On asymptotic extension dimension

ON ASYMPTOTIC EXTENSION DIMENSION DU ˇ SAN REPOV ˇ S AND MYKHA ILO ZARICHNYI Abstract. The aim of this pap er is to i n troduce an asymptotic coun terpart of the exten- sion dimension define d by Dranishnik o v. The main result establishes a relation betw een the asymptotic extensional dimension of a proper metric space and extension dimension of its Higson corona. 1. Introduction Asymptotic dimension defined by Gromov [7] has be e n a n o b ject of study in numerous publications (see exp ositor y pape r [2]). A metric spac e ( X , d ) is of a symptotic dimensio n ≤ n (written a sdim X ≤ n ) if for every D > 0 there ex ists a uniformly b ounded cov er U of X suc h that U = U 0 ∪ · · · ∪ U n , wher e ev ery family U i is D -disjoint, i = 0 , 1 , . . . , n . Recall that a family A of subsets of X is uniformly b ounde d if mesh A = sup { dia m A | A ∈ A} < ∞ and is ca lled D - disjoint if inf { d ( a, a ′ ) | a ∈ A, a ′ ∈ A ′ } > D , for every distinct A, A ′ ∈ A . The a symptotic dimension can b e c haracterized in different terms; in particular, in terms of extension of maps in to euclidean spaces. The aim of this pa p er is to in tro duce an a symptotic analogue of the extensio n dimension introduced by Dranishniko v in [3, 4]. 2. Preliminaries A typical metr ic is denoted by d . By N r ( x ) we denote the op en ball of ra dius r cen tered at a p oint x of a metric s pace. 2.1. Asymptotic category . A map f : X → Y betw een metric spa ces is called ( λ, ε )- Lipschitz for λ > 0, ε ≥ 0 if d ( f ( x ) , f ( x ′ )) ≤ λd ( x, x ′ ) + ε for every x, x ′ ∈ X . A ma p is called asymptotic al ly Lipschitz if it is ( λ, ε )- L ipschitz for so me λ, ε > 0. The ( λ, 0 )-Lipschitz maps are also called λ -Lipschitz , (1 , 0)-Lips chitz maps are also called short . A metr ic space X is called pr op er if every closed ball in X is compact. The asymptotic c ate gory A was in tro duced by Dranishniko v [4]. The o b jects of A are prop er metric spaces and the mor phisms a re prop er asymptotically Lipschitz maps. Recall that a map is called pr op er if the pr eimage of every compa ct set is compact. W e also need the notion of a c o arse map . A map betw een prop er metric spaces is called c o arse uniform if for ev ery C > 0 ther e is K > 0 such that for every x, x ′ ∈ X with d ( x, x ′ ) < C we hav e d ( f ( x ) , f ( x ′ )) < K . A ma p f : X → Y is ca lled m etric pr op er if the preimage f − 1 ( B ) Date : Nov ember 21, 2018. 2010 Mathematics Subje c t Classific ation. Prim ary: 54F45, 54C40; Secondary: 55M10. Key wor ds and phr ases. Asymptotic category , asymptotic extension dimension, Higson compactification, Higson corona, coarse uniform m ap, coarse CW complex, pr op er asimptotically Lipschitz map, sl owly oscil- lating map, almost geo decis space, Lipschitz neigh borho o d euclidean extensor, foliated warped cone. This research was suppor ted by the Slov enian Research Agency gran ts P1-0292-0101 and J1-2057-0101. 1 2 DU ˇ SAN RE POV ˇ S AND MYKHAILO ZARICHNYI is bo unded for every b ounded set B ⊂ Y . A map is c o arse if it is metric pr op er a nd coar se uniform. 2.2. Hi gson com pactification and Higs on corona. Let ϕ : X → R b e a function defined on a metric space X . F or every x ∈ X and every r > 0 let V r ϕ ( x ) = sup {| ϕ ( y ) − ϕ ( x ) | | y ∈ N r ( x ) } . A function ϕ is ca lled slow ly oscil lating whenever for every r > 0 we hav e V r ϕ ( x ) → 0 as x → ∞ (the latter means tha t for ev ery ε > 0 there exists a compact s ubs pa ce K ⊂ X such tha t | V r ϕ ( x ) | < ε for all x ∈ X \ K . Let ¯ X b e the compactifica tion of X that c o rresp o nds to the family o f a ll contin uous b ounded slowly oscilla tion functions. The Higson c or ona of X is the r emainder ν X = ¯ X \ X of this compactification. It is w ell- known that the Higs on corona is a functor from the category of pr op er metr ic space a nd coarse maps into the catego ry of compact Hausdorff s paces. In particular, if X ⊂ Y , then ν X ⊂ ν Y . F o r any subset A of X we denote b y A ′ its trace o n ν X , i. e. the intersection o f the clo s ure of A in ¯ X with ν X . Obviously , the se t A ′ coincides with the Higs on co rona ν A . 2.3. Cones . Let X be a metric space of diameter ≤ 1. The op en c one of X is the s et O X = ( X × R + ) / ( X × { 0 } ) endow ed with the metric (b y [ x, t ] w e denote the equiv alence class of ( x, t ) ∈ X × R + ): d ([ x 1 , t 1 ] , [ x 2 , t 2 ]) = | t 1 − t 2 | + min { t 1 , t 2 } d ( x 1 , x 2 ) . F o r a map f : X → Y of metric spaces we denote by O f : O X → O Y the map defined as O f ([ x, t ]) = [ f ( x ) , t ]. Prop ositi on 2.1 . If f : X → Y i s a Lipschi tz map than O f is an asymptotic al ly Lipschitz map. Pr o of. Supp ose a map f : X → Y is λ -Lipschitz. Then for a ny [ x 1 , t 1 ] , [ x 2 , t 2 ] ∈ O X we hav e d ( O f ([ x 1 , t 1 ]) , O f ([ x 2 , t 2 ])) = d ([ f ( x 1 ) , t 1 ] , [ f ( x 2 ) , t 2 ]) = | t 1 − t 2 | + min { t 1 , t 2 } d ( f ( x 1 ) , f ( x 2 )) ≤ λ ′ ( | t 1 − t 2 | + min { t 1 , t 2 } d ( x 1 , x 2 )) , where λ ′ = max { λ, 1 } .  The op en cone of a finite CW-complex is a coarse CW-complex in the sens e of [8]. Denote by α L : O L → R the function defined b y α L ([ x, t ]) = t . O bviously , α L is a short function. Let ˜ O L = { [ x, t ] ∈ O L | t ≥ 1 } . Denote by β L : ˜ O L → L the map β L ([ x, t ]) = x . Lemma 2.2 . The map β L is slow ly oscil lating. Pr o of. F or R > 0 , the R - ba ll centered at [ x, 0] is { [ x, t ] | t < R . If d ([ x, t ] , [ x 1 , t 1 ]) < K < R , then | t − t 1 | + min { t, t 1 } d ( x, x 1 ) < K , i.e. ( t − R ) d ( x, x 1 ) < R and d ( x, x 1 ) < K/ ( t − K ). Therefore, d ( β L ( x ) , β L ( x 1 )) < K/ ( R − K ) → 0 as R → ∞ .  Let ¯ β L : ˜ O L → L b e the (unique) extension of the map β L . Denote by η L : ν ˜ O L → L the restriction o f β L . Prop ositi on 2 .3. L et f : A → O L b e a pr op er asymptotic al ly Lipsch itz m ap define d on a pr op er close d subset A of a pr op er m etric sp ac e X . Ther e ex ists a neighb or go o d W of A in X , a pr op er asymptotic al ly Lipschi tz map g : W → O L with the fol lowing pr op erty: ther e exist c onstants λ, s > 0 such that α L ( g ( a )) ≤ λd ( a, X \ W ) + s . ON ASYMPTOTIC EXTENSION DIM E NSION 3 Pr o of. W e ma y assume that L is a s ubset of I n for some n and there exists a Lipsc hitz retraction r : U → L of a neighborho od U of L in I n . Since O I n is Lipschitz equiv a lent to R n +1 + , there ex is ts a ( λ ′ , s ′ )-Lipschitz extension ˜ g : X → O I n of g . Put W = ˜ g − 1 ( O U ) a nd ¯ g = ˜ g | W . F o r every a ∈ A and w ∈ X \ W we hav e d ( g ( a ) , ˜ g ( w ) ≤ λ ′ d ( a, w ) + s ′ ≤ λ ′ d ( a, X \ W ) + s. Suppo se that d ( L, I n \ U ) = c > 0 , then, since ˜ g ( w ) / ∈ C U , d ( g ( a ) , ˜ g ( w ) = | α L ( g ( a )) − α L ( ˜ g ( w )) | + min { α L ( g ( a )) , α L ( ˜ g ( w )) } d ( β L ( g ( a )) , β L ( ˜ g ( w ))) ≥| α L ( g ( a )) − α L ( ˜ g ( w )) | + c min { α L ( g ( a )) , α L ( ˜ g ( w )) } ≥ c ′ α L ( g ( a )) , where c ′ = min { c, 1 } . Then α L ( g ( a )) ≤ λd ( a, X \ W ) + s , where λ = λ ′ /c ′ , s = s ′ /c ′ .  3. Auxiliar y resul ts In this section we s hall collect some results needed for the pro of of the main re s ult. They are pr ov e d in [4] but it turns out that we hav e also to cover the case of functions with infinite v alues. A map f : X → R + ∪ {∞} is said to b e c o arsely pr op er if the preima ge f − 1 ([0 , c ]) is b ounded for e very c ∈ R + . Lemma 3.1. F or any function ϕ : X → R + with ϕ ( x ) → 0 as x → ∞ the function 1 /ϕ : X → R + ∪ {∞} is c o arsely pr op er. Prop ositi on 3.2. Le t f : X → R + ∪ {∞} b e a c o arsely pr op er fu n ction. Ther e exist s an asymptotic al ly Lipschi tz pr op er fun ction q : X → R + with q ≤ f . Pr o of. This was prov ed in [4] for the case of f : X → R + (see Prop os ition 3.5). That pr o of also works in our cas e.  Prop ositi on 3.3. L et f n : X → R + ∪ {∞} b e a se qu en c e of c o arsely pr op er funct ions. The n ther e ex ists a filtr ation X = ∪ ∞ n =1 A n and a c o arsely pr op er fun ction f : X → R + with f | A n ≤ n and f | ( X \ A n ) ≤ f n for every n . Pr o of. Let B n = ∪ n i =1 f − 1 i ([0 , n ]). the sets B i are b o unded a nd B 1 ⊂ B 2 ⊂ . . . . Therefore, there ex ist bo unded subsets A 1 ⊂ A 2 ⊂ . . . such that A n ∩ ( ∪ ∞ i =1 B i ) = B n and ∪ ∞ i =1 A i = X . F o r x ∈ A n \ A n − 1 , put f ( x ) = n . Obviously , f is coar sely prop er and f | A n ≤ n . Now supp ose that x / ∈ A n , then x / ∈ B n and therefore x / ∈ f − 1 n ([0 , n ]), i.e. f n ( x ) > n ≥ f | ( X \ A n ).  The fo llowing is a n easy mo dification of Lemma 3.6 from [4] and the pro of of it works in our cas e as well. Lemma 3.4 . Supp ose that f : A → R + ∪ {∞} is a c o arsely pr op er map define d on a close d subset A of a pr op er metric sp ac e X and g : W → R + is a pr op er asymptotic al ly Lipschitz map such t hat g ≤ f | W and ther e exist λ, s such that λd ( a, X \ W ) + s ≥ g ( a ) for every a ∈ A . Then ther e exists a pr op er asymptotic al ly Lipschitz map ¯ g : X → R + for which ¯ g ≤ f and ¯ g | A = g . 4 DU ˇ SAN RE POV ˇ S AND MYKHAILO ZARICHNYI 3.1. Alm ost geo des ic s paces. A metric s pa ce X is said to be almost ge o desic if there exists C > 0 such that for every tw o p oints x, y ∈ X there is a shor t map f : [0 , C d ( x, y )] → X with f (0) = x , f ( C d ( x, y )) = y . If in this definition C = 1, then we come to the well-known no tion of ge o desic space. W e are going to describ e a construction of em bedding of a discrete metric space X into an almost g eo desic s pace of the asy mptotic dimensio n min { asdim X , 1 } . F o r an unbounded discrete metr ic s pace X with base p oint x 0 define a function f : X → [0 , ∞ ) by the formula f ( x ) = d ( x, x 0 ). Cho ose a sequence 0 = t 0 < t 1 < t 2 < . . . in f ( X ) so that t i +1 > 2 t i for every i . T o every pair of p oints x, y ∈ f − 1 ([ t i , t i +1 ]), for so me i , attach the line segment [0 , d ( x, y )] a long its endp oints. Le t ˆ X is the unio n of X and all attached seg ments. W e endow ˆ X w ith the maximal metric that agrees with the initial metric on X and the standard metric on every attached seg ment. Note that since X is discrete and prop er, every s et f − 1 ([ t i , t i +1 ]) is finite and there fore ˆ X is a prop er metric space. Prop ositi on 3.5. The sp ac e ˆ X is almost ge o desic. Pr o of. Supp ose that x, y ∈ ˆ X , then x ∈ [ x 1 , x 2 ], y ∈ [ y 1 , y 2 ], where x 1 , x 2 , y 1 , y 2 ∈ X and [ x 1 , x 2 ], [ y 1 , y 2 ] are attached seg ment s. W e may supp ose that d ( x, y ) = d ( x, x 1 ) + d ( x 1 , y 1 ) + d ( y 1 , y ). Case 1 ): Ther e exists i such tha t x 1 , y 1 ∈ f − 1 ([ t i , t i +1 ]). Then [ x, x 1 ] ∪ [ x 1 , y 1 ] ∪ [ y 1 , y ] is a s e gment o f diameter d ( x, y ) that connects x and y in ˆ X . Case 2 ): f ( x 1 ) ∈ [ t i , t i +1 ], f ( y 1 ) ∈ [ t j , t j +1 ], where i 6 = j . Without loss of generality , w e may as s ume that i < j . Obviously , d ( x 1 , y 1 ) ≤ d ( x, y ). Since | t j − t j − 1 | ≤ d ( x 1 , y 1 ), w e see that | t j − t j − 1 | ≤ d ( x, y ). This implies that t j / 2 ≤ d ( x, y ), or equiv alently , t j ≤ d ( x, y ). Bes ides, d ( y 1 , f − 1 ([0 , t j − 1 ]))) ≤ d ( x 1 , y 1 ) ≤ d ( a, b ). F o r every k = i, i + 1 , . . . , j 1 choose z k ∈ f − 1 ( t k ). Then d ( y 1 , z j − 1 ) ≤ d ( y 1 , f − 1 ([0 , t j − 1 ]) + diam ( f − 1 ([0 , t j − 1 ])) ≤ d ( a, b ) + 2 t j − 1 ≤ d ( a, b ) + t j ≤ 3 d ( a, b ) . W e co nnec t x and y by the segment J = [ x, x 1 ] ∪ [ x 1 , z 1 ] ∪ ∪ j − 1 k = i [ z k , z k +1 ]) ∪ [ z j − 1 , y 1 ] ∪ [ y 1 , y ] . Then diam J ≤ d ( x, x 1 ) + d ( x 1 , z i +1 ) + j − 1 X k = i +1 d ( z k , z k +1 ) ! + d ( z j − 1 , y 1 ) + d ( y 1 , y ) = d ( x, y ) + 2 t i +1 + j − 1 X k = i +1 2 t k +1 + 5 d ( x, y ) + d ( x, y ) ≤ 7 d ( x, y ) + 2( t i +1 + · · · + t j ) ≤ 7 d ( x, y ) + 4 t j ≤ 15 d ( x, y ) .  W e need a version of the fact prov ed in [4 ] for geo desic spa ces. Prop ositi on 3.6 . L et f : X → Y b e a c o arse uniform map of an almost ge o desic sp ac e X . Then f is asymptotic al ly Lipschi ts. ON ASYMPTOTIC EXTENSION DIM E NSION 5 Pr o of. Let C be a consta nt from the definition of almost geo desic space. Supp os e x, y ∈ X , then there ex ists a sho rt map α : [0 , C d ( x, y )] → X suc h that α (0) = x , α ( C d ( x , y )) = y . There exis t p oints 0 = t 0 < t 1 < · · · < t k − 1 < t k = C d ( x, y ), where k ≤ [ d ( x, y )] + 1, such that | t i − t i − 1 | ≤ C for every i = 1 , . . . , k . Since f is co arse uniform, there exists R > 0 such that d ( f ( x ′ ) , f ( y ′ ) < R whenever d ( x ′ , y ′ ) ≤ C . Then d ( f ( x ) , f ( y )) ≤ k X i =1 d ( f ( α ( t i )) , f ( α ( t i − 1 ))) ≤ k R ≤ ([ d ( x, y )] + 1) R ≤ Rd ( x, y ) + 2 R.  4. A symptotic extension dimension Let P be an ob ject of the categ ory A . F or any ob ject X o f A the K u r atowski notation X τ P means the following: for every pr op er asymptotically Lipschitz map f : A → P defined on a close d subset A of X there is a prop er asymptotically Lipschitz extension of f onto X . Denote by L the class o f compact absolute Lipsc hitz neighborho od euclidea n extensors (ALNER). F ollowing [], we define a preor der relation ≤ on L . F or L 1 , L 2 ∈ L , we hav e L 1 ≤ L 2 if and only if X τ O L 1 implies X τ O L 2 for all pr op er metric spaces X . T his preo rder relation leads to the following equiv alence r e la tion ∼ o n L : L 1 ∼ L 2 if and only if L 1 ≤ L 2 and L 2 ≤ L 1 . W e denote by [ L ] the equiv alence class containing L ∈ L . F o r a prop er metric space X , we say that its asymptotic exten sion dimension do es not exc e e d [ O L ] (briefly as − ext − dim X ≤ [ O L ] whenever X τ O L . If as − ext − dim X ≤ [ O L ], then the equality as − ext − dim X = [ O L ] means the following. If we a lso have a s − ext − dim X ≤ [ O L ′ ], then [ O L ] ≤ [ O L ′ ]. By [4] (se e also [1 1]), the element [ ∗ ] is max imal. Theorem 4.1. L et L b e a c omp act metric ALNER . The fol lowing c onditions ar e e quivalent: (1) as − ext − dim X ≤ [ O L ] ; (2) ext - dim ν X ≤ [ L ] . Pr o of. Assume that as − ext − dim X ≤ [ O L ]. Let ϕ : C → L b e a map defined on a closed subset C of ν X . Since L ∈ ANE, there e xists an extension ϕ ′ : V → L of ϕ ov er a closed neighborho o d V of C in ¯ X = X ∪ ν X . Then V ar R ϕ ′ ( x ) → 0 as x → ∞ , for any fixed R > 0. By Le mma 3.1, the function f n : V ∩ X → R + ∪ {∞} , f n ( x ) = 1 V ar R ϕ ′ ( x ) , is coa rsely prop er, for every n ∈ N . By Pro p o sition 3 .3, there is a coarse ly pr op er function f : V ∩ X → R + and a filtration V ∩ X = ∪ ∞ n =1 A n such tha t f | A n ≤ n and f | ( X \ A n ) ≤ f n . By Prop os ition 3.5 from [4], there is an asy mptotically Lipschitz function q : V ∩ X → R + with q ≤ f . W e supp os e that q is ( λ, s )-Lipschitz for some λ, s > 0. Define the map g : V ∩ X → O L by the for mula g ( x ) = [ ϕ ′ ( x ) , q ( x )]. W e are g oing to check that the map g ( x ) is asymptotica lly Lipschitz. Let x, y ∈ V ∩ X and n − 1 ≤ d ( x, y ) ≤ n . Suppo se that x, y ∈ ( V ∩ X ) \ A n , then q ( x ) ≤ f n ( x ), q ( y ) ≤ f n ( y ). W e have d ( g ( x ) , g ( y )) = | q ( x ) − q ( y ) | + min { q ( x ) , q ( y ) } d ( ϕ ′ ( x ) , ϕ ′ ( y )) ≤ λd ( x, y ) + s + min { q ( x ) , q ( y ) } V ar n ϕ ′ ( x ) ≤ λd ( x, y ) + s + 1 . 6 DU ˇ SAN RE POV ˇ S AND MYKHAILO ZARICHNYI If x ∈ A n , then q ( x ) ≤ n and we obtain d ( g ( x ) , g ( y )) ≤ λd ( x, y ) + s + nd ( ϕ ′ ( x ) , ϕ ′ ( y )) ≤ λd ( x, y ) + s + n diam L ≤ λd ( x, y ) + s + ( d ( x, y ) + 1)diam L ≤ ( λ + diam L ) d ( x, y ) + ( s + diam L ) . W e argue s imila rly if y ∈ A n . Now, by the a ssumption, there is a n a symptotically Lipschitz extension ¯ g : X → O L o f g . Co nsider the comp osition η L ν ¯ g : ν X → O L . Obviously , η L ν ¯ g | C = ϕ . Let f : A → O L b e a n asymptotically Lipschitz map defined o n a prop er closed subset A o f a prop er metric space X . By P r op osition 2.2, there is a pr op er asymptotically Lipschitz map ˜ f : W → O L and c onstants λ, s such that α L f ( a ) ≤ λd ( a, X \ W ) + s for all a ∈ A . Denote by ϕ : ν X → L a n e x tension of the compos ition η L ν ˜ f . Since L is a n absolute neig hborho o d extensor, there exists an extens io n ψ : V → L of ϕ o nto a closed neig hborho o d o f ν X in the Higson co mpactification ¯ X . Extend ψ to a map ˆ ψ : ( V ∩ X ) ˆ → L as follows. Let J b e a segment attached to V with endp oints a and b . W e require that ˆ ψ linearly ma ps J onto a geo desic s e g ment in L with endp oints ψ ( a ) and ψ ( b ). W e show that ˆ ψ is a slo wly oscillating map. Since ψ is slowly os c illating, for e very ε > 0 and R > 0 there exists K > 0 such that V ar R ψ ( x ) < ε whenever d ( x, x 0 ) > K . Sup- po se that ˆ ψ is not slowly os c illating, then there exist R > 0, C > 0, and sequences ( x i 1 ), x i 2 in ( V ∩ X ) ˆ such that d ( x i 1 , x i 2 ) < R , x i 1 → ∞ , x i 2 → ∞ and d ( ˆ ψ ( x i 1 ) , ˆ ψ ( x i 2 )) > C for every i . W e assume that x i 1 ∈ [ a i 1 , b i 1 ], x i 2 ∈ [ a i 2 , b i 2 ], for every i , where a i 1 , b i 1 , a i 2 , b i 2 ∈ X ∩ V . Without loss o f genera lit y we may assume that a i 1 → ∞ and there e x ists C 1 > 0 such tha t d ( ˆ ψ ( x i 1 ) , ˆ ψ ( a i 1 )) > C 1 for every i . If d ( a i 1 , b i 1 ) < K for a ll i and some K > 0, then d ( ˆ ψ ( x i 1 ) , ˆ ψ ( a i 1 )) < d ( ˆ ψ ( a i 1 ) , ˆ ψ ( b i 1 )) → 0, and we obtain a contradiction. Therefore, we may as s ume that d ( a i 1 , b i 1 ) → ∞ . Then d ( a i 1 , x i 1 ) /d ( a i 1 , b i 1 ) < R/d ( a i 1 , b i 1 ) → 0 a nd there- fore, by the definition of the ma p ˆ ψ , d ( ˆ ψ ( x i 1 ) , ˆ ψ ( a i 1 )) /d ( ˆ ψ ( a i 1 ) , ˆ ψ ( b i 1 )) → 0. Then obviously d ( ˆ ψ ( x i 1 ) , ˆ ψ ( a i 1 )) → 0 a nd we obtain a contradiction. Since the map ˜ f is asy mptotically Lipschitz, there exists K > 0 such that for any a ∈ W we have diam ( α L ˜ f ( N 1 ( a )) + α L ˜ f ( a )diam ( ψ ( N 1 ( a )) ≤ K. Define the function r : ( X ∩ V ) ˆ → R + ∪ {∞} by the formula r ( x ) = K/ ( ψ ( N 1 ( x ))). W e have f ( a ) ≤ r ( a ) for every a ∈ A . The function r is asy mptotica lly prop er and by P rop osition 3.2, ther e exists a ( λ ′ , s ′ )-Lipschitz function ¯ f : X → R + , for some λ ′ , s ′ , with ¯ f ≤ r and ¯ f | A = α L f . Define a map g : ( X ∩ V ) ˆ → R by the formula g ( x ) = ( ψ ( x ) , ¯ f ( x )). Obviously , g | A = f . W e ar e going to show that g is a coarse unifor m ma p. Suppose x, y ∈ X , d ( x, y ) < 1, then d ( g ( x ) , g ( y )) ≤| ¯ f ( x ) − ¯ f ( y ) | + min { ¯ f ( x ) , ¯ f ( y ) } d ( ψ ( x ) , ψ ( y )) ≤ λ ′ + s ′ + K . Note that, since ¯ f is pr op er, g is also prop er. Since g is coa rse unifor m, by Pro p osition 3.6, g is as y mptotically Lipschit z.  Corollary 4.2. (Finite Sum Theorem) Su pp ose X is a pr op er metric s p ac e, X = X 1 ∪ X 2 , wher e X 1 , X 2 ar e close d su bsets of X with as − ext − dim X i ≤ [ O L ] , i = 1 , 2 , for some L ∈ L . Then as − ext − dim X ≤ [ O L ] . Pr o of. Since ν X = ν X 1 ∪ ν X 2 , the result follows from Theor em 4.1 and the finite sum theorem for e xtension dimension (see [2]).  ON ASYMPTOTIC EXTENSION DIM E NSION 7 5. Remarks and o pen questions Question 5. 1. Do es the equa lity as − ext − dim R n = S n hold? Question 5.2. Let L 1 , L 2 be finite poly hedra in euclidean spaces endow e d with the induced metric. Is the inequality [ L 1 ] ≤ [ L 2 ] introduced in [3] equiv alent to the inequality [ L 1 ] ≤ [ L 2 ] as in Section 4? One can define ana logue of the asymptotic extensio n dimensio n by us ing warped cones instead of op en cones. F o llowing [9] w e review this constructio n briefly . L e t F b e a foliation on a compact smo oth manifold V . Let N b e an y co mplementary subbundle to T F in T M . Cho ose Euclidean metrics g N in N and g F in T F . The foliate d warp e d c one O F is the manifold V × [0 , ∞ ) / V × { 0 } equipp ed with the metric induced for t ≥ 1 by the Riemannian metric g R + g F + t 2 g N . The metric structure o n any bounded neighborho o d o f the distinguished po int is ir relev ant. Question 5.3. Is the obtained warp ed cone an abso lute neighborho o d extensor in the as- ymptotic ca tegory? An affir mative answer to this question would allow us to define as y mptotic e x tension di- mension theo ry with the v a lues in warp ed c ones. References [1] G. Bell and A. N. Dranishniko v, O n asymptotic dimension of gr oups , Algebraic and Geometric T op ology 1 (2001) , 57–71. [2] G. Bell and A. N. Dranishniko v, Asymptotic d imension , preprin t on the A rchiv es h ttp://arxiv.org/abs/math /0703766 v2 [3] A. N. Dr anishniko v, O n the the ory of e xtensions of mappings of c omp act sp ac e s, (Russi an) Uspekhi Mat. Nauk 53 :5 (1998), 65–72; English transl. in Russian Math. Surveys 53 :5 (1998), 929–935. [4] A. N. Dranishniko v, Asymptotic top olo g y , Russian Math. Surveys 55 (2000), No 6, 71–116. [5] A. Dr anishniko v and J. Dydak, Extension dimension and e xtension typ es , T rudy Mat. Inst. Steklo v a 21 2 (1996), Otobrazh. i Razmer., 61–94; Engli sh transl. in Pro c. Steklov Inst. Math. 212 :1 (1996), 55–88. [6] A. Dranishniko v and J. Dydak, Extension the ory of sep ar able metrizable sp ac es with applic ations to dimension the ory , T rans. A mer. Math. Soc. 353 :1 (2001) 133–156. [7] M. Gromov, Asymptotic invariants of infinite gr oups , Geometric group theory , V ol. 2 (Sussex, 1991), London Math. Soc. Lecture Note Ser., vol. 182, Camb ridge Univ. Press, Cambridge, 1993, pp. 1-295. [8] P . Mi tc hener, Co arse homolo gy the ories , Algebr. Geom. T opol. 1 (2001) , 271–297 (electronic). [9] J. Ro e, F r om foliations to co arse geo metry and b ack , in: Analysis and geometry i n foliated manifolds, (X. M asa, E. M acias-Virg´ os, J.A.Al v arez L´ opez, Editors), W orld Scientific, Singapore 1995, pp. 195-206. [10] J. Ro e, Warp e d c ones and pr op erty A , Geometry & T opology 9 (2005) 163178 Publi shed: 6 January 2005, Corrected: 7 M arch 2005 [11] M. Sa wic ki, Absolute ext ensors and absolute neighb orho o d exte nsors in asymptotic c ate gories , T opology Appl. 150 :13 (2005), 59-78. F acul ty of Ma thematic s and Physics, an d F acul ty of Educa tion, University of Ljub ljana, Jad- ranska 19, 1000 Ljubljana, Slovenia E-mail addr ess : dusan.re povs@gue st.arnes.si Dep ar tment of Mechanics and Mat hema tics, L viv Na tional University, Universytetska 1, 7 9000 L viv, Ukraine E-mail addr ess : mzar@lit ech.lviv .ua