne cte d sp ac e, then π 1 ( X, x 0 ) is discr ete if and only if X is semilo c al ly simply c onne cte d. Prop ositio n 3.4. If π ( V , x 0 ) ⊂ H for some op en c over V of X , then t he identity function b X H → e X H is a home omorphism. Pro of. Let us show B H ([ α ] H , U , W ) = B H ([ α ] H , W ) if U is an op en cov er of X refining V a nd W is an element of U co ntaining α (1). C le a rly , B H ([ α ] H , W ) ⊂ B H ([ α ] H , U , W ), s o ass ume [ β ] H ∈ B H ([ 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 ]. Cho ose h 1 ∈ H so that [ h 1 ∗ α ] = [ α ∗ λ ] ( h 1 = [ α ∗ λ ∗ α − 1 ] ∈ π ( U , x 0 ) ⊂ H ). Now [ β ] = [ h ∗ α ∗ λ ∗ γ ] = [ h ∗ h 1 ∗ α ∗ γ ] and [ β ] H ∈ B H ([ α ] H , W ). Now we can show t he iden tit y function b X H → e X H is ope n: given an o pe n co v er W of X and g iven a path α from x 0 to x 1 pick an elemen t W of U = W ∩ V containing x 1 and notice B H ([ α ] H , U , W ) ⊂ B H ([ α ] H , W ).  Lemma 3. 5. If G ⊂ H ar e su b gr oups of π 1 ( X, x 0 ) , then t he pr oje ction p : e X G → e X H is op en. Pro of. It suffices to s how p ( B G ([ α ] G , U , V )) = B H ([ α ] H , U , V ). Clearly , p ( B G ([ α ] G , U , V )) ⊂ B H ([ α ] H , U , V ), so supp ose [ β ] H ∈ B H ([ 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 ).  W e a rrived a t the fundamen tal result for the new top ology on e X H : Theorem 3.6. Su pp ose G ⊂ H ar e sub gr oups of π 1 ( X, x 0 ) . If G is normal in π 1 ( X, x 0 ) , t hen H/ G , identifie d with t he fi b er p − 1 ([ ˜ x 0 ] H ) of t he pr oje ction p : e X G → e X H , is a top olo gic al gr oup and acts c ontinuously on e X G so t hat a) The natura l map ( H /G ) × e X G → e X G × e X G define d by ([ α ] G , [ β ] G ) 7→ ([ α ∗ β ] G , [ β ] G ) is an emb e dding, b) The quotient map fr om e X G to the orbit s p ac e c orr esp onds to the pr oje ction p : e X G → e X H . Pro of. The fiber F of the pro jection p : e X G → e X H is the set of cla sses [ α ] G such that [ α ] ∈ H , so it corre s po nds to H/G . Define µ : F × e X G → e X G as f ollows: given [ α ] G ∈ F and given [ β ] G ∈ e X G put µ ([ α ] G , [ β ] G ) = [ α ∗ β ] G . T o s e e µ is well defined assume [ γ 1 ] , [ γ 2 ] ∈ G . Now [ γ 1 ∗ α ∗ γ 2 ∗ β ] G [( α ∗ γ 2 ∗ α − 1 ) ∗ ( α ∗ β )] G = [ α ∗ β ] G as [ α ∗ γ 2 ∗ α − 1 ] ∈ G due to normality of G in H . Suppo se U is an op en cov er o f X , V , V 1 ∈ U , and 10 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA (1) [ α ] G ∈ F , [ β ] G ∈ e X G , (2) [ α 1 ] G ∈ B G ([ α ] G , U , V 1 ) ∩ F , and [ β 1 ] G ∈ B G ([ β ] G , U , V ). Thu s [ α 1 ] = [ g 1 ∗ α ∗ λ 1 ] for so me [ λ 1 ] ∈ π ( U , x 0 ) a nd [ g 1 ] ∈ G . Similarly , [ β 1 ] = [ g 2 ∗ β ∗ λ 2 ∗ γ ], where [ g 2 ] ∈ G , [ λ 2 ] ∈ π ( U , β (1 )), and γ is a path in V . Now, [ α − 1 1 ∗ β 1 ] G = [ λ − 1 1 ∗ α − 1 ∗ g − 1 1 ∗ g 2 ∗ β ∗ λ 2 ∗ γ ] G = [( λ − 1 1 ∗ α − 1 ∗ g − 1 1 ∗ g 2 ∗ α ∗ λ 1 ) ∗ λ − 1 1 ∗ α − 1 ∗ β ∗ λ 2 ∗ γ ] G = [ λ − 1 1 ∗ α − 1 ∗ β ∗ λ 2 ∗ γ ] G = [( α − 1 ∗ β ) ∗ ( β − 1 ∗ α ∗ λ − 1 1 ∗ α − 1 ∗ β ) ∗ λ 2 ∗ γ ] G ∈ B G ([ α − 1 ∗ β ] G , U , V ) as [ λ − 1 1 ∗ α − 1 ∗ g − 1 1 ∗ g 2 ∗ α ∗ λ 1 ] ∈ G and [ β − 1 ∗ α ∗ λ − 1 1 ∗ α − 1 ∗ β ] ∈ π ( U , ( α − 1 ∗ β )(1)). The ab ov e calculations amoun t to ρ (( F ∩ B G ( x, U , V 1 )) × B G ( y , U , V )) ⊂ B G ( ρ ( x, y ) , U , V ) , where ρ ( x, y ) := µ ( x − 1 , y ), whic h implies the following (1) F is a to po logical group, (2) µ is con tinuous, (3) ( x, y ) → ( µ ( x − 1 , y ) , y ) from F × e X G onto its imag e is op en. As the map in (3) is injective, it is an e mbedding . Hence ( x , y ) → ( µ ( x, y ) , y ) is an embedding. T o see b) use 3.5 or c heck it directly .  4. P a th lifting Definition 4. 1. A pointed map f : ( X , x 0 ) → ( Y , y 0 ) has the path lifting prop- ert y if an y path α : ( I , 0) → ( Y , y 0 ) has a lift β : ( I , 0) → ( X , x 0 ). A surjectiv e map f : X → Y has the path lifti ng prop ert y if fo r any path α : I → Y and any y 0 ∈ f − 1 ( α (0)) there is a lift β : I → X o f α s uch that β (0) = y 0 . Definition 4.2. A p ointed map f : ( X , x 0 ) → ( Y , y 0 ) has the u ni queness of path lifts prop erty if an y t w o paths α, β : ( I , 0) → ( X , x 0 ) are equal if f ◦ α = f ◦ β . A p ointed map f : ( X , x 0 ) → ( Y , y 0 ) has the unique path li fti ng prop ert y if it has bo th the path lifting prop erty and the unique ne s s of path lifts prop erty . A map f : X → Y has the uniqueness of path lifts prop ert y if any t wo pa ths α, β : I → X are equal if f ◦ α = f ◦ β and α (0) = β (0). A surjective map f : X → Y has the uni que path l ifting prop ert y if it ha s bo th the path lifting prop erty and the uniq uene s s of path lifts prop erty . Corollary 4.3. Suppp ose G ⊂ H ar e sub gr oups o f π 1 ( X, x 0 ) . If G is normal in π 1 ( X, x 0 ) , then the fol lowing c onditio ns ar e e quivalent: a) The natur al map e X G → e X H has the un iqueness of p ath lifts pr op erty, b) π 0 ( H/ G ) = H /G , i.e. H /G has t rivial p ath c omp onents. Pro of. a) = ⇒ b). If H /G has a non-tr iv ial path co mpo nent , then there is a non-trivial lift of the constant path at th e base-p o int of e X H . b) = ⇒ a). Suppos e α and β are tw o lifts of the sa me path γ in e X H and α (0) = β (0). B y 3.6 there is a path λ in H /G with the pr op erty λ ( t ) · α ( t ) = β ( t ) for ea ch t ∈ I . As λ (0) = 1 ∈ H /G and H /G has trivia l path comp onents, λ ( t ) = 1 ∈ H /G for all t ∈ I a nd α = β .  CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 11 Prop ositio n 4.4 . Suppp ose G ⊂ H ar e sub gr oups of π 1 ( X, x 0 ) . If G is normal in π 1 ( X, x 0 ) , then the fol lowing c onditio ns ar e e quivalent: a) H /G is a T 0 -sp ac e, b) H/ G is Hausdorff, c) Fib ers of the pr oje ction p : e X G → e X H ar e T 0 , d) Fib ers of the pr oje ction p : e X G → e X H ar e Hausdorff, e) F or e ach h ∈ H − G ther e is a c over U such that ( G · h ) ∩ π ( U , x 0 ) = ∅ , f ) G is close d in H . Pro of. In view of 3 .6, a) ≡ c) and b) ≡ d). a) = ⇒ e). Assume H /G is T 0 and h ∈ H − G . Since [ β ] G ∈ B G ([ α ] G , U , V ) is equiv ale nt to [ α ] G ∈ B G ([ β ] G , U , V ), there is an o p en cover U and V ∈ U containing x 0 such tha t G · h / ∈ B G ( G · 1 , U , V ). That means pre cisely ther e is no λ ∈ π ( U , x 0 ) such that G · h = G · λ , hence ( G · h ) ∩ π ( U , x 0 ) = ∅ . b) ≡ d) and a) ≡ c) follow from 3.6. e) = ⇒ d). Suppo se α, β are tw o paths in ( X , x 0 ) so that [ α ] H = [ β ] H but [ α ] G 6 = [ β ] G . choose h ∈ H − G satisfying [ h · α ] = [ β ]. Pick a n op en cover U of X satisfying G · h ∩ π ( U , x 0 ) = ∅ and let V ∈ U c ontain α (1). Suppo s e [ γ ] G ∈ B G ([ α ] G , U , V ) ∩ B G ([ β ] G , U , V ) and [ γ ] H = [ α ] H . Let h 0 ∈ H satisfy [ h 0 · α ] = [ γ ]. Cho os e λ 1 , λ 2 ∈ π ( U , α (1)) such that G · [ h 0 · α ] = G · α · λ 1 and G · [ h 0 · α ] = G · [ h · α ] · λ 2 . As G is no rmal in H , G · h = h · G = G · ( α · λ 1 · λ − 1 2 α − 1 ), a contradiction a s α · λ 1 · λ − 1 2 · α − 1 ∈ π ( U , x 0 ). b) = ⇒ a) is ob vious. e) ≡ f ). G b eing closed in H means existence, for each h ∈ H − G , of an op en cov er U such that G ∩ B ( h, U , V ) = ∅ for some V ∈ U co nt aining x 0 . That, in turn, is equiv a le n t to non-existence of λ ∈ π ( U , x 0 ) satisfying h · λ ∈ G , i.e . ( G · h − 1 ) ∩ π ( U , x 0 ) = ∅ .  Corollary 4.5. Supp ose G ⊂ H ar e sub gr oups of π 1 ( X, x 0 ) . If G is a normal sub gr oup of π 1 ( X, x 0 ) , then the fol lowing c onditio ns ar e e quivalent: a. H / G has trivial c omp onents, b. H /G has t rivial p ath c omp onents, c. G is close d in H . Pro of. b) = ⇒ c). Suppos e H /G has triv ia l path comp onents. In vie w of 4.4 it suffices to show H / G is T 0 to deduce G is closed in H . If ther e are tw o p oints u and v of H /G such that any ope n subset of H/ G either co nt ains both of them or contains none of them, then any function I → { u, v } ⊂ H/G is contin uous. Hence u = v as H /G do es not contain non-trivial paths. c) = ⇒ a ). Claim. If h 1 , h 2 ∈ H and G · f ∈ B H ( G · h 1 , U , V ) ∩ B H ( G · h 2 , U , V ) ∩ ( H /G ) for some o pen cov er U of X and so me V ∈ U containing x 0 , then G · h − 1 1 · h 2 ⊂ Gπ ( U , x 0 ). Pro of of Claim: G · f = G · h 1 · λ 1 and G · f = G · h 2 · λ 2 for some λ 1 , λ 2 ∈ π ( U , x 0 ). Now h 1 · G = h 2 · G · ( λ 2 · λ − 1 1 ) and ( h − 1 1 · h 2 ) · G ⊂ G · ( λ 1 · λ − 1 2 ) ⊂ Gπ ( U , x 0 ).  Suppo se G is clo sed in H a nd h ∈ H − G . By 4.4 there is a cover U such that ( G · h ) ∩ π ( U , x 0 ) = ∅ . If there is a connected subset C of H/ G containing G · h 1 h and G · h 1 for some h 1 ∈ H , we consider the open co ver { C ∩ B G ( z , U , V ) } z ∈ C of C and define the equiv alence r elation on C determined by that cov er ( z ∼ z ′ if there is a finite chain z = z 1 , . . . , z k = z ′ in C such that B G ( z i , U , V ) ∩ B G ( z i +1 , U , V ) ∩ C 6 = ∅ for a ll i < k ). Equiv alence cla sses of tha t rela tion are op en, hence closed and 12 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA m ust equa l C . Thus there is a finite chain h 1 , . . . , h k = h 1 · h in H s uch that B G ([ h i ] G , U , V ) ∩ B G ([ h i +1 ] G , U , V ) ∩ ( H/ G ) 6 = ∅ for a ll i < k . By Claim there a re elements g i ∈ G ( i < k ) so that g i · h − 1 i · h i +1 ∈ π ( U , x 0 ). B y normality of G in H there is g ∈ G satisfying g · k − 1 Q i =1 h − 1 i · h i +1 = g · h ∈ π ( U , x 0 ), a contradiction.  Theorem 4. 6. If G is a normal sub gr oup of π 1 ( X, x 0 ) , then t he fol lowing c ondi- tions ar e e quivalent: a. The endp oint pr oje ct ion p G : ( e X G , e x 0 ) → ( X , x 0 ) has the unique p ath lif ting pr op erty, b. G is close d in π 1 ( X, x 0 ) , c. π 1 ( p G ) : π 1 ( e X G , e x 0 ) → π 1 ( X, x 0 ) is a monomorphism and its image e quals G . Pro of. Put H = π 1 ( X, x 0 ) and obs e rve e X H is the Peanification of ( X , x 0 ) by 2.16. a) ≡ b). By 4.3 the gro up H/G has trivial path comp onents. Use 4.5. a) = ⇒ c). Given a lo op in ( e X G , e x 0 ) we may a ssume it is a c anonical lift o f a lo op α in ( X, x 0 ). F or that lift to b e a lo o p we m ust hav e [ α ] ∈ G . Thus the image of π 1 ( p G ) : π 1 ( e X G , e x 0 ) → π 1 ( X, x 0 ) equals G (ca nonical lifts of elements of G show that the image contains G ). If α is null-homotopic in ( X , x 0 ), then its canonical lift is null-homotopic as w ell. Th us π 1 ( p G ) : π 1 ( e X G , e x 0 ) → π 1 ( X, x 0 ) is a monomorphism. c) = ⇒ a). If H/ G has a non-trivial path comp onent (we use 4.3), then there is a path from the base-po in t to a diff erent p oint [ α ] G of H /G . Concatenating the canonical lift of α with the rev erse of that path giv es a lo o p in ( e X G , e x 0 ) whose image in π 1 ( X, x 0 ) is [ α ] / ∈ G , a contradiction.  Prop ositio n 4.7. Supp ose ( X , x 0 ) is a p ointe d top olo gic al sp ac e and H is a sub- gr oup of π 1 ( X, x 0 ) . The closur e of H in π 1 ( X, x 0 ) c onsists of al l element s g ∈ π 1 ( X, x 0 ) su ch that for e ach op en c over U of X ther e is h ∈ H and λ ∈ π ( U , x 0 ) satisfying g = h · λ . If H is a normal sub gr oup of π 1 ( X, x 0 ) , then so is its closur e. Pro of. Supp ose g ∈ π 1 ( X, x 0 ) and for each op en cov er U of X there is h ∈ H and λ ∈ π ( U , x 0 ) satisfying g = h · λ . Notice B ( g , U ) contains h , so g belongs to the closure o f H . If H is normal, then k · g · k − 1 = ( k · h · k − 1 ) · ( k · λ · k − 1 ) also belo ngs to the closure of H .  Corollary 4.8. The closur e o f the trivial sub gr oup of π 1 ( X, x 0 ) i n π 1 ( X, x 0 ) e quals T U ∈ C O V π ( U , x 0 ) , wher e C OV stands for the family of al l op en c overs of X . Example 4.9. The Harmo nic Arc hipe la go H A of Bogley and Sierads ki [2] is a Peano space suc h that π 1 ( X, x 0 ) equals T U ∈ C O V π ( U , x 0 ). Hence π 1 ( X, x 0 ) is the only closed subgroup o f π 1 ( X, x 0 ). H A is built by stretching disks B (2 − n , 2 − n − 2 ) to form cones ov er its boundar y with the v ertices at height 1 in the 3 -space. Corollary 4.10. Supp ose ( X , x 0 ) is a p ointe d top olo gic al sp ac e. The fol lowing sub gr oups of π 1 ( X, x 0 ) ar e close d: a) S u b gr oups H c ontaining π ( U , x 0 ) for some op en c over U of X , b) T U ∈ S π ( U , x 0 ) for any family S of op en c overs of X , CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 13 c) The kernel of π 1 ( f ) : π 1 ( X, x 0 ) → π 1 ( Y , y 0 ) for any map f : ( X , x 0 ) → ( Y , y 0 ) t o a p ointe d semilo c al ly simply c onne cte d sp ac e. d) The kernel of t he natur al homomorphism π 1 ( X, x 0 ) → ˇ π 1 ( X, x 0 ) fr om t he fundamental gr oup to t he ˇ Ce ch fundamental gr oup. Pro of. a) An y s ubgroup containing π ( U , x 0 ) is op en. Any op en subgroup of a top ological group is closed. b) easily follows from a). c) follows fr om 3.3 and 3 .1 as π 1 ( f ) : π 1 ( X, x 0 ) → π 1 ( Y , y 0 ) is c ontin uo us and π 1 ( Y , y 0 ) is discrete. d) follows from c). Indeed ˇ π 1 ( X, x 0 ) is defined (see [9] or [19]) as the inverse limit of an inv erse system { π 1 ( K s , k s ) } s ∈ S , where ea ch K s is a s implicial complex and there a re m aps f s : ( X , x 0 ) → ( K s , k s ) so that for t > s the ma p f s is homotopic to the comp osition of f t and the b onding map ( K t , k t ) → ( K s , k s ). That means the kernel of the natura l homomorphis m π 1 ( X, x 0 ) → ˇ π 1 ( X, x 0 ) is the intersection of kernels o f a ll π 1 ( f s ), s ∈ S .  The conce pt of a space X b eing hom otopically Hausdorff was in tro duced by Conner and Lamor eaux [8, Definition 1.1] to mea n that for any point x 0 in X and for any non-homo topically trivia l lo op γ a t x 0 there is a neighbo rho o d U of x 0 in X with the pr op erty that no loo p in U is homo topic to γ rel. x 0 in X . Subsequently , Fischer and Zastr ow [15]) defined a space X to b e homotopicall y Hausdorff relativ e to a subgroup H of π 1 ( X, x 0 ) if f or an y g / ∈ H and for an y path α origina ting at x 0 there is an op en neigh bo rho o d U of α (1) in X such that no element of H · g can b e express ed as [ α ∗ γ ∗ α − 1 ] for some lo op γ in ( U, α (1)). W e g eneralize this definition as follows: Definition 4.11. Supp ose G ⊂ H are subg roups of π 1 ( X, x 0 ). X is ( H , G ) - homotopically Hausdorff if fo r any h ∈ H \ G and any pa th α orig ina ting at x 0 there is an op en neigh bo rho o d U of α (1) in X s uch that no ne o f t he elemen ts of G · h can b e expresse d as [ α ∗ γ ∗ α − 1 ] for any loop γ in ( U, α (1)). Notice X being homo topically Hausdorff relative to H corresp onds to X being ( π 1 ( X, x 0 ) , H )-homotopically Hausdorff. Let us characterize the concept of b e ing ( H , G )-homotopically Hausdorff in terms of the basic topo logy on the fundament al group. Prop ositio n 4.12. If G ⊂ H ar e sub gr oups of π 1 ( X, x 0 ) , then X is ( H , G ) - homotopic al ly Hausdorff if and only if for every p ath α in X that terminates at x 0 the gro up h α ( G ) is close d in h α ( H ) in the b asic t op olo gy. Pro of. h α ( G ) b eing closed in h α ( H ) mea ns existence, for ea ch h ∈ H \ G , o f a neighborho o d U of x 1 = α (0) such that B ([ α ∗ h ∗ α − 1 ] , U ) ∩ ([ α ] · G · [ α − 1 ]) = ∅ . Th us, for every lo op γ in U at x 1 , ther e is no g ∈ G satisfying [ α ∗ h ∗ α − 1 ∗ γ − 1 ] = [ α ∗ g ∗ α − 1 ]. The last equalit y is equiv alent to [ g ∗ h ] = [ α − 1 ∗ γ ∗ α ] whic h completes the proo f.  Example 4.13 . P rop osition 4 .12 allows for an easy construction of subgro ups H of π 1 ( X, x 0 ) such that X is no t homotopica lly Hausdo rff relative to H . Namely , X = S 1 × S 1 × . . . and H = L Z ⊂ Q Z = π 1 ( X ). Let us show G b eing closed in H (in the new top olo gy) is a stronger condition than X b eing ( H , G )-homotopically Hausdor ff. 14 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA Lemma 4.14. Supp ose G ⊂ H ar e su b gr oups of π 1 ( X, x 0 ) . If G is close d in H , then X is ( H, G ) -homotopic al ly Hausdorff. Pro of. Given h ∈ H \ G pick an o pen co ver U and W ∈ U containing x 0 so that B ( h, U , W ) does not intersect G . Given a pa th α in X fr om x 0 to x 1 choose V ∈ U containing x 1 . Supp os e there is a lo op γ in ( V , x 1 ) so that [ α ∗ γ ∗ α − 1 ] = g · h for so me g ∈ G . Now [ α ∗ γ − 1 ∗ α − 1 ] ∈ π ( U , x 0 ) and g − 1 = h ∗ [ α ∗ γ − 1 ∗ α − 1 ] ∈ G ∩ B ( h, U , W ), a contradiction.  R emark 4 .15 . The pro of o f 4.14 suggests that the trivia l subgroup of π 1 ( X, x 0 ) being clo sed is philosophically related to the conce pt of X b eing strongly homo- topically Hausdorff (see [22]). Recall a metric space X is stro ngly homotopically Hausdorff if for any non-null-homotopic loop α in X ther e is an ǫ > 0 suc h that α is not freely homotopic to a loo p of diameter less than ǫ . Lemma 4.16. Given sub gr oups G ⊂ H of π 1 ( X, x 0 ) the fol lowing c onditions ar e e quivalent: a) The fib ers of the natur al pr oje ct ion p : b X G → b X H ar e T 0 , b) The fib ers of t he natu r al pr oje ction p : b X G → b X H ar e Hausdorff, c) X is ( H, G ) -homotopic al ly Hausdorff. Pro of. a) = ⇒ c). Supp ose h ∈ H \ G and α is a path in X from x 0 to x 1 . As [ h ∗ α ] G 6 = [ α ] G belo ng to the same fibe r of p , there is a neighbor ho o d U of x 1 so that [ h ∗ α ] G / ∈ B G ([ α ] G , U ) or [ α ] G / ∈ B G ([ h ∗ α ] G , U ). Notice [ h ∗ α ] G / ∈ B G ([ α ] G , U ) is equiv ale nt to [ α ] G / ∈ B G ([ h ∗ α ] G , U ). Supp ose there is a lo op γ in ( U, x 1 ) s o that g · h = [ α ∗ γ ∗ α − 1 ] for some g ∈ G . No w [ h ∗ α ] G = [ g · h ∗ α ] G = [ α ∗ γ ] G ∈ B G ([ α ] G , U ), a contradiction. c) = ⇒ b). Any t w o different elemen ts of the same fib er of p can be represented as [ h ∗ α ] G 6 = [ α ] G for some path α in X from x 0 to x 1 and some h ∈ H \ G . Cho ose a neig h b orho o d U of x 1 with the pr op erty that none of the elements of G · h ca n be expre s sed as [ α ∗ γ ∗ α − 1 ] for any lo op γ in ( U, x 1 ). Supp ose [ β ] G ∈ ( H/ G ) ∩ B G ([ α ] G , U ) ∩ B G ([ h ∗ α ] G , U ). That means existence of lo o ps γ 1 , γ 2 in ( U, x 1 ) so that [ β ] G = [ h ∗ α ∗ γ 1 ] G = [ α ∗ γ 2 ] G . Hence [ h ] G = [ α ∗ ( γ 2 ∗ γ − 1 1 ) ∗ α − 1 ] G , a contradiction.  Lemma 4.17. Suppp ose G ⊂ H a r e sub gr oups of π 1 ( X, x 0 ) , G is n ormal in π 1 ( X, x 0 ) , and X is ( H , G ) -homotopic al ly Hausdorff. If α, β : ( I , 0) → ( b X G , b x 0 ) ar e t wo c ontinuous lifts of the same p ath γ : ( I , 0) → ( b X H , b x 0 ) , then for every h ∈ H the set S = { t ∈ I | α ( t ) = h · β ( t ) } is close d. Pro of. Cho o se paths u t , v t in ( X , x 0 ) so that α ( t ) = [ u t ] G and β ( t ) = [ v t ] G for all t ∈ I . Assume [ u t ] G 6 = [ h · v t ] G for some t ∈ I . Pick a neig hborho o d U of x 1 = u t (1) so that [ v t ∗ u − 1 t ] · h · G 6 = [ v t ∗ γ ∗ v − 1 t ] · G for a n y lo o p γ in ( U, x 1 ). There is a neighborho o d V of t in I so that [ u s ] G ∈ B G ([ u t ] G , U ) and [ v s ] G ∈ B G ([ v t ] G , U ) for all s ∈ V . That means [ u s ] = [ g 1 ∗ u t ∗ γ 1 ] and [ v s ] = [ g 2 ∗ v t ∗ γ 2 ] for so me g 1 , g 2 ∈ G and so me paths γ 1 , γ 2 in U joining x 1 and u 1 (1) = v s (1). P ut γ = γ 1 ∗ γ − 1 2 and notice [ u s ∗ v − 1 s ] = [ g 1 ∗ u t ∗ v − 1 t ∗ ( v t ∗ γ ∗ v − 1 t ) ∗ g − 1 2 ]. As G is normal in π 1 ( X, x 0 ), there is g 3 ∈ G satisfying [ g 1 ∗ u t ∗ v − 1 t ∗ ( v t ∗ γ ∗ v − 1 t ) ∗ g − 1 2 ] = [ g 3 ∗ u t ∗ v − 1 t ∗ ( v t ∗ γ ∗ v − 1 t )] and that element cannot belong to G · h by the c hoice of U .  CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 15 Corollary 4.18. Suppp ose G ⊂ H ar e sub gr oups of π 1 ( X, x 0 ) . If H /G is c ount- able, G is normal in π 1 ( X, x 0 ) , and X is ( H , G ) -homotopic al ly Hausdorff, then the natur al map b X G → b X H has the uniqueness of p ath lifts pr op erty. Pro of. Pick representativ es h i ∈ H , i ≥ 1, o f all rig ht c osets of H /G so that h 1 = 1 . If α a nd β are t wo co nt inuous lifts in b X G of the sa me path in b X H , then each set S i = { t ∈ I | α ( t ) = h i · β ( t ) } is closed, they are disjoint, and their union is the whole interv al I . Hence only o ne of them is non-empty and it mu st b e S 1 . Thu s α = β .  5. Peano maps This section is about one of the main ingr edient s of o ur theo r y of cov ering maps for lpc- spaces. It amounts to the follo wing generalization of Peano spaces: Definition 5.1. A map f : X → Y is a P eano map if the family of path comp o- nent s of f − 1 ( U ), U op en in Y , forms a basis of neighbor ho o ds of X . Notice X is an lp c- s pace if f : X → Y is a Peano map. One ma y reword the ab ov e definition a s follows: X is a n lp c-space and lifts o f short paths in Y are shor t in X . Indeed, giv en a neig h b orho o d U of x 0 ∈ X there is a neighbor ho o d V of f ( x 0 ) in Y such that any path α in ( f − 1 ( V ) , x 0 ) (i.e. f ◦ α is contained in V , hence short) m ust be c o ntained in U . Prop ositio n 5.2. Any pr o duct of Pe ano maps is a Pe ano map. Pro of. Suppos e f s : X s → Y S , s ∈ S , are Peano maps. Obse r ve X = Q s ∈ S X s is an lpc-space. Given a neig hbo rho o d U of x = { x s } s ∈ S ∈ X , w e find a finite subset T o f S and neighbor ho o ds U s of x s in X s such that Q s ∈ S U s ⊂ U and U s = X s for s / ∈ T . Choose neighborho o ds V s of f s ( x s ) in Y s , s ∈ T , so that the path-comp onent of x s in f − 1 s ( V s ) is co ntained in U s . Put V s = X s for s / ∈ T and obser ve the path comp onent o f x in f − 1 ( V ), f = Q s ∈ S f s and V = Q s ∈ S V s , is contained in U .  Here is our basic class of Peano maps: Prop ositio n 5.3 . If H is a sub gr oup of π 1 ( X, x 0 ) , then the endp oint pr oje ction p H : b X H → X is a Pe ano map. Pro of. It suffices to s how that for any U op en in X the path co mpo nent of any [ α ] H in p − 1 H ( U ) is precis ely B H ([ α ] H , U ). It’s straightforward that B H ([ α ] H , U ) is path-co nnected so supp ose β is a path in p − 1 H ( U ) starting at [ α ] H . W e wish to show that β ([0 , 1 ]) ⊂ B H ([ α ] H , U ) . Let T = { t : β ( t ) ∈ B H ([ α ] H , U ) } . Now T is nonempty since β (0) = [ α ] H and op en a s the inv erse image of a n op en set. It suffices to prov e [0 , t ) ⊂ T implies [0 , t ] ⊂ T . Set β ( t ) = [ b ] H . Now p H β ([0 , 1]) ⊂ U so in particular p H ([ b ] H ) ∈ U. Consider B H ([ b ] H , U ) . There is an ε > 0 such that β ( t − ε, t ] ⊂ B H ([ b ] H , U ) . Pick s ∈ ( t − ε, t ). Then β ( s ) = [ c 1 ] H and [ b ] H = [ b 1 ] H such that c 1 ≃ b 1 ∗ γ 1 for s o me γ 1 with γ 1 [0 , 1] ⊂ U. But β ( s ) ∈ B H ([ α ] H , U ) so β ( s ) = [ c 2 ] H and [ α ] H = [ a 1 ] H such that c 2 ≃ a 1 ∗ γ 2 for some γ 2 with γ 2 ([0 , 1 ]) ⊂ U. Then b ≃ H b 1 ≃ c 1 ∗ γ − 1 1 ≃ H c 2 ∗ γ − 1 1 ≃ a 1 ∗ γ 2 ∗ γ − 1 1 ≃ H a ∗ γ 2 ∗ γ − 1 1 and ( γ 2 ∗ γ − 1 1 )([0 , 1 ]) ⊂ U so [ b ] H ∈ B H ([ α ] H , U ) and t ∈ T . Therefore T = [0 , 1] .  In analo gy to path lifting and unique path lifting pr op erties (see 4.1 and 4.2) one can in tro duce the corres po nding concepts for hedgehogs: 16 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA Definition 5. 4. A surjective map f : X → Y has the hedgehog lifting prop e rt y if for any map α : W s ∈ S I s → Y from a hedgehog and any y 0 ∈ f − 1 ( α (0)) there is a contin uous lift β : W s ∈ S I s → X of α suc h that β (0) = y 0 . Definition 5. 5. f : X → Y has the unique hed g ehog lifti ng prop ert y if it has bo th the hedgehog lifting prop erty and th e uniqueness of path lifts pr o p e rty . Theorem 5.6. If f : X → Y has the unique he dgeh o g lifting pr op erty, t hen f : l pc ( X ) → Y is a Pe ano map. Pro of. Assume U is open in X and x 0 ∈ U . Supp ose for each neigh bo rho o d V of f ( x 0 ) in X there is a path α V : ( I , 0) → ( f − 1 ( V ) , x 0 ) s uch that α V (1) / ∈ U . By 2.9 the wedge W V ∈ S f ◦ α V is a map g from a hedgeho g to Y (here S is the family of all neigh bo rho o ds of f ( x 0 ) in Y ). Its lift m ust be the w edge h = W V ∈ S α V . Ho wev er h − 1 ( U ) is not op en in l pc ( X ), a con tradiction.  Definition 5.7. Given a map f : X → Y of to p o lo gical spac es its P eano m ap P ( f ) : P f ( X ) → Y is f on X equipp ed with the top olo gy gener ated by path com- po nent s of sets f − 1 ( U ), U op en in Y . Notice that in the case of f = i d X the r ange P id X ( X ) of P ( id X ), wher e id X : X → X is the identit y map, is iden tical to l pc ( X ) a s defined in 2.2. Recall f : X → Y is a Hurewicz fibration if every comm utative diagram K × { 0 } α − − − − → X   y   y f K × I H − − − − → Y has a filler G : K × I → X (that means f ◦ G = H and G extends α ). If the ab ov e condition is sa tis fie d for K being any n -cell I n , n ≥ 0 (equiv alently , for any finite po lyhedron K ), then f is called a Se rre fibration . Notice fo r K b eing a p oint this is the classical path l ifting prop ert y . If the a bove condition is sa tisfied for K being any hedgeho g, then f is called a hedgehog fibration . If the ab ov e co ndition is sa tis fied for K b eing a ny Peano space, then f is called a P eano fibration . W e will modify those conce pts for maps betw een po in ted spaces as follows: Definition 5.8. A map f : ( X , x 0 ) → ( Y , y 0 ) is a Se rre 1 -fibration if any co m- m utative diagram ( I × { 0 } , ( 1 2 , 0 )) α − − − − → ( X , x 0 )   y   y f ( I × I , ( 1 2 , 0 )) H − − − − → ( Y , y 0 ) has a filler G : ( I × I , ( 1 2 , 0 )) → ( X , x 0 ) (that means f ◦ G = H and G extends α ). Observe Ser re 1-fibr ations ha ve the path lifting pr op erty in the sense that any path in Y starting at y 0 lifts to a path in X o riginating at x 0 . CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 17 Theorem 5.9. Supp ose ( T , z 0 ) g 1 − − − − → ( X , x 0 ) i   y   y f ( Z, z 0 ) g − − − − → ( Y , y 0 ) is a c ommutative diagr am in the top olo gic al c ate gory such that ( Z, z 0 ) is a Pe ano sp ac e and i is t he inclusion fr om a p ath-c onne cte d subsp ac e T of Z . If f is a Serr e 1 -fi br ation, then ther e is a c ontinuous lift h : ( Z , z 0 ) → ( P f ( X ) , x 0 ) of g extending g 1 if the image of π 1 ( g ) : π 1 ( Z, z 0 ) → π 1 ( Y , y 0 ) is c ontaine d in the image of π 1 ( f ) : π 1 ( X, x 0 ) → π 1 ( Y , y 0 ) . Pro of. F or e ach p oint z ∈ Z pic k a path α z in Z from z 0 to z and let β z be a lift of g : α z 7→ Y . In case of z = z 0 we pick the constant pa ths α z and β z . In case z ∈ T the path α z is contained in T and β z = g 1 ◦ α z . Define h : ( Z , z 0 ) → ( P f ( X ) , x 0 ) by h ( z ) = β z (1). Given a neigh bor ho o d U of g ( z ) in Y , let V b e the path co mpo nent of h ( z ) in f − 1 ( U ) a nd let W b e the path comp onent of g − 1 ( U ) c ontaining z . Our goal is to show h ( W ) ⊂ V a s that is sufficient for h : ( Z, z 0 ) → ( P f ( X ) , x 0 ) to b e contin uous. F or any t ∈ W choose a path µ t in W from z to t . Let γ b e a loo p in X at x 0 so tha t f ( γ ) is ho mo topic to g ( α z ∗ µ t ∗ α − 1 t ). Notice f ( β z ) is homotopic to f ( γ ∗ β t ) via a homotopy H so that H ( { 1 } × I ) ⊂ U . By lifting that homo topy 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 Pe ano map f : X → Y is a Pe ano fibr ation if and only if it is a S erre 1 -fibr ation. Pro of. Assume f : X → Y is a Peano map and a Serre 1-fibration (in the other direction 5.10 is left as an exer cise), g : Z × { 0 } → X is a map from a Peano space, and H : Z × I → Y is a ho motopy starting fro m f ◦ g . P ick z 0 ∈ Z and put x 0 = g ( z 0 , 0 ), y 0 = f ( x 0 ). Notice the image of π 1 ( g ) : π 1 ( Z × { 0 } , ( z 0 , 0 )) → π 1 ( Y , y 0 ) is contained in the image of π 1 ( f ). Use 5.9 to pr o duce an extension G : Z × I → X of g that is a lift of H .  6. P eano co vering maps 5.9 suggests the following concept: Definition 6.1. A map f : X → Y is calle d a Peano co v eri ng map if the fol- lowing conditio ns ar e satisfied: (1) f is a P eano map, (2) f is a Serre fibration, (3) The fib er s o f f have triv ial path components. Notice 3) ab ove can b e r eplaced by f having the unique path lifting prop erty (see 8.3). Also notice that, in case fibers of a P eano map f : X → Y are T 0 spaces, path-comp onents of fibers are triv ia l. Indeed, t w o p oints in a path-compo nent of a fiber a re alwa ys in any open set that con tains one of them. Prop ositio n 6.2. Any pr o duct of Pe ano c overing maps is a Pe ano c overi ng map. Pro of. Supp o se f s : X s → Y S , s ∈ S , are Peano cov ering maps. Put f = Q s ∈ S f s , X = Q s ∈ S X s , and Y = Q s ∈ S Y s . By 5.2 f is a Peano map. It is obvious f is a Se r re fibration and has the uniqueness of path lifting prop er ty .  18 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA Corollary 6. 3 . S upp ose f : ( X , x 0 ) → ( Y , y 0 ) is a Pe ano c overi ng map. If ( Z , z 0 ) is a Pe ano sp ac e, then any map g : ( Z, z 0 ) → ( Y , y 0 ) has a u nique c ontinuous lift h : ( Z, z 0 ) → ( X , x 0 ) if the image of π 1 ( g ) is c ontaine d in the image of π 1 ( f ) . Pro of. By 5 .9 a lift h exists a nd is unique by the uniqueness of path lifting prop erty .  Our basic example of Peano co vering maps is related to the basic top olog y : Theorem 6.4. If X is a p ath-c onne cte d sp ac e and x 0 ∈ X , then the fol lowing c onditions ar e e quivale nt: a. p H : ( b X H , b x 0 ) → ( X , x 0 ) has the unique p ath lifting pr op erty, b. p H : b X H → X is a Pe ano c overing map. Pro of. a ) = ⇒ b). In view of 5.3 and 8.4 it suffices to show p H : ( b X H , b x 0 ) → ( X, x 0 ) is a Serr e fibration. Supp ose f : ( Z , z 0 ) → ( X, x 0 ) is a map from a s imply connected Peano spac e Z (the case of Z = I n is of interest here). Ther e is a standar d lift g : ( Z , z 0 ) → b X H of f defined as g ( z ) = [ α z ] H , wher e α z is a path in Z from z 0 to z . If T is a path-co nnected subspace of Z con taining z 0 and h : ( T , z 0 ) → ( b X H , b x 0 ) is any c o ntin uo us lift of f | T , then h = g | T due to the uniqueness o f the path lifting prop erty o f p H . That prov es p H is a Serre fibration in view of 8.4. b) = ⇒ a) is ob vious.  Theorem 6.5. If f : X → Y is a map and X is an lp c-sp ac e, then the fol lowing c onditions ar e e quivale nt: a) f is a Pe ano c overing map, b) f is a Pe ano fibr ation and has the uniqueness of p ath lifting pr op erty, c) f is a he dgeho g fibr ation and has t he u n iqueness of p ath lifting pr op erty, d) F or any x 0 ∈ X and any map g : ( Z , z 0 ) → ( Y , f ( x 0 )) fr om a simply- c onne cte d Pe ano sp ac e ther e is a lift h : ( Z , z 0 ) → ( X , x 0 ) of g and that lift is un ique. Pro of. a) = ⇒ b). Supp ose H : Z × I → Y is a homoto py , Z is a Peano space, and G : Z × { 0 } → X is a lift of H | Z × { 0 } . Pick z 0 ∈ Z , put x 0 = G ( z 0 , 0 ) a nd y 0 = f ( x 0 ), and notice im ( π 1 ( Z × I , ( z 0 , 0 ))) ⊂ im ( π 1 ( f )). Using 5.9 there is a lift of H and that lift is unique, hence it ag rees with G on Z × { 0 } . b) = ⇒ c) is obvious. d) = ⇒ c) is obvious. a) = ⇒ b) fo llows from 5.9. c) = ⇒ a). Notice f has the unique hedgehog lifting prop er ty and is a Serre 1-fibration. By 5.6 f is a Peano ma p.  Corollary 6 . 6. Su pp ose f : X → Y and g : Y → Z ar e maps of p ath-c onne cte d sp ac es and Y is a Pe ano sp ac e. If any two of f , g , h = g ◦ f ar e Pe ano c overing maps, then so is the thir d pr ovide d its domai n is an lp c-sp ac e. Pro of. In view of 6.5 it amounts to verifying that the map has uniqueness o f lifts of simply-connected Peano s paces, an easy exercise.  Prop ositio n 6.7. Supp ose f : X → Y is a map. a. If f : X → Y i s a Pe ano c overing map, then f : f − 1 ( U ) → U is a Pe ano c overing map for every op en subset U of Y . CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 19 b. If every p oint y ∈ Y has a n eighb orho o d U su ch that f : f − 1 ( U ) → U is a Pe ano c overing map, then f is a Pe ano c overing map. Pro of. a ). f : f − 1 ( U ) → U is clearly a Peano map, is a fibration, and ha s the unique path lifting prop erty . b). f is a Ser re 1-fibration and path comp onents of fibers ar e trivial. If V is a n op en subse t of Y containing y we pick an o pe n subset U of X con taining f ( y ) such that f : f − 1 ( U ) → U is a Peano co v ering map. There is a n op en neighborho o d W of f ( y ) in U so that the pa th comp onent of y in f − 1 ( W ) is ope n and is con tained in V ∩ f − 1 ( U ). Tha t prov es f : Y → X is a Peano map.  In analog y to regular classica l cov ering maps let us introduce r egular Peano cov ering ma ps: Definition 6.8. A Peano covering map f : X → Y is regu l ar if lifts of lo o ps in Y are either alwa ys lo o ps of are alw ays no n-lo ops. Corollary 6.9. Given a map f : X → Y the fol lowing c onditions ar e e quivalent if X is p ath-c onne cte d: a) f is a r e gular Pe ano c overing m ap, b) f is a Pe ano c overing map and the image of π 1 ( f ) is a n ormal sub gr oup of π 1 ( Y , f ( x 0 )) for al l x 0 ∈ X , c) f : X → Y is a gener alize d c overing map in the sen s e of Fischer-Zastr ow. Pro of. a) = ⇒ b). If the image of π 1 ( f ) is not a normal subgro up o f π 1 ( Y , f ( x 0 )) for s ome x 0 ∈ X , then there is a lo op α in Y at y 0 = f ( x 0 ) tha t lifts to a lo op in X at x 0 and there is a lo op β in Y at y 0 such that β ∗ α ∗ β − 1 do es not lift to a lo op in X at x 0 . Let γ be a lift of α orig inating at x 0 . Let x 1 = β (1). Notice the lift of α origina ting at x 1 cannot b e a lo op, a contradiction. b) = ⇒ c). As im ( π 1 ( f )) is a normal subgroup H of π 1 ( Y , y 0 ), it do es nor dep end on the choice of the base-p oint of X in f − 1 ( y 0 ). Using 5.9 one gets f is a gener a lized cov ering ma p. c) = ⇒ a ). Since ea ch hedgehog is con tractible, f has the unique hedgehog lifting prop erty and is a Peano map by 5.6. It is also a Serre fibration, hence a P eano cov ering map. Also, as im ( π 1 ( f )) is a normal subgroup H of π 1 ( Y , y 0 ), it does nor depe nd on the choice of the bas e-p oint of X in f − 1 ( y 0 ). Hence a lo op in Y lifts to a lo o p in X if and only if it r epresents an elemen t o f H . Th us f is a regular Peano cov ering ma p.  In the remainder of this sectio n we will discuss the rela tion of Peano cov ering maps to classical cov ering maps. Prop ositio n 6.10. If f : Y → X is a Pe ano c overing ma p a nd U is an op en su bset of X su ch t hat every lo op in U is nul l-homotopic in X , then f − 1 ( V ) → P ( V ) is a a t rivial discr ete bun d le for every p ath c omp onent V of U . Pro of. Consider a path comp onent 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 equiv ale n t to P ( V ) → V .  Corollary 6.11. If X is a semilo c al ly simply c onne cte d Pe ano sp ac e, then f : Y → X is a Pe ano c overing map i f and o nly if it is a classic al c overing map and Y is c onne cte d. 20 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA Pro of. If f is a classical cov ering map and Y is connected, then Y is lo ca lly path-connected, f has unique path lif ting pr op erty and is a Serre 1 -fibration. Th us it is a P eano cov ering map. Suppo se f is a Peano cov ering map and x ∈ X . Cho ose a path-connected neighborho o d U of x in X such that any lo op in U is n ull- homotopic in X . By 6.10 U is ev enly cov ered b y f .  Corollary 6. 1 2. If f : Y → P ( X ) is a classic al c overing map, t hen f : Y → X is a Pe ano c overing map. Pro of. By 6.7, f : Y → P ( X ) is a Peano cov ering map. As the identit y function induces a P eano cov ering map P ( X ) → X , f : Y → X is a Peano cov e ring map b y 6.6.  Prop ositio n 6.13 . If f : Y → X is a Pe ano c overing map and X is p ath-c onne cte d, then al l fib ers of f have the s ame c ar dinality. Pro of. Given tw o po int s x 1 , x 2 ∈ X fix a path α from x 1 to x 2 and notice lifts of α establish bijectivit y of fib ers f − 1 ( x 1 ) and f − 1 ( x 2 ).  The following result has its origins in Lemma 2.3 of [8] and Pro po sition 6.6 o f [15]. Prop ositio n 6.1 4 . Su pp ose f : Y → X is a re gular Pe ano c overing map. If f − 1 ( x 0 ) is c ountable and x 0 has a c ountable b asis of neighb orho o ds in X , then ther e is a neighb orho o d U of x 0 in X such that f − 1 ( V ) → P ( V ) is a classic al c overing map, wher e V is the p ath c omp onent of x 0 in U . Pro of. Switc h to X being Peano b y co nsidering f : Y → P ( X ). Notice x 0 has a co unt able basis of n eighborho o ds and f is op en. Supp ose there is no op en subset U of X con taining x 0 such tha t U is ev enly cov ered. Tha t means path comp onents of f − 1 ( U ) are not mapp ed bijectively ont o their images. First, we plan to s how there is a neighbo rho o d U of x 0 in X such that the image of π 1 ( U, x 0 ) → π 1 ( X, x 0 ) is contained in the image of π 1 ( f ) : π 1 ( Y , y 0 ) → π 1 ( X, x 0 ). In particular, there is a lift of P ( U, x 0 ) → ( Y , y 0 ) of the inclusion induced map P ( U, x 0 ) → ( X , x 0 ). Suppo se no such U exists. By inductio n we will find a ba sis o f neighbor ho o ds { U i } of x 0 in X and elements [ α i ] ∈ π 1 ( U i , x 0 ) that ar e not contained in the image of π 1 ( U i +1 , x 0 ) → π 1 ( X, x 0 ) and who se lifts a re not lo o ps and end at p oints y i such that y i 6 = y j if i 6 = j . Given a neighborho o d U i pick a lo op α i in ( U i , x 0 ) whose lift (as a path) in ( Y , y 0 ) is not a loo p and ends at y i 6 = y 0 . There is a neighborho o d U i +1 of x 0 in U i such that the no path comp onents of f − 1 ( U i +1 ) contains both y 0 and some y j , j ≤ i . Pick a lo o p α i +1 in ( U i +1 , x 0 ) whose lift is not a lo op. As in [21] o ne can crea te infinite co ncatenations α i (1) ∗ . . . ∗ α i ( k ) ∗ . . . for any increasing sequence { i ( k ) } k ≥ 1 . By lo ok ing at lifts of those infinite conca tenations, there a re t wo different infinite concatenations α i (1) ∗ . . . ∗ α i ( k ) ∗ . . . and α j (1) ∗ . . . ∗ α j ( k ) ∗ . . . whose lifts end at the same point y ∈ f − 1 ( x 0 ). Pick the smalles t k 0 so that i ( k 0 ) 6 = j ( k 0 ). W e may ass ume i ( k 0 ) < j ( k 0 ) and conclude ther e are lo ops β in ( U k 0 +1 , x 0 ) and γ in ( Y , y 0 ) so that α i ( k 0 ) ∼ f ( γ ) ∗ β i which case the lift of α i ( k 0 ) in ( Y , y 0 ) ends in t he pa th comp onent o f f − 1 ( U i ( k 0 )+1 ) containing y 0 , a con tradiction. As f is a regular Peano co vering map, we can find lifts ( U, x 0 ) → ( Y , y ) o f the inclusion map ( U, x 0 ) → ( X , x 0 ) for any y ∈ f − 1 ( x 0 ).  CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 21 7. P eano subgroups Definition 7.1. Supp ose ( X , x 0 ) is a pointed path-connected space. A subgro up H of π 1 ( X, x 0 ) is a Peano subgroup of π 1 ( X, x 0 ) if there is a Peano co vering map f : Y → X such that H is the image of π 1 ( f ) : π 1 ( Y , y 0 ) → π 1 ( X, x 0 ) f or some y 0 ∈ f − 1 ( x 0 ). Prop ositio n 7.2. If H is a Pe ano su b gr oup of π 1 ( X, x 0 ) , then X is homotopic al ly Hausdorff r elative to H . In p articular, H is close d in π 1 ( X, x 0 ) e quipp e d with the b asic top olo gy. Pro of. Cho ose a Peano cov ering map f : Y → X so that im ( π 1 ( f )) = H for some y 0 ∈ f − 1 ( x 0 ). If g ∈ π 1 ( X, x 0 ) \ H and α is a pa th in X from x 0 to x 1 , then lifts of α and g · α end in tw o different p oints y 1 and y 2 of the fib er f − 1 ( x 1 ) and there is a neig hborho o d U of x 1 in X such that no path comp onent of f − 1 ( U ) contains b o th y 1 and y 2 . Supp ose there is a loo p γ in ( U, x 1 ) with the pr op erty [ α ∗ γ ∗ α − 1 ] ∈ H · g . In that cas e the lifts of b oth α ∗ γ and g · α end at y 2 . Since the lift o f α ends in the same pa th comp onent of f − 1 ( U ) a s the lift of α ∗ γ , b oth y 1 and y 2 belo ng to the same comp onent of f − 1 ( U ), a contradiction. Use 4.12 t o conclude H is closed in π 1 ( X, x 0 ) equipped with the basic topolog y .  R emark 7.3 . In case of H b eing the trivial subgroup, L e mma 2.10 of [1 5] se e ms to imply that X is homotopically Hausdorff but the pro of of it is v alid only in a sp ecial case. Prop ositio n 7.4. If H is a Pe ano sub gr oup of π 1 ( X, x 0 ) , then a ny c onjugate of H is a Pe ano sub gr oup of π 1 ( X, x 0 ) . Pro of. Cho ose a Peano cov ering map f : Y → X so that im ( π 1 ( f )) = H for some y 0 ∈ f − 1 ( x 0 ). Suppo se G = g · H · g − 1 and ch o ose a lo o p α in ( X , x 0 ) representing g − 1 . Let β b e a pa th in ( Y , y 0 ) that is the lift of α . Put y 1 = β (1) and notice the image of π 1 ( f ) : π 1 ( Y , y 1 ) → π 1 ( X, x 0 ) is G .  Prop ositio n 7.5. Supp ose ( X, x 0 ) i s a p ointe d p ath-c onne cte d top olo gic al sp ac e. If f : ( Y , y 0 ) → ( X , x 0 ) is a Pe ano c overing map with image of π 1 ( f ) e qual H , then f is e quivalent to the pr oje ction p H : b X H → X . Pro of. Define h : ( b X H , b x 0 ) → ( Y , y 0 ) by choosing a lift b α of every path α in X starting a t x 0 and declaring h ([ α ] H ) = b α (1). Note h is a bijection. Given y 1 = b α (1) and g iven a neighbo rho o d U o f y 1 in Y cho ose a neighbo rho o d V of f ( y 1 ) = α (1) in X so that the path comp onent of f − 1 ( V ) cont aining y 1 is a subset o f U . Observe B H ([ α ] H , V ) ⊂ h − 1 ( U ) which prov es h is contin uous. Conv ersely , given a neig hborho o d W of α (1) in X the image h ( B H ([ α ] H , W )) of B H ([ α ] H , W ) e q uals the path comp onent o f b α (1) in f − 1 ( W ) and is open in Y .  Theorem 7.6. If X is a p ath-c onne cte d sp ac e, x 0 ∈ X , and H is a sub gr oup of π 1 ( X, x 0 ) , then the fol lowing c onditio ns ar e e quivalent: a. H is a Pe ano sub gr ou p of π 1 ( X, x 0 ) , b. The endp oint pr oje ction p H : ( b X H , b x 0 ) → X is a Pe ano c overing map, c. The image of π 1 ( p H ) : π 1 ( b X H , b x 0 ) → π 1 ( X, x 0 ) is c ontaine d in H , d. p H : ( b X H , b x 0 ) → ( X , x 0 ) has the unique p ath lifting pr op erty. 22 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA Pro of. c) ≡ d) is done in 2.18. b) ≡ d) is cont ained in 6.4. a) = ⇒ b) fo llows from 7.5. b) = ⇒ a) holds as c) implies the image of π 1 ( p H ) is H .  Let us state a straightforward consequence of 7.6: Corollary 7. 7. If X is a p ath-c onne cte d sp ac e and x 0 ∈ X , t hen t he fol lowing c onditions ar e e quivale nt: a. The endp oint pr oje ction p : b X → X is a Pe ano c overing map, b. π 1 ( p ) : π 1 ( b X , b x 0 ) → π 1 ( X, x 0 ) is trivial, c. b X is simply c onne cte d, d. p : ( b X , b x 0 ) → ( X , x 0 ) has the unique p ath lifting pr op erty. Corollary 7.8. Close d and normal su b gr oups of π 1 ( X, x 0 ) ar e Pe ano su b gr oups of π 1 ( X, x 0 ) . Pro of. By 4.6 the endpo int pro jection p H : ( e X H , e x 0 ) → X has unique path lifting prop er ty . Since p H : ( b X H , b x 0 ) → X has path lifting pr op erty , this implies p H : ( b X H , b x 0 ) → X ha s the unique path lifting prop erty .  Corollary 7.9. If H ( s ) is a Pe ano sub gr oup of π 1 ( X, x 0 ) for e ach s ∈ S , then G = T s ∈ S H ( s ) is a Pe ano sub gr oup of π 1 ( X, x 0 ) . Pro of. The pr o jection p G : ( b X G , b x 0 ) → ( X , x 0 ) factors through p H ( s ) : ( b X H ( s ) , b x 0 ) → ( X , x 0 ) for each s ∈ S . Ther efore im ( π 1 ( p G )) ⊂ T s ∈ S H ( s ) = G and 6.4 (in conjunction with 2.18) says G is a P eano subgroup of π 1 ( X, x 0 ).  Corollary 7.10. F or e ach p ath-c onne cte d sp ac e X ther e is a universal Pe ano c ov- ering map p : Y → X . Thus, for e ach Pe ano c overing map q : Z → X and any p oints z 0 ∈ Z and y 0 ∈ Y satisfying q ( z 0 ) = p ( y 0 ) , ther e is a Pe ano c overing map r : Y → Z so that r ( y 0 ) = z 0 . Mor e over, the image of π 1 ( Y ) is normal in π 1 ( X ) . Pro of. L e t H b e the in tersection of all Peano subgro ups of π 1 ( X, x 0 ) by 7.9 and 7.4 it is a no rmal Peano subg roup of π 1 ( X, x 0 ). Put Y = b X H and use 6.3.  It would be of in terest to characterize path-co nnec ted s paces X admitting a universal Peano cov ering that is simply co nnected (that a mounts to b X b eing simply connected). Here is an equiv a lent pro blem: Problem 7.11 . Char acterize p ath-c onne cte d sp ac es X so that t he trivial gr oup is a Pe ano su b gr oup of π 1 ( X, x 0 ) . So far the following classes of spaces belong to that catego ry: (1) Any pro duct of spaces admitting simply connected Peano co v er (see 6.2). (2) Subsets of closed surfaces: it is pr ov ed in [1 4] that if X is a n y s ubs et of a closed surface, then π 1 ( X, x 0 ) → ˇ π 1 ( X, x 0 ) is injectiv e. (3) 1- dimensional, compa ct and Hausdorff, or 1 - dimensional, separa ble and metrizable: π 1 ( X, x 0 ) → ˇ π 1 ( X, x 0 ) is injective b y [11, Corollary 1.2 and Final Remar k]. It is shown in [10] (see pro of of Theo r em 1.4) that the pro jection b X → X has the uniqueness of path- lifting prope r ty if X is 1- dimensional and metriz a ble. See [6] for results on the fundamental group of 1-dimensiona l spaces. CO VERING MAPS FOR LOCALL Y P A TH-CONNECTED SP A CES 23 (4) T rees of manifolds: If X is the limit of an in verse system of closed PL- manifolds o f some fixed dimens ion, whose cons ecutive terms are obta ined by connect summing with c losed PL-manifolds, which in turn ar e trivializ ed by the b onding maps, then X is called a tree of manifolds. E very tree of manifolds is path-connected and lo cally path-connected, but it need not b e semilo cally simplyconnected at an y one of its p oints. T rees o f manifolds arise as b oundaries of cer tain Coxeter gr oups and as bounda ries of certain negatively curved geo desic spaces [13]. It is shown in [13] tha t if X is a tree of manifolds (with a cer tain densenes s of the a ttachmen ts in the case of surfaces), then π 1 ( X, x 0 ) → ˇ π 1 ( X, x 0 ) is injectiv e. Notice Example 2.7 in [15] gives X so that p : b X → X does not ha ve the unique path lifting prop erty (one can construct a simpler example with X being the Har - monic Archipelago). How ev er, X is not homotopically Hausdorff. Problem 7.12. Is ther e a homotopi c al ly H ausdorff sp ac e X such that p : b X → X do es not have the uniqueness of p ath lifting pr op erty? Corollary 7.13. Supp ose H is a normal sub gr oup o f π 1 ( X, x 0 ) . If ther e is a Pe ano sub gr oup G of π 1 ( X, x 0 ) c ontaining H such that G/H is c ountable, t hen H is a Pe ano sub gr oup of π 1 ( X, x 0 ) if and only if X is homotopic al ly Hausdorff r elative to H . Pro of. By 7.2, X is homotopically Hausdorff relative to H if H is a Peano subgroup of π 1 ( X, x 0 ). Suppo se X is homotopically Hausdorff relative to H . Given t wo lifts in b X H of the same pa th in X , their co mpo s ition with b X H → b X G are the same by 7.6. By 4.18 the t w o lifts a re identical and 7 .6 says H is a Peano subgr oup of π 1 ( X, x 0 ).  Corollary 7.14. Supp ose H is a normal sub gr oup of π 1 ( X, x 0 ) . If π 1 ( X, x 0 ) /H is c oun t able, then H is a Pe ano sub gr oup of π 1 ( X, x 0 ) if and only if X is homotopic al ly Hausdorff rel ative t o H . 8. A ppendix: Pointed versus un pointed In this section we dis cuss relations betw een p ointed a nd unpo int ed lifting prop- erties. Lemma 8. 1. If f : ( X , x 0 ) → ( Y , y 0 ) has the uniqueness of p ath lifts pr op ert y and X is p ath-c onne cte d, then f : X → Y has the uniquen ess of p ath lifts pr op ert y. Pro of. Given t wo pa ths α a nd β in X orig inating at the same p oint a nd satisfying f ◦ α = f ◦ β , choo se a path γ in X from x 0 to α (0). Now f ◦ ( γ ∗ α ) = f ◦ ( γ ∗ β ), so γ ∗ α = γ ∗ β a nd α = β .  Lemma 8.2. If f : ( X , x 0 ) → ( Y , y 0 ) has the unique p ath lifting pr op erty and X is p ath-c onne cte d, t hen f : X → Y has t he unique p ath lifting pr op erty. Pro of. In view of 8.2 it suffices to show f : X → Y is surjective and has the path lifting pro per ty . If y 1 ∈ Y , we pick a path α from y 0 to y 1 and lift it to ( X , x 0 ). The endp oint o f the lift maps to y 1 , hence f is sur jective. Supp ose α is a path in Y and f ( x 1 ) = α (0). Cho ose a pa th β in X from x 0 to x 1 and lift ( f ◦ β ) ∗ α to a path γ in ( X , x 0 ). Due to the uniqueness of path lifts prop erty of f : ( X, x 0 ) → ( Y , y 0 ) 24 N. BRODSKIY, J. DYD AK, B. LABUZ, AND A. MITRA one has γ ( t ) = β (2 t ) fo r t ≤ 1 2 . Hence γ ( 1 2 ) = x 1 and λ defined as λ ( t ) = γ ( 1 2 + t 2 ) for t ∈ I is a lift of α originating from x 1 .  Lemma 8. 3 (Lemma 15.1 in [17]) . If f : X → Y is a Serr e 1 - fibr ation, then f has the unique p ath lifting pr op erty if and only if p ath c omp onents of fib ers of f ar e trivial. Pro of. Supp ose the fibers of f have trivial path compo nent s a nd α, β a re tw o lifts o f the same path in Y that orig inate at x 1 ∈ X . Let H : I × I → Y b e the standard homotopy from f ◦ ( α − 1 ∗ β ) to the consta nt path at f ( x 1 ). There is a lift G : I × I → X o f H starting fro m α − 1 ∗ β . As pa th comp onents of f are trivia l, α = β due to the wa y t he standar d homoto p y H is defined.  Lemma 8.4. Su pp ose n ≥ 1 . If f : ( X , x 0 ) → ( Y , y 0 ) is a Serr e n - fibr ation, b oth X and Y ar e p ath-c onne ct e d, and f has the uniqueness of p ath lifts pr op erty, then f : X → Y is a S err e n -fibr ation. Pro of. Suppo s e H : I n × I → Y is a homoto p y and G : I n × { 0 } → X is its partial lift. Cho os e a path α in X from x 0 to G ( b, 0), where b is the center o f I n . 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Munkres, T op olo gy , Pren tice Hall , Upper Saddle Ri ve r, NJ 2000. [21] J.P a wliko wski , The fundamental gr oup of a co mp act metric sp ac e , Pro ceedings of the Amer- ican Mathematical So ciety , 12 6 (1998), 3083–3087. [22] D.Repov ˇ s and A.Zastrow, Shape injectivity is not implied by being strongly homotopically Hausdorff, pr eprint . [23] E. Spanier, Algebr aic top olo gy , McGra w-Hill , New Y ork 1966. University of Tennessee, Knoxville, TN 37996 E-mail addr ess : bro dskiy@@mat h.utk.edu University of Tennessee, Knoxville, TN 37996 E-mail addr ess : dyd ak@@math.u tk.edu University of Tennessee, Knoxville, TN 37996 E-mail addr ess : lab uz@@math.u tk.edu University of Tennessee, Knoxville, TN 37996 E-mail addr ess : ajm itra@@math .utk.edu