Simultaneous Z/p-acyclic resolutions of expanding sequences

SIMUL T ANEOUS Z /p -A CYCLIC RESOLUTIONS OF EXP ANDING SEQUENCES LEONARD R. R UBIN AND VERA TONI ´ C ∗ Abstra ct. W e pro ve the follo wing theorem. Theorem : L et X b e a nonempty c om p act metrizable sp ac e, let l 1 ≤ l 2 ≤ . . . b e a se quenc e in N , and l et X 1 ⊂ X 2 ⊂ . . . b e a se quenc e of nonempty close d subsp ac es of X such t hat for e ach k ∈ N , dim Z /p X k ≤ l k . Then ther e exists a c omp act metrizable sp ac e Z , having close d subsp ac es Z 1 ⊂ Z 2 ⊂ . . . , and a (surje ctive) c el l-like map π : Z → X , such that for e ach k ∈ N , (a) dim Z k ≤ l k , (b) π ( Z k ) = X k , and (c) π | Z k : Z k → X k is a Z /p -acyclic map. Mor e over, ther e is a se quenc e A 1 ⊂ A 2 ⊂ . . . of close d subsp ac es of Z such that for e ach k , dim A k ≤ l k , π | A k : A k → X is surje ctive, and f or k ∈ N , Z k ⊂ A k and π | A k : A k → X is a UV l k − 1 -map. It is not required that X = S ∞ k =1 X k or th at Z = S ∞ k =1 Z k . This result generalizes the Z /p - resolution theorem of A. D ranishniko v and runs parallel to a similar theorem of S. Ageev, R. Jim´ enez, and the first author, who studied the situation where the group was Z . 1. Introduction The goal of th is p ap er is to pr ov e the follo wing theorem. Theorem 1.1. L et X b e a nonempty c omp act metrizable sp ac e, let l 1 ≤ l 2 ≤ . . . b e a se quenc e in N , and let X 1 ⊂ X 2 ⊂ . . . b e a se quenc e of nonempty close d subsp ac es of X such that f or e ach k ∈ N , d im Z /p X k ≤ l k . Then ther e exists a c omp act metrizable sp ac e Z , having close d subsp ac es Z 1 ⊂ Z 2 ⊂ . . . , and a (surje c tive) c el l-like map π : Z → X , such that for e ach k ∈ N , (a) dim Z k ≤ l k , (b) π ( Z k ) = X k , and (c) π | Z k : Z k → X k is a Z /p -acyclic ma p. Mor e over, ther e is a se quenc e A 1 ⊂ A 2 ⊂ . . . of close d subsp ac es of Z such th at f or e ach k , dim A k ≤ l k , π | A k : A k → X is surje ctive, and for k ∈ N , Z k ⊂ A k and π | A k : A k → X is a UV l k − 1 -map. The second Sectio n will con tai n some tec h n ical results necessary for the pro of of Th eo- rem 1.1, and the pro of will b e describ ed in th e th ir d Section. In Section 4 w e will outline a pro of of a case of T heorem 1.1 that w as suggested to u s b y an anonymous referee. Unfortun ately , this technique cannot b e used to prov e the most difficult cases of Theorem 1.1, nor do es it ha v e th e p otentia l for generalizatio n for those Date : 26 Jan uary , 2013. 2010 Mathematics Subje ct Classific ation. Primary 55M10, 54F45; Secondary 55P20. Key wor ds and phr ases. Cell-lik e map, cohomological dimension, CW-complex, d imension, Edwards- W alsh resolution, Eilenberg-MacLane complex, G -acyclic map, inverse sequence, simplicial complex, UV k - map. ∗ P art of this paper wa s written while the second author w as a p ost-do ctoral fello w at Nipissing Universit y , Department of Computer Science and Mathematics, North Bay , Ontario, Canada. 1 2 L. Rubin, V . T oni ´ c groups G whose resolutions require a domain space of dimension n + 1, if the range space had dim G ≤ n ([Le]). F or example, the theorem that follo ws is an immediate consequence of Th eorem 1.1, bu t it cannot b e p ro v en using th e tec hnique describ ed in Section 4. Theorem 1.2. L et n ∈ N and let ( X i ) b e a se quenc e of (not ne c essarily neste d) close d subsets of the Hilb ert c u b e Q with dim Z /p X i ≤ n for al l i . Then ther e exists a c omp act metrizable sp ac e Z , a c el l-lik e map π : Z → Q , and a se quenc e ( Z i ) of close d subsets of Z such that ∀ i , (a) dim Z i ≤ n , and (b) π | Z i : Z i → X i is a surje ctive Z /p -acyclic map. This theorem provides a cell-lik e resolution of the Hilb ert cub e Q and sim ultaneously Z /p -acyclic r esolutions o v er an y F σ -collec tion whatsoeve r of suc h X i . Let u s pr o cee d by explaining some terms that migh t b e unfamiliar to the reader. Basic facts ab out cell-lik e spaces and maps can b e foun d in [Da]. A map π : Z → X is called c el l-like if for eac h x ∈ X , π − 1 ( x ) has the shap e of a p oin t. T o detect that a compact metrizable space Y h as the shap e of a p oint, it is s u fficien t to p ro v e that there is an inv erse sequence ( Z i , p i +1 i ), of compact metrizable spaces Z i , whose limit is homeomorphic to Y and s u c h that for infi n itely many i ∈ N , p i +1 i : Z i +1 → Z i is n ull-homotopic. It is also sufficien t to sho w that ev ery map of Y to a CW-complex is null-homotopic. A map π : Z → X b etw een top ological spaces is called G - acyclic ([Dr]) if all its fib ers π − 1 ( x ) ha v e trivial red uced ˇ Cec h cohomology with resp ect to a giv en ab elian group G , or, equiv alen tly , ev ery map f : π − 1 ( x ) → K ( G, n ) is null-homotopic. Note that a map π : Z → X b eing cell-lik e implies that π is also G -acyclic. T o detect that a compact metrizable space Y has trivial reduced ˇ Cec h cohomolog y with resp ect to th e group G , it is s ufficien t to pro ve that there is an inv erse sequence ( Z i , p i +1 i ) of compact p olyhedra Z i whose limit is homeomorphic to Y , suc h that for in finitely man y i ∈ N , the map p i +1 i : Z i +1 → Z i induces the zero-homomorphism of cohomology grou p s H m ( Z i ; G ) → H m ( Z i +1 ; G ), for all m ∈ N . A map π : Z → X is called a UV k - map ([Da]) if eac h of its fi b ers h as prop erty UV k . This means that eac h embed ding π − 1 ( x ) ֒ → A into an ANR A h as prop ert y UV k : for ev ery 0 ≤ r ≤ k and ev ery neighborh o o d U of π − 1 ( x ) in A , there exists a n eigh b orho o d V of π − 1 ( x ) in U suc h that ev ery map of S r in to V is null- homotopic in U . In order to p ro v e that π is a UV k -map, it is su fficien t to show that, f or all x ∈ X , there is an inv erse sequence ( Z i , p i +1 i ) of compact p olyhedra Z i , whose limit is homeomorphic to π − 1 ( x ) and suc h that ∀ i ∈ N , if 0 ≤ r ≤ k , then an y map h : S r → Z i is n ull-homotopic. It is w ell-kno wn that cell-lik e compacta h a v e prop ert y UV k for all k . A map g : X → | K | b et wee n a space X an d a p olyhedr on | K | is called a K - mo dific ation of a m ap f : X → | K | if w henev er x ∈ X and f ( x ) ∈ σ , f or some σ ∈ K , then g ( x ) ∈ σ . This is equiv alen t to the follo wing: w henev er x ∈ X and f ( x ) ∈ ◦ σ , for some σ ∈ K , then g ( x ) ∈ σ . The pro of of T h eorem 1.1 uses some tec hniques dev elop ed by A. Dranishniko v in the pro of of the follo wing theorem, which can b e found as Th eorem 8.7 in [Dr]. Theorem 1.3. F or every c omp act metrizable sp ac e X with dim Z /p X ≤ n , ther e exists a c omp act metrizable sp ac e Z and a surje ctiv e map π : Z → X such that π is Z /p -acyclic and dim Z ≤ n . W e will show in Remark 3.3 that our Th eorem 1.1 is a generalizatio n of this th eorem. Dranishniko v used Edwa rds–W alsh complexes and resolutions, and so shall w e. Simultan eous Z /p -acyclic r esolutions 3 The follo wing definition of Ed w ards–W alsh co mplexes (EW-complexes) and resolutions, as well as results ab out them, can b e found in [Dr], [D W ] or [KY]. F or G = Z , these resolutions were f orm ally form ulated in [W a]. Definition 1.4. L et G b e an ab elian gr oup, n ∈ N and L a simplicial c omplex. An Ed w ards– W alsh resolution of L in dimension n is a p air (EW( L, G, n ) , ω ) c onsisting of a CW -c omplex EW( L, G, n ) and a c ombinatorial map ω : EW( L, G, n ) → | L | (that is, ω − 1 ( | L ′ | ) is a sub c omp lex, for e ach sub c omplex L ′ of L ) such tha t: (i) ω − 1 ( | L ( n ) | ) = | L ( n ) | and ω | | L ( n ) | is the identity map of | L ( n ) | onto itself, (ii) for every simplex σ of L with d im σ > n , the pr ei mage ω − 1 ( | σ | ) i s an Ei lenb er g– MacL ane sp ac e of typ e ( L G, n ) , wher e the sum L G is finite, and (iii) for every su b c omplex L ′ of L and eve ry map f : | L ′ | → K ( G, n ) , the c omp osition f ◦ ω | ω − 1 ( | L ′ | ) : ω − 1 ( | L ′ | ) → K ( G, n ) extends to a map F : EW( L, G, n ) → K ( G, n ) . W e usu ally refer to the CW-complex EW( L, G, n ) as an Edwar ds–Walsh c omplex , and to the map ω itself as an Edwar ds–Walsh pr oje ction . Remark 1.5. L et L ′ b e a sub c omplex of L , K b e the sub c omp lex ω − 1 ( | L ′ | ) of E W ( L, G, n ) , and ω L ′ = ω | ω − 1 ( | L ′ | ) : ω − 1 ( | L ′ | ) → | L ′ | . Then ( K, ω L ′ ) is an Edwar ds–Walsh r esolution of the form ( E W ( L ′ , G, n ) , ω L ′ ) . A d iscussion ab out the existence of Edwa rds–W alsh resolutions, as we ll as th eir construc- tion, can b e found in [Dr], [D W], [KY], [W a]. F or our needs, it is enough to k n o w that when G is either Z or Z /p , Edw ards–W alsh resolutions exist for an y simplicial complex L . In particular, we sh all briefly d escrib e the construction of (EW ( L, Z /p, n ) , ω ) for a finite- dimensional simp licial complex L . If dim L ≤ n , define EW ( L, Z /p, n ) = L ( n ) = L , and ω = id L . If dim L = n + 1, w e start with L ( n ) and the iden tit y map id L ( n ) , and pro ceed by building a K ( Z /p, n ) on ∂ σ , for eac h ( n + 1)-simplex σ of L , and we b uild ω b y extend in g ∂ σ ֒ → σ o ver th is n ewly attac h ed K ( Z /p, n ). In this wa y , ω − 1 ( | σ | ) = K ( Z /p, n ), ∀ ( n + 1)- simplex σ of L . If d im L > n + 1, then we shall distinguish the cases n ≥ 2 and n = 1. In b oth of these cases ou r constru ction is ind u ctiv e. If n ≥ 2 and d im L > n + 1, then th e skelet on L ( n +1) is dealt with as describ ed ab o v e, i.e., b y att ac hing a K ( Z /p, n ) to ∂ σ , for ea c h ( n + 1)-simplex σ ∈ L ( n +1) . T h is repr esen ts the basis of our indu ctiv e construction. F or the step of our in ductiv e constru ction, let k > n + 1. Then for an y k -simplex σ of L , we ha v e that π n ( ω − 1 ( | ∂ σ | )) = L Z /p , wh ere this s u m is finite. So ω − 1 ( | σ | ) will b e obtained from ω − 1 ( | ∂ σ | ) by attac hing cells of dim ≥ n + 2 in order to k ill off the h igher homotop y groups of ω − 1 ( | ∂ σ | ), and ac hiev e that ω − 1 ( | σ | ) = K ( L Z /p, n ). If n = 1 and dim L > 2, then the 2-ske leton L (2) is dealt w ith as describ ed ab ov e, th at is, by attac hin g a K ( Z /p, 1) to ∂ σ , for eac h 2-simplex σ ∈ L ( n +1) . T o b e more precise, this means attac hing a 2-cell u s ing a map of degree p from the b oundary of the 2-cell to ∂ σ , for every 2-simplex σ of L , and then pro ceeding b y attac hing cells of d im ≥ 3 to form a K ( Z /p, 1) on top of eac h of these Mo ore spaces. Ho w ev er, the ab o ve m en tioned 2-cells are not the only ones that get attac hed here; w e will ha v e to attac h more of these. Namely , when k > 2, then for any k -simplex σ of L , there will b e 2-cells γ ⊂ ω − 1 ( | σ | ) \ ω − 1 ( | ∂ σ | ), attac hed by a map ∂ γ → ω − 1 ( | ∂ σ | ) represent ing a commuta tor in π 1 ( ω − 1 ( ∂ σ )). This is to ensure that π 1 ( ω − 1 ( | σ | )) = L Z /p . W e pro ceed by at tac hing cells of dimen sion ≥ 3 to ac hiev e that ω − 1 ( | σ | ) = K ( L Z /p, 1). The follo wing fact is p ro v en in [Dr] as Lemma 8.1, and (iv Z /p ) is clear from our construc- tion ab o ve. 4 L. Rubin, V . T oni ´ c Lemma 1.6. F or the gr oups Z and Z /p , for any n ∈ N and for any simplicial c omplex L , ther e is an Edwar ds–Walsh r esolution ω : E W( L, G, n ) → | L | with the additional pr op erty for n > 1 : (iv Z ) the ( n + 1 ) -skeleton of EW ( L, Z , n ) is e qual to L ( n ) ; (iv Z /p ) the ( n + 1) -skeleton of EW ( L, Z /p, n ) is obtaine d fr om L ( n ) by attaching ( n + 1) -c el ls by a map of de gr e e p to the b oundary ∂ σ , for every ( n + 1) -dimensional simpl ex σ . Here are some other prop erties follo wing from the construction of Edw ards-W alsh com- plexes for the groups Z /p . Remark 1.7. L e t L b e a simplicial c ompl ex, let σ b e any simplex of L with dim σ > n , and let (EW ( L, Z /p, n ) , ω ) b e an Edwar ds-Walsh r esolution of L . A c c or ding to R emark 1.5, ω − 1 ( | σ | ) = EW ( σ , Z /p, n ) and fr om the c onstruction of EW( L, Z /p, n ) , we have that the numb er of summands in π n ( ω − 1 ( | σ | )) ∼ = L Z /p is less than or e qual to the numb er of the ( n + 1) -fac es of σ . F rom this Remark and our constru ction, w e get: Corollary 1.8. L et σ b e a simplex with dim σ > n , taken as a simplicial c omp lex, and let (EW( σ, Z /p, n ) , ω ) b e an Edwar ds-Walsh r esolution of σ . Then (I) H n ( | σ ( n ) | ) ∼ = L r 1 Z , and (I I) H n (EW( σ, Z /p, n )) ∼ = L r 1 Z /p , wher e r ≤ the numb er of al l ( n + 1) -fac es of σ . Mor e over, (I I I) we c an cho ose τ 1 , . . . , τ r to b e some ( n + 1) -fac es of σ so that the images h 1 , . . . , h r of the gener ator s of H n ( ∂ τ 1 ) , . . . , H n ( ∂ τ r ) , induc e d by the i nclusions ∂ τ i ֒ → σ ( n ) , form a b asis of H n ( | σ ( n ) | ) . Then if q 1 , . . . , q r ar e the images of the gene r ator s of H n ( ∂ τ 1 ) , . . . , H n ( ∂ τ r ) , induc e d by the inclusions ∂ τ i ֒ → EW ( σ , Z /p, n ) , and λ ∗ = H n ( λ ) is induc e d by the i nc lu si on λ : σ ( n ) ֒ → EW ( σ, Z /p, n ) , we get that q 1 = λ ∗ ( h 1 ) , . . . , q r = λ ∗ ( h r ) form a b asis of H n (EW( σ, Z /p, n )) . The follo wing lemma is pr ov en in [Dr] as Lemma 8.2. It concerns (approxima tely) lifting maps thr ou gh EW-complexes: Lemma 1.9. L et X b e a c omp act metrizable sp ac e with d im G X ≤ n , and let L b e a finite simplicial c omplex. Then for every E dwar ds–Walsh r esolution ω : EW ( L, G, n ) → | L | , and for eve ry map f : X → | L | , ther e exists a map f ′ : X → EW( L, G, n ) such that (i) f ′ | f − 1 ( | L ( n ) | ) = f | f − 1 ( | L ( n ) | ) , and (ii) ω ◦ f ′ is an L -mo dific ation of f . Our pr imary construction will b e done in the Hilber t cub e Q – our space X is compact metrizable, and Q is un iversal for all compact metrizable spaces. Let the Hilb ert cub e Q = ∞ Y i =1 I b e end o w ed with th e metric ρ su ch that if x = ( x i ), y = ( y i ), then ρ ( x, y ) = ∞ X i =1 | x i − y i | 2 i . As u sual, I = [0 , 1]. F or an y i ∈ N it will b e con v enien t to write Q = I i × Q i in factored form. In this case, an y su bset E of I i will alw a ys b e treated as E × { 0 } ⊂ Q . W e sh all use p i : Q → I i for coord in ate p ro jectio n. In some of the pr o ofs that follo w w e will use s tabilit y theory , ab out whic h more details can b e found in § VI.1 of [HW]. Namely , we will use th e consequ en ces of Theorem VI.1. from [HW]: if X is a s eparable metrizable space with d im X ≤ n , then for an y map f : X → I n +1 all v alues of f are u nstable. A p oint y ∈ f ( X ) is called an unstable value of f if for ev ery δ > 0 there exists a map g : X → I n +1 suc h that: Simultan eous Z /p -acyclic r esolutions 5 (1) d ( f ( x ) , g ( x )) < δ for ev ery x ∈ X , and (2) g ( X ) ⊂ I n +1 \ { y } . Moreo ver, this map g can b e chosen s o that g = f on the complement of f − 1 ( U ), for an y op en n eigh b orho o d U of y , and so that g is homotopic to f (see Corollary I.3.2.1 of [MS]). The follo wing lemma is a form of th e h omotopy extension theorem with con trol, and can b e foun d in [AJR ] as Lemma 2.1. Lemma 1.10. L et f : X → R b e a map of a c omp act p olyhe dr on X to a sp ac e R , X 0 b e a close d subp olyhe dr on of X , and U b e an op en c over of R . Supp ose that F : X 0 × I → R is a U - homoto py of f | X 0 . Then ther e exists a U -homoto py H : X × I → R of f such that H | X 0 × I = F : X 0 × I → R . Notation. W e w ill use the follo wing n otation. Let x b elong to a metric space X and let δ > 0. Then by N ( x, δ ) w e sh all m ean the closed δ -neigh b orho o d of x in X . Usually there will b e no am biguit y , bu t notice that for x ∈ Q , p i ( x ) ∈ I i so N ( p i ( x ) , δ ) will alwa ys refer to the closed δ -neighb orh o o d of p i ( x ) in I i , ev en though p i ( x ) migh t also b e con tained in some su bsets of I i . If σ is a simp lex in a triangulat ion τ of a p olyhed ron P , then N ( σ, δ ) will stand for the op en δ -neigh b orh o od of σ in P . Whenev er ( P i , g i +1 i ) is an inv erse sequence, T i ⊂ P i and g i +1 i ( T i +1 ) ⊂ T i for eac h i , then w e shall write ( T i , g i +1 i ) for the indu ced inv erse sequ ence, using the same notation for th e b ondin g maps as long as no confusion can arise. Whenev er P is a p olyhedron, τ is a triangulation of P , and k ≥ 0, then P ( k ) will d enote the subp olyhedron of P triangulated by the k -ske leton of τ , i.e., P ( k ) = | τ ( k ) | . If R is a subp olyhedron of P and we hav e to build an Edwards-W alsh complex on τ | R , we will write EW( R, Z /p, n ) instead of EW ( τ | R , Z /p, n ), to k eep matters simpler. 2. Technical lemmas The follo wing type of result is a lemma w hic h is technical, but wh ic h will help us find certain maps and u nderstand their fib ers. This lemma can b e foun d in [AJR] as Lemma 3.1. Once th e correct conditions are foun d on the construction of said maps, then T h eorem 1.1 will follo w r eadily . Lemma 2.1. Supp ose that for e ach i ∈ N we have sele cte d n i ∈ N , a c omp act subset P i ⊂ I n i , δ i > 0 , ε i > 0 , and a map g i +1 i : P i +1 → P i so that: (i) if u , v ∈ Q and ρ ( u, v ) ≤ ε i +1 , then ρ ( p n i ( u ) , p n i ( v )) < δ i , (ii) n i < n i +1 , (iii) 9 2 n i < ε i , (iv) ρ ( g i +1 i ( x ) , p n i ( x )) < δ i for al l x ∈ P i +1 , (v) δ i < 1 2 n i − 1 , and (vi) P i +1 × Q n i +1 ⊂ P i × Q n i . Put X = ∞ \ i =1 P i × Q n i , P = ( P i , g i +1 i ) , and Z = lim P . Then for e ach z = ( a 1 , a 2 , . . . ) ∈ Z ⊂ ∞ Y i =1 P i , and asso ciate d se que nc e ( a i ) in Q , (a) ( a i ) is a Cauchy se quenc e in Q whose limit lies in X , and (b) the fu nc tion π : Z → X given by π ( z ) = lim i →∞ ( a i ) is c ontinuous. Fix x ∈ X and f or e ach i ∈ N , let B x,i = N ( p n i ( x ) , 2 δ i ) ∩ P i , B # x,i = N ( p n i ( x ) , ε i ) ∩ P i . Then, (c) B x,i ⊂ B # x,i and g i +1 i ( B # x,i +1 ) ⊂ B x,i . 6 L. Rubin, V . T oni ´ c If we let P x = ( B x,i , g i +1 i ) and P # x = ( B # x,i , g i +1 i ) , then, (d) lim P x = lim P # x , and (e) π − 1 ( x ) = lim P x . In addition, su pp ose we ar e give n, for e ach i ∈ N , a close d subsp ac e T i ⊂ P i in such a manner that g i +1 i ( T i +1 ) ⊂ T i . Put T = ( T i , g i +1 i ) and Z ′ = lim T ⊂ Z . F or x ∈ X , let S x,i = B x,i ∩ T i , T x = ( S x,i , g i +1 i ) ; se t ˜ π = π | Z ′ : Z ′ → X . Then, (f ) ˜ π − 1 ( x ) = lim T x , and (g) if S x,i 6 = ∅ for e ach i , then ˜ π − 1 ( x ) 6 = ∅ . A helpful d iagram for Lemma 2.1: . . . P i o o _    P i +1 _    p n i | o o g i +1 i s s . . . o o Z π   ✤ ✤ ✤ . . . P i × Q n i ? _ o o P i × Q n i +1 ? _ o o . . . ? _ o o X Before pr o ceeding, note th at if L is a simplicial complex, K a CW-complex, and f : | L | → K a map, then we sa y that f is c el lular if it is cellular with resp ect to the CW-structure induced on | L | b y L and the given one of K, i.e., f tak es the (simplicial) n -sk eleton of L to the (CW) n -skelet on of K , ∀ n . The follo wing Corollary is a v ersion of Corollary 3.2 from [AJR], adapted for the Z /p - case. When u s ed (in the pro of of the m ain theorem), A k can b e replace d by Z k (not just b y A k of Theorem 1.1). Corollary 2.2. Supp ose in L emma 2.1 that for e ach i ∈ N , P i = | τ i | is a nonempty subp olyhe dr on of I n i having a triangulation e τ i , with a sub division τ i with mesh τ i < δ i , so that for every simplex γ of e τ i , τ i | γ is c ol lapsible. M or e over, assume that g i +1 i is a simplicial map (in p articular, for al l k ≥ 0 , g i +1 i ( P ( k ) i +1 ) ⊂ P ( k ) i , wher e τ i +1 and τ i ar e the r elevant triangulations). L et l 1 ≤ l 2 ≤ . . . b e a se quenc e in N , and let T k = ( P ( l k ) i , g i +1 i ) , and A k = lim T k . Then A 1 ⊂ A 2 ⊂ . . . , and for e ach k ≥ 1 , (I) dim A k ≤ l k and π | A k : A k → X is surje ctive. Assume further that for e ach x ∈ X and i ∈ N , ther e is a c ontr actible p olyhe dr on P x,i which is the close d star of a vertex in the triangula tion e τ i , such that B x,i ⊂ P x,i ⊂ B # x,i . Then (I I) π : Z → X is a c el l-like map, and (I I I) for e ach k ∈ N , π | A k : A k → X is a UV l k − 1 -map. Supp ose now that al l of the ab ove statements ar e true, and let k ∈ N . If for i nfinitely many indexes i we have that f or al l x ∈ X , ω ◦ ¯ f i ( P x,i +1 ) ⊂ P x,i , and g i +1 i | P x,i +1 ≃ ω ◦ ¯ f i | P x,i +1 , wher e ω : EW( P i , Z /p, l k ) → P i is an E dwar ds–Wal sh pr oje ction, and ¯ f i : P i +1 → EW( P i , Z /p, l k ) is a c el lular map, then (IV) π | A k : A k → X is a Z /p - acyclic map. Before showing the pro of of Corollary 2.2, we will state and pro v e some lemmas which will b e usefu l for its pro of. Simultan eous Z /p -acyclic r esolutions 7 Lemma 2.3. L et n ∈ N , and let P = | L | and Q = | M | b e c omp act p olyhe dr a with d im P , dim Q ≥ n + 1 . F or any ( n + 1) -simplex τ e of M , let h e and q e b e the images of a gener ator of H n ( ∂ τ e ) under the homomorphism s of H n ( ∂ τ e ) induc e d by the inclusions ∂ τ e ֒ → | M ( n ) | and ∂ τ e ֒ → EW( M , Z /p, n ) , r esp e ctively. L et µ , ν and λ b e the inclusi ons as shown in the up c oming diagr am, and let f : | L | → EW( M , Z /p, n ) b e a c el lular map making this diagr am c ommutative. Mor e over, let M b e such th at: (I) H n ( | M ( n ) | ) ∼ = L r 1 Z , and (I I) H n (EW( M , Z /p, n )) ∼ = L r 1 Z /p , wher e r ≤ the numb er of al l ( n + 1) -simplexes of M ; and (I I I) we c an cho ose some ( n + 1) -simplexes τ 1 , . . . , τ r of M so that { h 1 , . . . , h r } forms a b asis of H n ( | M ( n ) | ) , and so that { q 1 , . . . , q r } forms a b asis of H n (EW( M , Z /p, n )) . Then for any ( n + 1 ) -simplex σ ∈ L , with H n ( ∂ σ ) = h g i , we have: (a) f ◦ ν ◦ µ is nul l-homotopic, so (b) H n ( f | | L ( n ) | ◦ µ )( g ) = P r e =1 ε e h e , wher e ε e ≡ 0 (mo d p ) , for e ∈ { 1 , . . . , r } . EW( M , Z /p, n ) ω   | L | f 4 4 ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ ✐ | M | | L ( n ) | 0 P ν b b ❉ ❉ ❉ ❉ ❉ ❉ ❉ ❉ f | / / | M ( n ) | 4 T g g ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ ❖ / O λ _ _ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ❃ ∂ σ ?  µ O O Pr o of : Sin ce ∂ σ is con tained in σ , wh ic h is con tractible, the inclusion ν ◦ µ : ∂ σ ֒ → | L | is n ull-homotopic. Therefore f ◦ ν ◦ µ is n ull-homotopic, so (a) is tru e. T o prov e (b), notice that f b eing a cellular map implies f ( | L ( n ) | ) ⊂ EW( M , Z /p, n ) ( n ) = | M ( n ) | . It is clear that f ◦ ν ◦ µ = λ ◦ f | | L ( n ) | ◦ µ . So (a) implies 0 = H n ( f ◦ ν ◦ µ )( g ) = H n ( λ ◦ f | | L ( n ) | ◦ µ )( g ) . F rom (I I I) we get that H n ( f | | L ( n ) | ◦ µ )( g ) = P r e =1 ε e h e , for s ome ε e ∈ Z , and therefore H n ( λ ◦ f | | L ( n ) | ◦ µ )( g ) = H n ( λ )( r X e =1 ε e h e ) = r X e =1 ε e q e = 0 , whic h means that ε e ≡ 0 (mo d p ), ∀ e ∈ { 1 , . . . , r } .  Some form of the follo wing lemma w as used b y v arious authors. Lemma 2.4. L et n ∈ N , P = | e L | b e a c omp act p olyhe dr on with dim P ≥ n + 1 and f M b e the close d star of a vertex fr om e L (0) . L et L b e a su b division of e L such that for ev ery simplex σ of e L , L | | σ | is a c ol lapsible simplicial c omp lex. L et M b e the simplicial c omplex that L induc es on | f M | , i.e., M = L | | f M | (sub divide d vertex sta r). Then (I) H n ( | M ( n ) | ) ∼ = L r 1 Z , and (I I) H n (EW( M , Z /p, n )) ∼ = L r 1 Z /p , wher e r ≤ the numb er of al l ( n + 1) -simplexes of M . Mor e over, 8 L. Rubin, V . T oni ´ c (I I I) we c an cho ose τ 1 , . . . , τ r to b e some ( n + 1) -simplexes of M so that the images h 1 , . . . , h r of the gener ators of H n ( ∂ τ 1 ) , . . . , H n ( ∂ τ r ) , induc e d by the inclusions ∂ τ i ֒ → M ( n ) , form a b asis of H n ( | M ( n ) | ) . Then i f q 1 , . . . , q r ar e the images of the gener- ators of H n ( ∂ τ 1 ) , . . . , H n ( ∂ τ r ) , induc e d by the inclusions ∂ τ i ֒ → EW ( M , Z /p, n ) , and H n ( λ ) is induc e d by the inclusion λ : M ( n ) ֒ → EW( M , Z /p, n ) , we get that q 1 = H n ( λ )( h 1 ) , . . . , q r = H n ( λ )( h r ) form a b asis of H n (EW( M , Z /p, n )) . W e will omit the pro of to sa v e space. On the w a y to proving this Lemma, one can first use Corollary 1.8 (conta ining the statemen t analogous to this one, bu t for a simplex) in ord er to pro v e analo gous statemen ts for a (non-sub d ivided) verte x star, and then for a sub divided simplex with a collapsible su b division. Then Lemma 2.4 can b e prov en b y first pr o ving its statemen t for dim M = n + 1, and then, b y induction, sh o wing it is tru e for dim M = n + k + 1. T h e general s tep of induction w ould utilize another induction, on the num b er of ( n + k + 1)-simplexes of f M , as w ell as a Ma y er-Vietoris sequence. W e u sed a collapsible su b division on simplexes of f M so that we could organize the p ro cess of ind uction. The information ab out the existence of sub divisions of a triangulation on a simplicial complex, in w hic h a simplex with a new sub division is still collapsible can b e found in [Gl]. Remark 2.5. W hen M is a sub divide d vertex star fr om L emma 2.4, then L emma 2.3 is true for Q = | M | and | M ( n ) | i s ( n − 1) -c onne cte d. Pro of of Corollary 2.2 : Su rely dim A k ≤ l k . Let x ∈ X . Apply Lemma 2.1 with T i = P ( l k ) i and S x,i = B x,i ∩ P ( l k ) i . Then T b ecome s T k and Z ′ = lim T k = lim( P ( l k ) i , g i +1 i ) = A k . Note that the representa tion of X implies that p n i ( X ) ⊂ P i , ∀ i ∈ N . This fact, together with mesh τ i < δ i , can b e used to c h ec k th at B x,i m ust con tain a v ertex of τ i , so S x,i 6 = ∅ . Therefore (g) of Lemma 2.1 sho ws that (I) is tru e. P art (c) of Lemma 2.1 and th e fact that B x,i ⊂ P x,i ⊂ B # x,i ∀ i ∈ N , show that ∀ i ∈ N , g i +1 i ( P x,i +1 ) ⊂ P x,i , so P ′ x := ( P x,i , g i +1 i ) is an in v erse sequence. Clearly (see (d) and (e) of Lemma 2.1), lim P ′ x = π − 1 ( x ). No w P ′ x is an in v erse sequence of con tractible p olyhedra. Hence (I I) is true. T o get at (I I I ), fi rst observ e th at b y (f ) of Lemma 2.1, the fib er ( π | A k ) − 1 ( x ) is the limit of the inv erse sequence ( S x,i , g i +1 i ). On the other hand , for ea c h i ∈ N , B x,i ⊂ P x,i ⊂ B # x,i , g i +1 i ( P ( l k ) i +1 ) ⊂ P ( l k ) i , and g i +1 i ( B # x,i +1 ) ⊂ B x,i . S o one ded u ces that g i +1 i ( P ( l k ) x,i +1 ) ⊂ g i +1 i ( B # x,i +1 ) ∩ g i +1 i ( P ( l k ) i +1 ) ⊂ B x,i ∩ P ( l k ) i ⊂ P x,i ∩ P ( l k ) i = P ( l k ) x,i . Th us P ′ ( l k ) x := ( P ( l k ) x,i , g i +1 i ) is an inv erse sequence of compact p olyhedr a. Since S x,i ⊂ P ( l k ) x,i and g i +1 i ( P ( l k ) x,i +1 ) ⊂ B x,i ∩ P ( l k ) i = S x,i , it is clear that lim P ′ ( l k ) x is the same as the limit of the inv erse sequence ( S x,i , g i +1 i ), i.e., that ( π | A k ) − 1 ( x ) = lim ( S x,i , g i +1 i ) = lim ( P ( l k ) x,i , g i +1 i ) . W e sh all sh ow that for eac h i ∈ N , if 0 ≤ r ≤ l k − 1 and h : S r → P ( l k ) x,i is a map, then h is homotopic to a constan t map. Since dim S r = r < l k , h is homoto pic in P ( l k ) x,i to a map Simultan eous Z /p -acyclic r esolutions 9 that carries S r in to P ( l k − 1) x,i (see remark ab out stabilit y theory). But P x,i is contrac tible, so th e inclus ion P ( l k − 1) x,i ֒ → P ( l k ) x,i is null-homoto pic. This sh o ws that h : S r → P ( l k ) x,i is n ull-homotopic. So all fib ers of π | A k are UV l k − 1 . T o pro v e (IV), w e need to show that an y fib er of π | A k is Z /p -acyclic , i.e., for infinitely man y ind exes i , the map g i +1 i | P ( l k ) x,i +1 : P ( l k ) x,i +1 → P ( l k ) x,i induces the zero-homomorphism of cohomology groups H m ( P ( l k ) x,i ; Z /p ) → H m ( P ( l k ) x,i +1 ; Z /p ), for all m ∈ N (we need not w orry ab out m = 0 b ecause the P ( l k ) x,i ’s are ( l k − 1)-connected, so their reduced zero- cohomology groups are trivial). W e will b e fo cus ing on those indexes i for which g i +1 i | P x,i +1 ≃ ω ◦ ¯ f i | P x,i +1 , as mentioned in the conditions of Corollary 2.2. It is, in fact, enough to s ho w that the map g i +1 i | P ( l k ) x,i +1 : P ( l k ) x,i +1 → P ( l k ) x,i induces the ze ro- homomorphism of homology group s w ith Z /p -co efficien ts. Here is why this is true. Notice that eac h of P x,i +1 and P x,i is a closed verte x star (in the coarser triangulation), su b divided so that eac h original s im p lex of the v ertex star is collapsible as a simplicial complex. So Lemma 2.4 (for n = l k ) is true for b oth | M | = P x,i +1 and | M | = P x,i . Therefore prop ert y (I) of Lemma 2.4 is true f or b oth P ( l k ) x,i +1 and P ( l k ) x,i , and b oth are ( l k − 1)-connected. Therefore b y th e Universal Co efficien ts T heorem f or homology and cohomology w e ha v e H m ( P ( l k ) x,i +1 ; Z /p ) ∼ = H m ( P ( l k ) x,i +1 ) ⊗ Z /p, ∀ m ≥ 1 , and H m ( P ( l k ) x,i +1 ; Z /p ) ∼ = Hom( H m ( P ( l k ) x,i +1 ) , Z /p ) , ∀ m ≥ 1 , and for P ( l k ) x,i analogously , and these expressions are n on -zero on ly f or m = l k . So if g i +1 i | P ( l k ) x,i +1 : P ( l k ) x,i +1 → P ( l k ) x,i induces the zero-homomorphism H l k ( g i +1 i ; Z /p ) : H l k ( P ( l k ) x,i +1 ; Z /p ) → H l k ( P ( l k ) x,i ; Z /p ), th en for any ϕ ∈ Hom( H l k ( P ( l k ) x,i ) , Z /p ), we ha v e ϕ ◦ H l k ( g i +1 i ) = 0 ∈ Hom( H l k ( P ( l k ) x,i +1 ) , Z /p ), that is, th e induced h omomorphism H l k ( g i +1 i ; Z /p ) : H l k ( P ( l k ) x,i ; Z /p ) → H l k ( P ( l k ) x,i +1 ; Z /p ) is the zero-homomorphism. So let u s sho w that H l k ( g i +1 i ; Z /p ) : H l k ( P ( l k ) x,i +1 ; Z /p ) → H l k ( P ( l k ) x,i ; Z /p ) is the zero- homomorphism. Before pro cee ding, note that by Remark 1.5, giv en an EW-resolution ω : EW( P i , Z /p, l k ) → P i , w e kno w that ω − 1 ( P x,i ) = EW ( P x,i , Z /p, l k ), so ω | ω − 1 ( P x,i ) : E W ( P x,i , Z /p, l k ) → P x,i is also an EW-resolution. Let σ b e any ( l k + 1)-simplex of P x,i +1 , and let g σ b e a generator of H l k ( ∂ σ ). Let µ : ∂ σ ֒ → P ( l k ) x,i +1 , ν : P ( l k ) x,i +1 ֒ → P x,i +1 , and λ : P ( l k ) x,i ֒ → EW ( P x,i , Z /p, l k ) b e the inclusions. Notice th at ω ◦ ¯ f i ( P x,i +1 ) ⊂ P x,i implies th at ¯ f i ( P x,i +1 ) ⊂ EW( P x,i , Z /p, l k ), and since ¯ f i is 10 L. Rubin, V . T oni ´ c a cellular m ap , we also ha v e ¯ f i ( P ( l k ) x,i +1 ) ⊂ EW ( P x,i , Z /p, l k ) ( l k ) = P ( l k ) x,i . EW( P i , Z /p, l k ) ω   EW( P x,i , Z /p, l k ) ω |   6 V i i ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ ❘ P i +1 ¯ f i 9 9 g i +1 i / / P i P x,i +1 1 Q c c ● ● ● ● ● ● ● ● ● g i +1 i | / / ¯ f i | 6 6 P x,i 6 V i i ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ ❙ P ( l k ) x,i +1 4 T ν f f ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ g i +1 i | / / ¯ f i | - - P ( l k ) x,i 4 T f f ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ ◆ , L λ Y Y ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ✹ ∂ σ ?  µ O O Since Lemma 2.3 is tru e for | M | = P x,i and n = l k , we h a v e ¯ f i | P x,i +1 ◦ ν ◦ µ = λ ◦ ¯ f i | P ( l k ) x,i +1 ◦ µ is null-homotopic, and H l k ( ¯ f i | P ( l k ) x,i +1 ◦ µ )( g σ ) = r X e =1 ε e h e ∈ H l k ( P ( l k ) x,i ) , where ε e ≡ 0 (mo d p ) . By Lemma 2.4 applied to P x,i +1 with n = l k , we can select σ 1 , . . . , σ s to b e some ( l k + 1)- simplexes of P x,i +1 so that the images g 1 , . . . , g s of the generators of H l k ( ∂ σ 1 ) , . . . , H l k ( ∂ σ s ) induced b y the inclusions ∂ σ j ֒ → P ( l k ) x,i +1 form a basis for H l k ( P ( l k ) x,i +1 ). Then for an y g ∈ H l k ( P ( l k ) x,i +1 ), H l k ( ¯ f i | P ( l k ) x,i +1 )( g ) = H l k ( ¯ f i | P ( l k ) x,i +1 )( s X j =1 m j g j ) = s X j =1 m j ( r X e =1 ε j,e h e ) , where m j ∈ Z , and ε j,e ≡ 0 (mo d p ). Finally , s in ce we kno w that g i +1 i | P x,i +1 ≃ ω ◦ ¯ f i | P x,i +1 and ω | P ( l k ) x,i = id , we ha v e that g i +1 i | P ( l k ) x,i +1 ≃ ¯ f i | P ( l k ) x,i +1 . Th erefore H l k ( g i +1 i | P ( l k ) x,i +1 ) = H l k ( ¯ f i | P ( l k ) x,i +1 ), so the last equation implies th at H l k ( g i +1 i | P ( l k ) x,i +1 ; Z /p ) is the zero-homomorphism.  3. Pr oof of Theorem 1.1 Pro of of Th eorem 1.1 : Cho ose a function ν : N → N such that for eac h i ∈ N , (i) ν ( i ) ≤ i , and (ii) ν − 1 ( i ) is infinite. One may assu m e that X ⊂ Q = Hilb ert cu b e. W e are going to prov e the existence for eac h k ∈ N ∪ {∞} of a certain sequence S j = ( n j , ( P k j ) , ε j , δ j , ( e τ k j )) , ( τ k j )) , j ∈ N , of en tities, and a sequence of maps ( g j +1 j ) , j ∈ N , such that: • n j ∈ N ; • P 1 j ⊂ P 2 j ⊂ · · · ⊂ P ∞ j are compact su bp olyhedr a of I n j ; • ε j , δ j > 0; Simultan eous Z /p -acyclic r esolutions 11 • e τ ∞ j is a triangulation of P ∞ j , τ ∞ j is a sub d ivision of e τ ∞ j , e τ k j = e τ ∞ j | P k j is a triangulation of P k j , τ k j = τ ∞ j | P k j is a su b division of e τ k j , (w e will consider P ∞ j = ( P ∞ j , τ ∞ j ) and P k j = ( P ∞ j , τ k j )); • g j +1 j : P ∞ j +1 → P ∞ j is a simp licial map r elativ e to τ ∞ j +1 and τ ∞ j . A diagram that might h elp: P 1 1 = P ∞ 1 ⊂ I n 1  _   P 1 2 g 2 1 | 6 6   / / P 2 2 = P ∞ 2 g 2 1 O O ⊂ I n 2  _   P 1 3 g 3 2 | 6 6   / / P 2 3 g 3 2 | 6 6   / / P 3 3 = P ∞ 3 g 3 2 O O ⊂ I n 3  _   . . . g 4 3 | 7 7 g 4 3 | 6 6 g 4 3 | 6 6 . . . g 4 3 O O . . .  _   P 1 j   / / P 2 j   / / · · ·   / / P j − 1 j   / / P j j = P ∞ j g j j − 1 O O ⊂ I n j  _   P 1 j +1   / / g i +1 i | 6 6 P 2 j +1   / / g i +1 i | 7 7 · · ·   / / P j − 1 j +1   / / g i +1 i | 7 7 P j j +1   / / g i +1 i | 6 6 P j +1 j +1 = P ∞ j +1 g j +1 j O O ⊂ I n j +1 W e sh all require that for eac h j ∈ N and k ∈ N : (1) j > 1 n j − 1 < n j ; (2) j ≥ 1 if j ≤ k < ∞ , then P k j = P ∞ j and P r j ⊂ in t I n j P r +1 j whenev er r < j ; (3) j ≥ 1 X ⊂ int Q ( P ∞ j × Q n j ) ⊂ N ( X, 2 j ), and, whenev er k < j , X k ⊂ in t Q ( P k j × Q n j ) ⊂ N ( X k , 2 j ); (4) j > 1 p n j − 1 ( P k j ) ⊂ in t I n j − 1 P k j − 1 ; (5) j > 1 if u , v ∈ Q and ρ ( u, v ) ≤ ε j , then ρ ( p n j − 1 ( u ) , p n j − 1 ( v )) < δ j − 1 ; (6) j ≥ 1 9 2 n j < ε j ; (7) j ≥ 1 δ j < 1 2 n j − 1 ; (8) j ≥ 1 τ ∞ j | | γ | is collapsible ∀ γ ∈ e τ ∞ j and mesh τ ∞ j < δ j 2 ; (9) j ≥ 1 if x ∈ X , then there exists a con tractible sub p olyhedron P ∞ x,j of P ∞ j , wh ic h is th e closed star of a vertex in the triangulation e τ ∞ j , i.e., P ∞ x,j = St( v , e τ ∞ j ) for some v ∈ ( e τ ∞ j ) (0) , and su c h that N ( p n j ( x ) , 2 δ j ) ∩ P ∞ j ⊂ P ∞ x,j ⊂ N ( p n j ( x ) , ε j ) ∩ P ∞ j ; ( P ∞ x,j is considered with the triangulation τ ∞ j , so it is a sub d ivided v ertex star); if k < j , and x ∈ X k , then there exists a contract ible sub p olyhedron P k x,j of P k j , whic h is the closed s tar of a vertex in the triangulation e τ k j , i.e., P k x,j = St( v , e τ k j ) for some v ∈ ( e τ k j ) (0) , and suc h th at N ( p n j ( x ) , 2 δ j ) ∩ P k j ⊂ P k x,j ⊂ N ( p n j ( x ) , ε j ) ∩ P k j ; ( P k x,j is considered with the triangulation τ k j ). Th is statemen t is also tr u e when k ≥ j , b ecause then P k x,j = P ∞ x,j , P k j = P ∞ j and X k ⊂ X ; (10) j > 1 whenev er x ∈ P ∞ j there exists a simplex σ of τ ∞ j − 1 suc h that g j j − 1 ( x ) ∈ σ , and p n j − 1 ( x ) lies in N ( σ , δ j − 1 2 ) (and therefore, it follo w s from here and (8) j − 1 that, ρ ( g j j − 1 ( x ) , p n j − 1 ( x )) < δ j − 1 / 2 + δ j − 1 / 2 = δ j − 1 for all x ∈ P ∞ j ); (11) j > 1 g j j − 1 ( P k j ) ⊂ P k j − 1 ; and 12 L. Rubin, V . T oni ´ c (12) j > 1 g j j − 1 | P ν ( j − 1) j ≃ ω ◦ ¯ f j − 1 , where ω : EW( P ν ( j − 1) j − 1 , Z /p, l ν ( j − 1) ) → P ν ( j − 1) j − 1 is an Ed wards– W alsh pro jection, and ¯ f j − 1 : P ν ( j − 1) j → EW( P ν ( j − 1) j − 1 , Z /p, l ν ( j − 1) ) is a cellular map. Moreo ver, f or all x ∈ X ν ( j − 1) , we h a v e that ω ◦ ¯ f j − 1 ( P ν ( j − 1) x,j ) ⊂ P ν ( j − 1) x,j − 1 , and g j j − 1 | P ν ( j − 1) x,j ≃ ω ◦ ¯ f j − 1 | P ν ( j − 1) x,j . EW( P ν ( j − 1) j − 1 , Z /p, l ν ( j − 1) ) ω   EW( P ν ( j − 1) x,j − 1 , Z /p, l ν ( j − 1) ) ω |   7 W j j ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ P ν ( j − 1) j ¯ f j − 1 8 8 g j j − 1 | / / P ν ( j − 1) j − 1 P ν ( j − 1) x,j 2 R d d ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ ❍ g j j − 1 | / / ¯ f j − 1 | 5 5 P ν ( j − 1) x,j − 1 7 W j j ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ ❯ Before pro ving the existence of su c h d ata, let us see why they w ould imply th e conclusion of Theorem 1.1. F or eac h i ∈ N , let P ∞ i corresp ond to P i from the statemen t of Lemma 2.1. Applying (5), (1), (6), (1 0) and (7), one sees that the conditions (i)–(v) of Lemma 2.1 are clearly tru e. C ondition (4) i +1 implies (vi) and one ma y use (3) to see that X = ∞ \ i =1 P ∞ i × Q n i . Let Z := lim( P ∞ i , g i +1 i ) . Surely Z is a metrizable compactum, and w e get the map π : Z → X defined by the form ula giv en in Lemma 2.1 (b). T o s ee that π is su rjectiv e, for eac h i ∈ N let T i = P i = P ∞ i (in Lemma 2.1 ). According to the notation of the last part of Lemma 2.1, one sees that for x ∈ X , S x,i = B x,i = N ( p n i ( x ) , 2 δ i ) ∩ P ∞ i (while B # x,i = N ( p n i ( x ) , ε i ) ∩ P ∞ i ). No tice that the fi rst part of (2) i together with (3) i implies (13) p n i ( X ) ⊂ in t I n i P ∞ i , and ∀ k ∈ N , p n i ( X k ) ⊂ in t I n i P k i . So p n i ( x ) ∈ P ∞ i and therefore p n i ( x ) ∈ B x,i , sho wing that the latter is n ot empty . Th e map ˜ π is the same as π in this setting, so (g) of Lemma 2.1 sho ws that π is sur jectiv e. One then chec ks that all the hypotheses of Corollary 2.2 except for the very last one (whic h we do not n eed y et) are also satisfied. Th us (I)–(I I I) of Corollary 2.2 h old true, so π is a cell-lik e m ap, and we are assured of the existence of the closed sub s paces A k , k ≥ 1, where A k := lim(( P ∞ i ) ( l k ) , g i +1 i ) , as required by Theorem 1.1 s o that dim A k ≤ l k , and wh en k ∈ N , π carries A k in a UV l k − 1 manner onto X . W e must identify the closed subs p aces Z 1 ⊂ Z 2 ⊂ . . . of Z , pro v e th ey satisfy (a)–(c) of Theorem 1.1, and sh o w that Z k ⊂ A k when k ∈ N . Fix k ∈ N . In the last part of Lemma 2.1, instead of putting T i = P ∞ i , as we j u st did to obtain Z , π , and the sets A k , this time put T i = ( P k i ) ( l k ) . Usin g (11), th e fact that τ k i = τ ∞ i | P k i , and that g i +1 i is simplicial from τ ∞ i +1 to τ ∞ i , one sees that Simultan eous Z /p -acyclic r esolutions 13 (14) g i +1 i (( P k i +1 ) ( l k ) ) ⊂ ( P k i ) ( l k ) . No w let Z k := lim(( P k i ) ( l k ) , g i +1 i ) , i.e., T k = (( P k i ) ( l k ) , g i +1 i ), and Z k = lim T k . Using (2) we see that P k i ⊂ P ∞ i for all i ∈ N . Of course, ( P k i ) ( l k ) ⊂ ( P ∞ i ) ( l k ) , and we deduce that Z k ⊂ A k as requested in Theorem 1.1. Moreo ver, dim Z k ≤ dim A k ≤ l k , s o (a) of Theorem 1.1 has b een resolve d. It is also clear that Z 1 ⊂ Z 2 ⊂ . . . as requir ed by Th eorem 1.1. Next put ˜ π k = π | Z k : Z k → X . If ( a 1 , a 2 , . . . ) is a thread of Z k , then a i ∈ P k i for eac h i ∈ N . T aking into acco unt (b ) of Lemma 2.1, as well as (3) i whic h implies that (15) X k = ∞ \ i =1 P k i × Q n i , one sees that ˜ π k ( Z k ) ⊂ X k . Supp ose no w that x ∈ X k . With the c hoice of T i = ( P k i ) ( l k ) , the sets S x,i in the last part of Lemma 2.1 b ecome S x,i = B x,i ∩ ( P k i ) ( l k ) . If w e can sho w that for eac h i ∈ N , S x,i 6 = ∅ , then (g) of L emma 2.1 would yield ˜ π k ( Z k ) ⊃ X k . Ind eed, it is sufficient to sho w that B # x,i ∩ ( P k i ) ( l k ) 6 = ∅ , since B # x,i +1 ∩ ( P k i +1 ) ( l k ) maps in to S x,i under g i +1 i (see (c) of Lemma 2.1 and (14)). Because of (15), x ∈ P k i × Q n i , so p n i ( x ) ∈ P k i . Applying (6) i –(8) i , we find a vertex v ∈ ( P k i ) (0) ⊂ ( P k i ) ( l k ) suc h that ρ ( p n i ( x ) , v ) < δ i 2 < 1 2 n i < ε i . This means v ∈ B # x,i ∩ ( P k i ) ( l k ) , i.e., B # x,i ∩ ( P k i ) ( l k ) 6 = ∅ . Therefore (b) of Th eorem 1.1 is true. Finally , after r eplacing A k from the statement of Corollary 2.2 with Z k , the u ltimate condition of Corollary 2.2, in v olving infi nitely many ind exes, is n o w op erative b ecause of (i) and (ii) of this section, and (12) for ν ( i − 1) = k . If w e app ly (IV) of Corollary 2.2, then w e find th at ˜ π k = π | Z k : Z k → X k is a Z /p -acyclic map. Th us, our p ro of of Theorem 1.1 will b e complete once w e ha v e obtained th e information in statemen ts (1)–(12). Inductiv e construction b egins : F or the basis of the indu ction ( j = 1), w e choose n 1 = l 1 and P k 1 = I n 1 = I l 1 for all k ∈ N ∪ {∞} . T h us (2) 1 and (3) 1 are satisfied. Next c ho ose an y ε 1 > 9 2 l 1 , so (6 ) 1 is sati sfied. It remains to pro duce δ 1 > 0 and triangulations e τ ∞ 1 and τ ∞ 1 of P ∞ 1 = I l 1 so that (7) 1 –(9) 1 are satisfied. Begin b y taking a triangulation e τ ∞ 1 of P ∞ 1 suc h that mesh e τ ∞ 1 < ε 1 2 . T h e op en stars of the v ertices in e τ ∞ 1 form a co v er for P ∞ 1 = I l 1 . Note that these op en stars are truly op en sets in I l 1 . F or an y x ∈ X , th er e exists a vertex v of e τ ∞ 1 suc h that p l 1 ( x ) ∈ St( v , e τ ∞ 1 ). Note that for an y y ∈ St( v, e τ ∞ 1 ), ρ ( y , p l 1 ( x )) ≤ 2 mesh e τ ∞ 1 < ε 1 , so St( v , e τ ∞ 1 ) ⊂ N ( p l 1 ( x ) , ε 1 ) = N ( p l 1 ( x ) , ε 1 ) ∩ P ∞ 1 . Since U := { St( v , e τ ∞ 1 ) | v ∈ ( e τ ∞ 1 ) (0) } is a co v er for P ∞ 1 whic h is compact, let λ b e a Leb esgue num b er of U . Pic k a δ 1 > 0 s u c h that 4 δ 1 < min n λ, 4 2 l 1 − 1 o . No w (7) 1 is also satisfied. Then for any x ∈ X , the closed ball N ( p l 1 ( x ) , 2 δ 1 ) is contai ned in some St( v , e τ ∞ 1 ), for a v ertex v ∈ ( e τ ∞ 1 ) (0) . Pic k one suc h star, and call its closure P ∞ x, 1 . Notice that P ∞ x, 1 is con tractible. Th us we get (9 ) 1 for x ∈ X : N ( p l 1 ( x ) , 2 δ 1 ) = N ( p l 1 ( x ) , 2 δ 1 ) ∩ P ∞ 1 ⊂ P ∞ x, 1 ⊂ N ( p l 1 ( x ) , ε 1 ) ∩ P ∞ 1 . Fin ally , choose a triangulation τ ∞ 1 so that it refines e τ ∞ 1 , and so that (8) 1 is satisfied. Assume that we ha v e completed the construction of S j for 1 ≤ j ≤ i , and g j +1 j for 1 ≤ j ≤ i − 1. Cho ose an op en co v er V of P ∞ i ha ving the prop ert y that mesh V < δ i 2 . Then select a finer op en co v er W suc h that an y tw o W -near m aps of any space int o P ∞ i are V -homotopic. Let τ b e a sub division of τ ∞ i suc h that N ( St( v , τ ) , ˜ ε ) lies in an elemen t 14 L. Rubin, V . T oni ´ c of W , f or ev ery verte x v ∈ τ (0) , wh ere ˜ ε > 0 is chosen so that: for an y pr incipal simplex σ of the triangulation τ , all of the p oints of the op en neigh b orho o d N ( σ, ˜ ε ) are at most one (principal) simplex aw ay from σ (i.e., if u ∈ N ( σ, ˜ ε ) \ σ , th en u ∈ γ = a n eigh b oring principal simplex of σ ). (Su r ely this ˜ ε exists b ecause P ∞ i is compact. Also, it is cle ar that ˜ ε ≤ mesh τ , and that it w ould b e enough to c ho ose τ s o that 2(mesh τ + ˜ ε ) < some fixed Leb esgue num b er of W . Also note that τ can b e chosen so that τ | | γ | is still collapsible, ∀ γ ∈ e τ ∞ i .) If i = 1, replace τ ∞ 1 b y τ , but con tinue to use the notation τ ∞ 1 for it. Note that pr op erties (8) 1 and (9) 1 , wh ic h are the only ones affected by this c h ange, are still true. If i > 1, c hoose a m ap µ : P ∞ i → P ∞ i whic h is simplicial from τ to τ ∞ i and whic h is a simplicial appro ximation to th e ident it y on P ∞ i . Then the map g i i − 1 ◦ µ is sim p licial fr om τ to τ ∞ i − 1 , and ¯ f i − 1 ◦ µ | P ν ( i − 1) i is cellular with resp ect to the triangulation on P ν ( i − 1) i induced b y τ for ¯ f i − 1 . I f we replace g i i − 1 b y g i i − 1 ◦ µ , ¯ f i − 1 b y ¯ f i − 1 ◦ µ | P ν ( i − 1) i , and τ ∞ i b y τ , then all the conditions (1)–(1 2) for in d ex i still prev ail (the only ones affected b eing (8) i –(12) i ). S o w e assume that these replacemen ts hav e b een made, bu t contin u e to use g i i − 1 , ¯ f i − 1 and τ ∞ i to denote th e resp ectiv e b onding map, cellular map in (12) i and triangulation. Construction of t he p olyhedra P k i +1 and the b onding map g i +1 i b egins . App ly Lemma 1.9 to X ν ( i ) , which has dim Z /p X ν ( i ) ≤ l ν ( i ) , where ν ( i ) ≤ i , and (u sing (13)) the map p n i | X ν ( i ) : X ν ( i ) → P ν ( i ) i , to p r o duce a map f ′ : X ν ( i ) → EW ( P ν ( i ) i , Z /p, l ν ( i ) ) such that for an y x ∈ X ν ( i ) , when p n i ( x ) lies in a particular simplex of P ν ( i ) i , then so do es ω ◦ f ′ ( x ). Th ere is a pr incipal simp lex σ x of P ν ( i ) i that con tains b oth ω ◦ f ′ ( x ) and p n i ( x ). W e can extend f ′ o v er an op en neigh b orho o d e U of X ν ( i ) in th e Hilb ert cub e Q , to get a map f ′′ : e U → EW( P ν ( i ) i , Z /p, l ν ( i ) ). EW( P ν ( i ) i , Z /p, l ν ( i ) ) ω   e U f ′′ 8 8 X ν ( i ) ? _ o o p n i | / / f ′ = = ③ ③ ③ ③ ③ ③ ③ ③ ③ ③ P ν ( i ) i No w w e can find a neigh b orho o d U of X ν ( i ) in e U suc h that: (16) for any u ∈ U , ω ◦ f ′′ ( u ) and p n i ( u ) b elong to th e op en ˜ ε –neigh b orho o d of some principal simplex σ x of P ν ( i ) i . Here is ho w w e fin d U : sin ce p n i is con tin uous (on Q ⊃ e U ), for any x ∈ X ν ( i ) , and for the ab o v e ˜ ε , ther e exists an op en neighborh o o d e Q x of x in e U such that p n i ( e Q x ) ⊂ N ( σ x , ˜ ε ). Since ω ◦ f ′ ( x ) ∈ σ x , then f ′ ( x ) ∈ ω − 1 ( σ x ) ⊂ ω − 1 ( N ( σ x , ˜ ε )). Now f ′′ ( x ) = f ′ ( x ), so th e con tin uit y of f ′′ guaran tees an op en neighborh o o d ¯ Q x of x with f ′′ ( ¯ Q x ) ⊂ ω − 1 ( N ( σ x , ˜ ε )). Of course, ω ◦ f ′′ ( ¯ Q x ) ⊂ N ( σ x , ˜ ε ). No w let Q x := e Q x ∩ ¯ Q x and d efine U := S x ∈ X Q x . Clearly this U has the needed prop ert y . Using the uniform con tin uit y of p n i on Q , c ho ose ε i +1 so that (5) i +1 holds: if u, v ∈ Q are su c h that ρ ( u, v ) < ε i +1 , then ρ ( p n i ( u ) , p n i ( v )) < δ i . In order to choose n i +1 : notice that one ma y find m 0 ∈ N such that if m ≥ m 0 , then X ⊂ p m ( X ) × Q m ⊂ N ( X, 2 i +1 ), and for all k ≤ i , X k ⊂ p m ( X k ) × Q m ⊂ N ( X k , 2 i +1 ). Simultan eous Z /p -acyclic r esolutions 15 Define n i +1 > m ax { l i +1 − 1 , n i , m 0 , log 2 ( 9 ε i +1 ) } . T his en sures that prop erties (1) i +1 and (6) i +1 hold. No w is the time to c ho ose compact p olyhed ra P ∞ i +1 = P i +1 i +1 , P i i +1 , . . . , P ν ( i ) i +1 , . . . , P 1 i +1 in I n i +1 . First note that there is an op en neigh b orho o d e V of p n i +1 ( X ) in I n i +1 suc h that e V × Q n i +1 ⊂ N ( X , 2 i +1 ). Ch o ose a compact p olyhedr on P ∞ i +1 ⊂ I n i +1 so that (17) p n i +1 ( X ) ⊂ in t I n i +1 P ∞ i +1 ⊂ P ∞ i +1 ⊂ e V , and P ∞ i +1 ⊂ p − 1 n i (in t I n i ( P ∞ i )) . This can b e done b ecause (3) i implies (13) ∞ i , i.e., p n i ( X ) = p n i ( p n i +1 ( X )) ⊂ int I n i ( P ∞ i ), so p n i +1 ( X ) ⊂ p − 1 n i (in t I n i ( P ∞ i )). Note that (17) implies pr op erties (3) i +1 and (4) i +1 for P ∞ i +1 . T o satisfy the first p art of (2) i +1 , we n ame P k i +1 = P ∞ i +1 for all k ≥ i + 1. Let us no w choose P k i +1 , for k = i, i − 1 , . . . , 1, whic h w e d o by a do wnw ard recurs ion. If k > ν ( i ), then here is how we m ak e our c hoice: find an op en neigh b orho o d e V k of p n i +1 ( X k ) in I n i +1 suc h that f V k × Q n i +1 ⊂ N ( X k , 2 i +1 ). Cho ose a compact p olyhedron P k i +1 ⊂ I n i +1 so that (18) p n i +1 ( X k ) ⊂ in t I n i +1 P k i +1 ⊂ P k i +1 ⊂ f V k , and P k i +1 ⊂ p − 1 n i (in t I n i ( P k i )) T in t I n i +1 ( P k +1 i +1 ). This can b e done b eca use (3) i implies (13) k i , i.e., p n i ( X k ) = p n i ( p n i +1 ( X k )) ⊂ int I n i ( P k i ), so p n i +1 ( X k ) ⊂ p − 1 n i (in t I n i ( P k i )). Also note that p n i +1 ( X k ) ⊂ in t I n i +1 ( P k +1 i +1 ), b ecause b efore w e reac h the construction of P k i +1 , P k +1 i +1 is already co nstructed so that (13) k +1 i +1 is true, so p n i +1 ( X k +1 ) ⊂ in t I n i +1 ( P k +1 i +1 ), and also recall that X k ⊂ X k +1 ⊂ X . Note that (18) imp lies p r op erties (2) i +1 (the s econd part), (3) i +1 and (4) i +1 for P k i +1 , when ν ( i ) < k ≤ i . F or k = ν ( i ), w e require the ab o v e men tio ned prop erties and, additionally , that P ν ( i ) i +1 × Q n i +1 ⊂ U , wh ere U is the neigh b orho o d of X ν ( i ) indicated in (16). F or k < ν ( i ), p ro ceed w ith the construction of P k i +1 as in the case of i ≥ k > ν ( i ). Conclude that prop erties (2) i +1 –(4) i +1 are no w true for all k ∈ { 1 , 2 , . . . , i } ∪ {∞} for wh ic h they app ly . Let ˜ f := f ′′ | P ν ( i ) i +1 × Q n i +1 ◦ i : P ν ( i ) i +1 → EW( P ν ( i ) i , Z /p, l ν ( i ) ), where i : P ν ( i ) i +1 → P ν ( i ) i +1 × Q n i +1 is th e in clusion. Cho ose δ i +1 and triangulations e τ ∞ i +1 and τ ∞ i +1 for P ∞ i +1 , which are also triangulating all P k i +1 for k < i (where e τ k i +1 := e τ ∞ i +1 | P k i +1 and τ k i +1 := τ ∞ i +1 | P k i +1 ), so th at (7) i +1 , (8) i +1 and (9) i +1 hold. Here is ho w this is done: b egin by taking a triangulation e τ ∞ i +1 of P ∞ i +1 , which also triangulates all P k i +1 , s u c h that mesh e τ ∞ i +1 < ε i +1 2 . The op en stars in e τ k i +1 of th e v ertices of e τ k i +1 form a co v er U k i +1 = { St( v, e τ k i +1 ) | v ∈ ( e τ k i +1 ) (0) } for P k i +1 , where k ∈ { 1 , 2 , . . . , i } ∪ {∞} . Note that for x ∈ X , p n i +1 ( x ) has to b elong to some St( v , e τ ∞ i +1 ). Then for an y y ∈ St( v, e τ ∞ i +1 ), ρ ( y , p n i +1 ( x )) ≤ 2 mesh e τ ∞ i +1 < ε i +1 , so St( v , e τ ∞ i +1 ) ⊂ N ( p n i +1 ( x ) , ε i +1 ) ∩ P ∞ i +1 . Analogously , since for x ∈ X k , p n i +1 ( x ) h as to b elo ng to some St( v , e τ k i +1 ) ⊂ St( v , e τ ∞ i +1 ), we get St( v , e τ k i +1 ) ⊂ N ( p n i +1 ( x ) , ε i +1 ) ∩ P k i +1 , for k ∈ { 1 , 2 , . . . , i } . On the other hand, s in ce P k i +1 is compact for k ∈ { 1 , 2 , . . . , i } ∪ {∞} , eac h co v er U k i +1 of P k i +1 has a Leb esgue num b er λ k i +1 , k ∈ { 1 , 2 , . . . , i } ∪ {∞} . Th us it is enough to pic k a δ i +1 > 0 such that 4 δ i +1 < min  n λ k i +1 : k ∈ { 1 , 2 , . . . , i } ∪ {∞} o ∪  4 2 n i +1 − 1  . 16 L. Rubin, V . T oni ´ c No w (7) i +1 is satisfied. Also, for an y x ∈ X k , N ( p n i +1 ( x ) , 2 δ i +1 ) ∩ P k i +1 is con tained in some St( v, e τ k i +1 ), for a verte x v ∈ ( e τ k i +1 ) (0) . Pic k one s uc h star, and call its closur e P k x,i +1 . Notice that P k x,i +1 is contract ible. T h us w e get (9) i +1 for k < i + 1: N ( p n i +1 ( x ) , 2 δ i +1 ) ∩ P k i +1 ⊂ P k x,i +1 ⊂ N ( p n i +1 ( x ) , ε i +1 ) ∩ P k i +1 . Analogously , w e get (9) i +1 for k = ∞ and x ∈ X . Finally , c ho ose a triangulation τ ∞ i +1 so that it refin es e τ ∞ i +1 , and so that (8) i +1 is satisfied. No w that we hav e a triangulation for P ∞ i +1 , and therefore for P ν ( i ) i +1 to o, tak e a cellular appro ximation ¯ f i : P ν ( i ) i +1 → EW( P ν ( i ) i , Z /p, l ν ( i ) ) of ˜ f : P ν ( i ) i +1 → EW( P ν ( i ) i , Z /p, l ν ( i ) ). Since P ν ( i ) i +1 × Q n i +1 ⊂ U , (16) is v alid for any u ∈ P ν ( i ) i +1 , th at is, ω ◦ f ′′ ( u, 0) and p n i ( u, 0) = p n i ( u ) b elong to the ˜ ε -neigh b orho o d of the same p rincipal simplex σ ∈ τ ∞ i . W e also know that ω ◦ f ′′ ( u, 0) b elo ngs to a pr incipal simplex γ whic h is a neigh b or of σ (the choice of ˜ ε makes sure that γ and σ are neigh b ors). Note that ω ◦ f ′′ ( u, 0) = ω ◦ f ′′ ◦ i ( u ) = ω ◦ ˜ f ( u ) ∈ γ . No w ω ◦ ¯ f i ( u ) also b elongs to γ , b ecause ¯ f i is a cellular appro ximation of ˜ f , and the prop erties of the Edwards–W alsh resolution ω guarantee that ˜ f ( u ) ∈ ω − 1 ( γ ) imp lies that ¯ f i ( u ) ∈ ω − 1 ( γ ). So we ha v e found a s implex γ of τ ∞ i suc h that ω ◦ ¯ f i ( u ) ∈ γ , and p n i ( u ) b elongs to the ˜ ε - neigh b orho o d of the closed star of a verte x v that is a common ve rtex of γ and σ . T herefore ω ◦ ¯ f i : P ν ( i ) i +1 → P ν ( i ) i and p n i | P ν ( i ) i +1 : P ν ( i ) i +1 → P ν ( i ) i are W -near, and therefore V -homotopic. According to Lemma 1.10 there exists a con tin uous extension ϕ : P ∞ i +1 → P ∞ i of ω ◦ ¯ f i suc h that ϕ and p n i | P ∞ i +1 are V -homotopic, and therefore V -near. EW( P ν ( i ) i , Z /p, l ν ( i ) ) ω   P ν ( i ) i   / / P ∞ i P ν ( i ) x,i +1   / / P ν ( i ) i +1 _  i   ¯ f i 9 9 ⑤ ④ ③ ② ② ① ✇ ✈ ✈ ✉ t s s ˜ f ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ; ; ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ p n i | ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ 7 7 ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥ ♥   / / P ∞ i +1 p n i | s s s s s s s s s s 9 9 s s s s s s s s s s g i +1 i < < ϕ 6 6 U P ν ( i ) i +1 × Q n i +1 f ′′ | @ @ ? _ o o X ν ( i ) p n i | C C ? _ o o With th is, (4) i +1 , and the fact that w e could hav e c hosen V as fine as we wish, we may assume that ϕ ( P k i +1 ) ⊂ P k i , for all 1 ≤ k ≤ ∞ . Finally , making τ ∞ i +1 finer if necessary (but so th at the pr op erties of colla psibilit y required in (8) i +1 are still preserved), take g i +1 i : P ∞ i +1 → P ∞ i to b e a simplicial app ro ximation of ϕ . Therefore, for an y u ∈ P ∞ i +1 , th ere exists a s implex σ ∈ τ ∞ i suc h that g i +1 i ( u ), ϕ ( u ) ∈ σ . W e also know that ρ ( ϕ ( u ) , p n i ( u )) < mesh V < δ i 2 , so p n i ( u ) ∈ N ( σ, δ i 2 ), i.e., prop erty (10) i +1 is true. Prop ert y (11) i +1 is true b ecause g i +1 i is a simp licial appr o ximation of ϕ . F or p rop ert y (12) i +1 , fi rst notice that g i +1 i | P ν ( i ) i +1 ≃ ϕ | P ν ( i ) i +1 = ω ◦ ¯ f i . Also, ω ◦ ¯ f i and p n i | P ν ( i ) i +1 b eing W -near imp lies th at for all x ∈ X ν ( i ) , ω ◦ ¯ f i ( P ν ( i ) x,i +1 ) ⊂ P ν ( i ) x,i . T o see w h y , tak e any u ∈ B ν ( i )# x,i +1 := N ( p n i +1 ( x ) , ε i +1 ) ∩ P ν ( i ) i +1 , i.e., ρ ( u, p n i +1 ( x )) < ε i +1 ; by (5) i +1 , Simultan eous Z /p -acyclic r esolutions 17 ρ ( p n i ( u ) , p n i ( x )) < δ i . T herefore, sin ce mesh( W ) < δ i 2 , ρ ( ω ◦ ¯ f i ( u ) , p n i ( x )) ≤ ρ ( ω ◦ ¯ f i ( u ) , p n i ( u )) + ρ ( p n i ( u ) , p n i ( x )) < δ i 2 + δ i < 2 δ i , so ω ◦ ¯ f i ( u ) ∈ B ν ( i ) x,i := N ( p n i ( x ) , 2 δ i ) ∩ P ν ( i ) i . T h us ω ◦ ¯ f i ( B ν ( i )# x,i +1 ) ⊂ B ν ( i ) x,i . S ince P ν ( i ) x,i +1 ⊂ N ( p n i +1 ( x ) , ε i +1 ), ω ◦ ¯ f i ( P ν ( i ) x,i +1 ) ⊂ P ν ( i ) x,i , to o. Also, ϕ ( P ν ( i ) x,i +1 ) = ω ◦ ¯ f i ( P ν ( i ) x,i +1 ) ⊂ P ν ( i ) x,i , s o g i +1 i , b eing a simplicial approxima tion of ϕ , h as the prop ert y g i +1 i ( P ν ( i ) x,i +1 ) ⊂ P ν ( i ) x,i . Finally , g i +1 i | P ν ( i ) x,i +1 ≃ ϕ | P ν ( i ) x,i +1 = ω ◦ ¯ f i | P ν ( i ) x,i +1 , so prop erty (12) i +1 holds.  Remark 3.1. Note that fr om our c onstruction of Z , it fol lows that in gener al Z is infinite dimensional. Remark 3.2. If we take 1 < 2 < . . . < m < . . . i nste ad of l 1 ≤ l 2 ≤ . . . ≤ l m ≤ . . . , the The or em 1.1 b e c omes p ar al lel to the r esult for dim Z fr om [AJR] . If l i = l i +1 but X l i ( X l i +1 , we get A i = A i +1 , but Z i ( Z i +1 . What if the sequence of n onempt y closed sub spaces X 1 ⊂ X 2 ⊂ . . . of the compact metrizable space X from the statemen t of Th eorem 1.1 is finite, that is, we are giv en X 1 ⊂ X 2 ⊂ · · · ⊂ X m ⊂ X ? And wh at if X itself is replaced by an X m , i.e., we hav e X 1 ⊂ X 2 ⊂ · · · ⊂ X m = X , where for eac h k ∈ { 1 , 2 , . . . , m } , dim Z /p X k ≤ l k ? In either of these cases, T heorem 1.1 yields a compact metrizable sp ace Z with closed subspaces Z 1 ⊂ Z 2 ⊂ · · · ⊂ Z m ⊂ Z , as w ell as a cell-lik e map π : Z → X w ith all of the prop erties men tioned in Th eorem 1.1, but w e can adapt th e pro of so that it would u se f ew er p olyhedra. Namely , here are the c hanges that somewhat simplify the pro of of T heorem 1.1 in b oth of the fi nite cases menti oned ab ov e. First, tak e a f unction ν : N → { 1 , 2 , . . . , m } suc h that (i) and (ii) are still satisfied. Second, c hange the conditions (2) j ≥ 1 and (3) j ≥ 1 from the original pr o of to the follo wing: (2) ′ j ≥ 1 if k ≥ min { j, m + 1 } then P k j = P ∞ j , and P r j ⊂ in t I n j P r +1 j whenev er r < min { j, m + 1 } ; (3) ′ j ≥ 1 X ⊂ int Q ( P ∞ j × Q n j ) ⊂ N ( X, 2 j ), and, whenev er k < min { j, m + 1 } , X k ⊂ in t Q ( P k j × Q n j ) ⊂ N ( X k , 2 j ); This will ens u re that we p r o duce only m + 1 sequences of p olyhedra ( P k j ) j ∈ N , k ∈ { 1 , . . . , m + 1 } , rather than the countably man y sequences that w ere required in the original pro of f or X 1 ⊂ X 2 ⊂ · · · ⊂ X m ⊂ · · · ⊂ X . The rest of the pr o of is the same, pro vided th at th e change in ind exes from (2) ′ is tak en in to account in the remainder of the pro of. It is worth noting th at, in the case when X = X m , the prop erty (3) ′ implies that w e can tak e P ∞ j = P m j , ∀ j . Still, Z and Z m w ould b e different, since Z m = lim(( P m i ) ( l m ) , g i +1 i ), and Z = lim( P ∞ i , g i +1 i ) = lim( P m i , g i +1 i ). Also , the map π | Z m : Z m → X is a surj ectiv e Z /p -acyclic m ap, w h ile π : Z → X is cell-lik e. Remark 3.3. In p articular, for m = 1 and X = X 1 such that d im Z /p X 1 ≤ l 1 , The or em 1.1 pr o duc es a c omp act metrizable sp ac e Z 1 with dim Z 1 ≤ l 1 , and a surje ctive Z /p -acyclic map π : Z 1 → X 1 . So The or em 1.1 is inde e d a gener alization of Dr anishnikov’s r esolution The or em 1.3. 4. Pr oof of a p ar ticular case of Theorem 1.1 What follo ws is an outline of a p ro of for a particular case that Th eorem 1.1 is co vering, namely f or the case wh en the sequence l 1 ≤ l 2 ≤ . . . of upp er b ounds for dim Z /p do es 18 L. Rubin, V . T oni ´ c not b ecome p ermanently stationary at any p oin t. This pro of was suggested to us by an anon ymous referee. I t do es not w ork if this sequence is ev en tually co nstan t, that is, if the spaces X i k eep c hanging, but fr om some p oin t i 0 on we h a v e l i 0 = l i 0 +1 = . . . . F or the sak e of simplicit y , let us s u pp ose that l 1 < l 2 < l 3 . . . since the pro of of this case can b e adjusted to w ork for all ca ses in whic h the sequence is not ev en tually constan t. Let X 1 ⊂ X 2 ⊂ . . . b e a sequence of non emp t y closed su bspaces of a compact metrizable space X su c h that dim Z /p X k ≤ l k , ∀ k ∈ N . Apply Dranishniko v’s Theorem 1.3 to X 1 in order to build a compact metrizable space Z 1 and a Z /p -acyclic map q 1 : Z 1 → X 1 suc h that dim Z 1 ≤ l 1 . L et Y 1 = X ∪ M ( q 1 ) b e the u n ion of X and the mapping cyllinder of q 1 . Notice that the pro jection p 1 : Y 1 → X is cell-lik e and that dim Z /p M ( q 1 ) ≤ l 1 + 1 ≤ l 2 , whic h mak es dim Z /p X 2 ∪ M ( q 1 ) ≤ l 2 . In order to pro duce Z 2 and q 2 , apply T heorem 1.3 to X 2 ∪ M ( q 1 ), with the exception of r equiring that q 2 has the prop ert y that q 2 | q − 1 2 ( Z 1 ) is a homeomorphism on to Z 1 . Then put Y 2 = X ∪ M ( q 2 ) and p 2 : Y 2 → X to b e the pr o j ection. Keep the p ro cedure inductive ly and define Z as the inv er s e limit of the inv erse sequen ce Y 1 ← Y 2 ← . . . ← Y k ← . . . Referen ces [AJR] S. A geev, R. Jim´ enez and L. Ru bin, C el l-li ke r esolutions in the str ongly c ountable Z -dimensional c ase , T opology and its Appls. 140 (2004), 5–14. [Da] R . Daverma n, De c omp ositions of Manifol ds , Academic Press, Orlando, Florida, 1986. [Dr] A. N. Dranishn iko v, Cohomolo gi c al dimension the ory of c omp act metric sp ac es , T op ology Atlas Invited Con tributions, http://at.yorku. ca/t/a/i/c/ 43.pdf [DW] J. Dydak and J. W alsh, Compl exes that arise i n c ohomolo gic al dimension the ory: a unifie d appr o ach , J. London Math. Soc. (2) (48), no. 2 (1993), 329-347 . [Ed] R . D. Edwa rds, A the or em and a question r el ate d to c ohomolo gic al dimension and c el l-like maps , Notices Amer. Math. Soc. 25 (1978), A58–A59. [Gl] L. C. Glaser, Ge ometric al c ombinatorial top olo gy, V ol. I , V an Nostrand Math. St u dies, 1970. [HW] W. Hurewicz and H. W allman, Dimension The ory , Princeton Universit y Press, 1948. [Ku] V. I. Kuz’minov, Homolo gic al Di mension The ory , Russian Math. Surveys 23 (1968), 1–45. [KY] A. Koy ama and K. Y okoi, Cohomolo gic al dimension and acyclic r esolutions , T op ology and its Appls. 120 (2002), 175–204. [Le] M. Levin, A cyclic r esolutions for arbitr ary gr oups , I sr. J. Math. 135 (2003), 193–204. [MS] S . Marde ˇ si ´ c and J. Segal, Shap e The ory , North-Holland, Amsterdam, 1982. [Sp] E. Spanier, Algebr aic T op olo gy , McGraw-Hill, New Y ork, 1966. [W a] J. W alsh, Shap e The ory and Ge ometric T op olo gy , Lecture Notes in Mathematics, volume 870, Sp ringer V erlag, Berlin, 1981, pp. 105–118. Dep ar tment of Ma thema tics, U niversity of Oklahoma, 601 Elm A ve , room 423, Norman, Oklahoma 73019, USA E-mail addr ess : lrubin@ou.edu Dep ar tment of Ma thema tics, B en Gurion Universi ty of the Negev, P.O.B. 653, B e’er Shev a 84105, Israe l E-mail addr ess : vera.tonic@gmai l.com