Vlume 13, 1988 Pages 73 82 http://tplgy.auburn.edu/tp/ QUASICOMPONENTS AND SHAPE THEORY by Jerzy Dydak and Manuel Alns Mrón Tplgy Prceedings Web: http://tplgy.auburn.edu/tp/ Mail: Tplgy Prceedings Department f Mathematics & Statistics Auburn Uniersity, Alabama 36849, USA E-mail: tplg@auburn.edu ISSN: 0146-4124 COPYRIGHT c by Tplgy Prceedings. All rights resered.
TOPOLOGY PROCEEDINGS Vlume 13 1988 73 QUASICOMPONENTS AND SHAPE THEORY Jerzy Dydak and Manuel Alns Mrn K. Brsuk [BJ (p. 214) cnstructed a functr A: SH ~ TOP frm the shape categry f cmpacta t the CM tplgical categry such that A(X) is the space f cmpnents f X. Mreer, fr eery fundamental sequence f = {fk,x,y} frm X t Y and fr eery cmpnent C f X, {fk,c,a(c)} is a fundamental sequence frm C t A[fJ(C). Seeral authrs (see [Ba l,2,3 J, [GJ and [SJ) tried t generalize this result t nn-cmpact spaces (lcally cmpact r metrizable). In the present paper we shw that the crrect setting fr pssible generalizatins is the shape analgue f quasicmpnents. Gien a space X let ~X be the set f all quasicmpnents f X with the qutient tplgy. PX: X ~ ~x is the prjectin and S: HTOP ~ Sh dentes the shape functr (see [D-SJ) frm the hmtpy categry t the shape categry. Fr any map a: X ~ Y and~-x E ~X there is a ~nique element f 4Y cntaining a(x O ). That means the map pya: X -+ 6Y factrs thrugh 6X, s there is a cntinuus map ~ (a) : 6X ~ ~y such that 6(a)px = pya.. Mreer, a hmtpic t b ilnplies ~(a) = ~(b) Thus we hae a functr 6: HTOP ~ TOP frm the hmtpy categry t the tplgical categry. If ne wants t get a functr ~: SH ~ TOP frm the shape categry, the natural way is
74 Dydak and Mrn t take the Cech system {XU,PUV,COV} f X (XU are the,neres f numerable cerings U f X) and define ~(X) as the inerse limit l~m{~xu'~(pu),cov} (with the inerse limit tplgy) f the system {~Xu'~(PUV),COV}. Therem 1. If eery pen cer f ~X admits an pen refinement cnsisting f mutually disjint sets, then the natural map ~X ~ ~(X) is a hmemrphism. Prf. The map ~X ~ ~(X) is always ne-t-ne. Indeed, if F and G are tw different quasicmpnents f X, then there is an pen-clsed set U in X with U cntaining F and X-U cntaining G. The cering U = {U,X-U} determines Xu such that F and G are sent t tw different pints f ~XU XU. Claim. If C is a clsed set in ~X, then the image f C in ~(X) is equal t rr{~pu(c): U E CO V} n ~(X). Prf f Clatm~ Obiusly the image f C in ~(X) is cntained in rr{~pu(c):u E COV} n 1(x). Suppse {C ' U E COV} E rr{~pu(c): U E COV} n 1(x). Then u {pu-l(c )' u E COV} is a system f pen-clsed sets in X. U If its intersectin has a mutual pint with C, we are dne. S let us assume n {pu -1 (C U )' U E COV} C X-C. Ntice -1 that the sets X-PU (CUl, U E COV, are pen-clsed in X. Thus there is a refinement W f {X - C} U {X - pu-l(c U )' U E COV} cnsisting f mutually disjint pen-clsed sets. By the definitin f the Cech system f X, W E COV, Xw is the nere f Wand PW: X ~ X w is an enumeratin f W.
TOPOLOGY PROCEEDINGS Vlume 13 1988 75 In ur case X is a O-dimensinal cmplex and PW: X + X w w maps each element U f W nt the ertex U f X. Obsere that ~xw = x and C must be a ertex U f X W On the w w -1 ther hand U is cntained in X-PV (C V ' fr sme V. Thus PV(C ) is disjint with C V ' a cntradictin. W Ntice that fr each Xu the set ~Pu(C) the discrete space ~xu (here PU: X + Xu are the prjectins). Since rr{~pu(c): y is clsed in U E COV} is clsed, we infer ~X + ~(X) is clsed. Taking C = X we get that ~X + ~(X) is nt. y Crllary 1. The natural map ~X + y ~(X) is a hmemrphism if ne f the filwing anditins is satisfied: a. ~X is paraampaat and dim~x = 0, b. X is laally ampaat metrizable and eaah ampnent f X is ampaat. Prf. Case a) fllws frm Therem 3 in [E] (p. 278). In case b) X is a tplgical sum f cmpact metrizable spaces (see [Ba 2 ], p. 258). Remark. In [M] (prpsitin 1.4) it is shwn that each quasicmpnent f X is cnnected prided X is nrmal, PX: X + ~X is clsed and ind(~x) =. Therem 2. Suppse f: X + Y is a shape mrphism, Y is paraampaat, dim~y = 0 and Py: Y + ~Y is alsed. If B is a alsed subset f ~Y and A C ~X n 1(f)-1(B), then there exists a unique shape mrphism
76 Oydak and Mrn -1-1 f:px (A) ~ Py (B) such that S(j)f O = f S(i)3 where -1-1 i: Px (A) ~ X and j: py (B) ~ Yare inclusins. Therem 2 is a simple cnsequence f the fllwing: Lemma 1. Suppse B is a subset f a paracmpact spaae Y suah that eaah neighbrhd f B in Y cntains an pen-alsed neighbrhd f B in Y. Let {YU,PUV~GOV} 'Y be the Cech system f Y. Fr eaah aering U E COV cnsider the subamplex N(UIB) (equal t the nere f U restricted t B) f Y U and let B U be the unin f all ampnents f pints f N(UIB) in Y U Then the natural pr-hmtqpy map B ~ {B U ' U E COV} satisfies the antinuity anditin. Prf. COV is L~e set f all numerable cerings f Y, Y is the nere N(U) f U and PU: Y ~ u Y is an U enumeratin f U (see [0-5J, pp. 20-22). The induced maps B ~ B will be dented by que We need t shw tw U prperties: a. Fr each map f: B ~ K E ANR(M) there is U E :COV and f U : B U ~ K with f hmtpic t fuqu. b. If fu,h U : B U ~. K are maps such that fuqu is hmtpic t huqu' then there is V ~ U with fuqu is hmtpic t huqvu' where q-vu: B-V ~ BU Since we can change K up t hmtpy type we may assume K is a cmplete metric space, and therefre it is an ANE fr paracmpact spaces (see [O-KJ).
TOPOLOGY PROCEEDINGS Vlume 13 1988 77 If f: B ~ K, we extend f t f': V ~ K, where V is an pen-clsed set cntaining B, and then we extend f' t f": Y ~ K. Since Y ~ {Y ' U E A} satisfies the cntinuity U cnditin, there is U E COV and gu: Y ~ K with fn hmu tpic t guqu. If fu,h U : B U ~ Nw take f U = gulb. K are maps such that fuqu is hmtpic t huqu' chse an pen-clsed neighbrhd V f B in Y with qu(v) cntained in BU. By extending the hmtpy between fuqu and huqu we may assume that thse tw maps are hmtpic as maps frm V t K. Let W (ul) u (UIY - V). Define a map s: Y ~ K (t: Y W W ~ K) as fupwu n N(Ul) (gupwu n N(UIV» and cnstant n N(UIY - V). Then spw is hmtpic t tpw' s there is V ~ Wwith spvw is hmtpic t tpvw. we are dne. Since N(Ul) = B W ' A simple cnsequence f Therem 2 is the fllwing: CrZZary. Suppse f: X ~ Y is a shape ismrphism f paracmpaat spaces. If bth maps Px: X ~ ~X, Py: y ~ 6Y are clsed and dim6x = dim6y = 0, then fr eery X E ~X there is a shape equialence fa: X ~ Y ~(f) (X ) such that S(j)f O ~ j: YOc. Yare inczusins. fs(i), where~: XO~ X and Here is a partial cnerse t the abe Crllary: Therem 3. Suppse f: X ~ Y is a azsed map f paracmpaat spaces such that fr eery X E ~X the
78 Oydak and Mr6n restrictin f O : X ~ YO = ~(XO) f f is a shape equialence and ~(f): ~(X) ~ ~(Y) is a bijectin. If Py: Y ~ ~Y is clsed and di~y = 0, then f is a shape equialence. Prf. Replacing Y by the mapping cylinder f f we may assume that f is the inclusin f a clsed set X int Y. We need t shw that fr each map a: X ~ K E ANR(M) there is an extensin a': Y ~ K. Since we can change K up t hmtpy type we may assume K is a cmplete metric space, and therefre it is an ANE fr paracm~act spaces (see [O-K]). Fr each X E ~x we can extend alx t YO E ~y cntaining X O Subsequently we extend a frm X t an pen-clsed neighbrhd V f YO in Y. Nw, chse a refinement f {V} cnsisting f mutually disjint pen sets and it is clear hw t define an extensin f a. If a,b: Y ~ K are tw maps such that alx and blx are hmtpic, we prceed similarly as abe t shw that a and b are hmtpic. Remark. Therem 3 is a generalizatin f Therem 4.5.5 in [D-5] (p. 62). Therem 4. Suppse X is a paracmpact space such that p : X ~ 6X is clsed and di~x =. If fr eery X X E ~X the defrmatin dimensin def-dim(x ) is Zess than O r equaz t n, then def-dim(x) ~ n. Prf. It suffices t shw that fr eery simplicial cmplex K (with the metric tplgy) any map f: X ~ hmtpic t a map g with g(x) C K(n) K is
TOPOLOGY PROCEEDINGS Vlume 13 1988 79 Claim. If def-dim(y) ~ n, then any map U K(P) x [ g: Y ~ p=l p,) is hmtpic t a map which alues lie in un K(P) x [p 00). p=l ' Prf f Claim. Since the identity map U OO K (p) x [p 00) ~ u K(P) x [p,) p=l w ' p=l is a hmtpy euqia1ence (here K w means K with the weak tplgy) we can factr 9 up t hmtpy thrugh L = U;=l Kw(P) x [p,) (L is cnsidered with the natural CW structure) and using the fact that def-dim(y) < n we can push 9 int L(n) which is cntained in L up K(P) x [p 00) n p=l ' Suppse g: X ~ K. Since the prjectin ~: L' = U;=l K(P) x [p,) ~ K is a hmtpy equialence (~ induces ismrphisms f hmtpy grups at each pint), there is g': X ~ LV with g' ~ TIg. Gien any quasicmpnent X f X, g'lx is hmtpic t g with g (X ) cntained in Ln. Since L' is an ANE fr paracmpact spaces (see [H], p. 63), we can find an pen clsed neighbrhd V f X in X such that g'l is hmtpic t a map w~th alues in Ln. Finally we can find a cering V f X cnsisting f mutually disjint pen sets such that g' I is hmtpic t a map with alues in L fr eery V n in V. Nw, it is clear that g' is hmtpic t a map gil with alues in L. Then g"~ is hmtpic t g and its n alues are in K(n) Remark. pred by S. Nwak [N]. In the cmpact metrizable case Therem 4 was
80 Dydak and Mrn Therem 5. Suppse X is a paracmpact space such that PX: X ~ ~X is clsed and di~x =. If F is a clsed cering f X such that PX -1 (px(r» F and eery B E F is mable (unifrmly mable)~ then X is mabze (unifrmly mable). Prf. We will pre nly the unifrmly mable case. The pr~f f the mable case is similar. Let {Xu,PUV,COV} be the Cech system f X. By Lemma 1, the natural pr-hmtpy map B ~ {B ' U E CO'V} satisfies the U cntinuity cnditin fr each B E F. Thus fr each cering U E COV and fr each B E F there is a cering V{B) E COV, V(B) ~ U, and a shape mrphism gb: BV(B) ~ X such that S[PU]gB = S[PV(B)U]. Chse an pen refinement W f {(PV(B» -1 (BV(B»: B E F} cnsisting f mutually disjint pen sets. Nw, we define an pen refinement V f U as fllws: gien W in W we chse B E F with -1 W C (PV{B» (BV{B» and declare the elements f V intersecting W t be precisely W n V{B). Ntice that N(W n V) is cntained in BV(B)' s that we can piece tgether gbis t get g: N(V) = X ~ X with S[PU]g S[PVU]. Remarks. Therem 5 is related t Therem 3.1 f [M]. In [D-S-S] there is an example f a nn-mable metrizable space X such that each f its quasicmpnents is mable. The same example shws that ne cannt drp bth f the hyptheses (px clsed and dim~x = 0) in Therem 4.
TOPOLOGY PROCEEDINGS Vlume 13 1988 81 References [BO] K. Brsuk, Thery f shape~ Plish Scientific Publishers, Warsaw 1975. [Ba ] B. J. Ball, Shapes f saturated subsets f cmpacta~ l Cllq. Math. 24 (1974), 241-246. [Ba ] B. J. Ball, Quasicmpactificatins and shape thery~ 2 Pacific J. Math. 84 (1979), 251-259. [Ba ] B. J. Ball, Partitining shape-equialent spaces~ 3 Bull. Acad. Pl. Sci. 29 (1981), 491-497. [D-K] J. Dydak and G. Kzlwski, A generalizatin f the Viet;;'-;{s--Begle therem~ (1988), 209-212. Prc. Amer. Math. Sc. 102 [D-S-S] [D-S] J. Dydak and J. Segal, Shape thery: An intrductin~ 1978, 1-150.' Lecture Ntes in Math. 688, Springer Verlag, J. Dydak, J. Segal and Stanislaw Spiez, A nnmable space with mable cmpnents~ Prceedings f ~he Arner. Math. Sc. (t appear) [E] R. Engelking, Outline f general tplgy~ Nrth [G] Hlland Publishing C., Amsterdam 1968. S. Gdlewski, On cmpnents f MANR-spaces~ Fund. Math. 114 (1981), 87-94. [H] S. T. Hu, Thery f retracts~ WayneuState Uniersity [M] [N] Press, Detrit, 1965. M. A. Mrn, Upper semicntinuus decmpsitins and mability in metric spaces~ 35 (1987), 351-357. Bull. Acad. Pl. Sci. S. Nwak, Sme prperties f fundamentaz dimensin~ Fund. Math. 85 (1974), 211-227. [S] J. M. R. Sanjurj, On a therem f B. J. Ball~ Acad. Pl. Sci. 33 (1985), 177-180. Bull. Uniersity f Tennessee Knxille, TN 37996 and
82 Dydak and M~6n E. T. S. de Ingeniers de Mntes Uniersidad Plitecnica de Madrid Ciudad Uniersitaria Madrid-28040, SPAIN