In [4], J. Ford and J. W. Rogers introduced the notion of refinable maps and the proved that each refinable map from a continuum to locally connected continuum is monotone [4, Corollary 1.2]. ..

Kato Hisao

加藤 久男

Tsukuba Repository  (Tulips-R)

Refinable maps onto locally n-connectedcompacta著者 Kato Hisao雑誌名 Tsukuba journal of mathematics巻 4号 1ページ 83-87発行年 1980-06URL http://hdl.handle.net/2241/7144TSUKUBA J. MATH.Vol. 4 (1980).83-87REFINABLE MAPS ONTO LOCALLY^-CONNECTED COMPACTAByHisao KatoIn [4], J. Ford and J.W. Rogers introduced the notion of refinable maps andthey proved that each refinable map from a continuum to a locally connectedcontinuum is monotone [4, Corollary 1.2]. In [5, Theorem 2.2], we proved thateach refinable map from a compactum to an FANR induces a shape equivalence.In this paper we shall prove that if a map r: X-* Y between compacta is refin-able and FeLC"1 (n^O), then r~＼y)^ACn for each jeF. Moreover if Y is anANR, then r is a CE-map.It is assumed that all spaces are metrizable and maps are continuous. Aconnected compactum is a continuum. A map /: X―> Y between compacta is ane-mapping, s>0, if / is surjective and diamf~1(y)<e for each jeF, If x and yare points of a metric space, d{x, y) denotes the distance from x to y. A mapr: X-*Y between compacta is refinable[4] if for any s>0 there is an s-mapping/: X^Y such that d(r,f)―swp{d(r(x), f(x))＼x<BX}<s. Such a map / is calledane-refinement of r. Note that every refinable map is surjective,every near homeo-morphism is refinable and if there is a refinable map from a compactum I to acompactum Y, then X is F-like. But simple examples show that any converseassertions of them are not true. A space X is locally n-connected (XgLC71) iffor each x^X and an open neighborhood U of x in X, there is an open set Vwith xeFct/ such that each map h: Sk-+V is null-homotopic in U for OSk^n,where Sk denotes the ^-sphere. A compactum X in the Hilbert cube Q is appro-ximatively n-connected (X<eACu) if for each open neighborhood U of X in Qthere is an open neighborhood VdU of X in Q such that each map h: Sk~^V isnull-homotopic in U for O^k^n (see [2]). A map f:X―*Y between compactais a CE-map if / is surjective and f~＼y) is an FAR (see [2]) for each y^Y.The following lemma is well-known.Lemma 1 ([7, Lemma 1]). Let f be a map from a compactum X to an ANRY and s>0. Then there is a positive number <5>0 such that if gx is any 8-map-Received November 22, 1979.84 Hisao Katoping from X to any compactum Z, then there is a map g.2:Z―*Y such thatd(f, gigi)<e.Lemma 2. Let X and Y be closed subsets of AR-spaces M and N respectively,and let f: M-+N be an extension of a map f: X-+Y. If X and Y are locally n-connected and f: (X, x)―*(Y,f{x)) induces a zero-homomorphism nk{f): 7ik(X, x)-^Xk(y,f(x)) for O^k^n, then for each open neighborhood V of Y in N there isan open neighborhood U of X in M such that Kk(f＼U): Kk{U, x)-*Tck(V,f(x)) is azero-homomorphism.Proof. By [3, Theorem 8.7] the natural morphisms ik: 7ik(X,x)-^pro-n k(X, x)and jk: ick{Y, y)-+pro-nk(Y, y) are isomorphisms for O^fe^n. Since jkXk(f)=pro-7Tk(f)ik,pro-7r/,(/):pro-7rfe(Z,x)-^pro-7r^(F,y) is a zero-homomorphism, whichimplies the existence of U in the statement of Lemma.Theorem. Let X and Y be compacta and r: X―*Y be a refutablemap. IfY&LCn (n^O), then r'＼y)^ACn for each ye^Y. Moreover if Y is an ANR,then r is a CE-map.Proof. Since X is a compactum, X can be embedded into the Hilbert cubeQ. Let y&Y and let G be any open neighborhood of r~＼y) in Q. Choose acompact ANR U such that r'＼y)d＼ntQUc:Uc:G. Since U is a compact ANR,there is a positive number £!>0 such that any ^-near maps to U are hornotopic.Let ei^=d(r-＼y),Q-U)=mf{d(x1,xi)＼x1^r-1(y),xi^Q~U}>0. Since Y^LCn,there is a sequence V1, V2, V3, ･･･ of open sets in Y such that(1) V{DV%^V1-DVt-D-"t(2) r＼Vi={y}(3) each map h: Sk-+Vi+1 (O^fe^n) is null-homotopic in Vt.Since r is refinable, there are maps r{- X-^Y such that each rt is an (l/i)-refine-ment of r and(4) ri(r~＼y))CVi+2.for each i.Then we shall show that lim ＼j'i~＼Vi)~]~r~~~i(y).In fact,suppose, on the contrary,that there is a sequence xnier;)(FB.) such that lim xni=x0 and r(xo)^y. Choosean open neighborhood W of x0 in X such that r(W) C S3(r(x0)),where 5 ―(l/4)d(r(x0),y)>0 and for a set A Sd(A) denotes the ^-neighborhood of A. By (2),choose a sufficientlylarge integer n* such that xn,^W, d(r, rn,)<8 and Fn,cSo-(v).Refinable maps onto locally n-connected compacta 85Then r(.rB,)eS3(r(*o)) and rrei(xn..)eVnidSd{y), henced{r(x0), y)^d(r(x0), r(x≫t))+d(iixnt), rni(xni))+d(rni(xn.), y)<d+d+8=3d, which implies the contradiction.Let 0<£<Min{£i, £2}- Since lim {rjl{Vi)~＼= r~＼y), there is a natural number iosuch that(5) rT＼Vt)czSB,a(r-Ky)) for each t^i0.By Lemma 1, there is a natural number m^i0 such that there is a map gm : Y―>Qsuch that(6) d{ix, gmrm)<s/3 , where ix : X-^Q is the inclusion.Then we shall show(7) gm{Vm)CLgm(Vm)CLU.In fact, for each xerm'(Fm), by (5) and (6) we haved(gmrm(x), r-＼y))^d{gmrm{x), x)+ d(x, r~Ky))<e/3+e/3<e ,hence gmrm(x)^S^rKy))dU.Now, take two yii2-spaces M and N containing Vm+l and Vm respectively asclosed subsets, and let i:M-*N be an extension of the inclusion i: Fm+1-*VOT.Since U is an ANR, by (7) there is an open neighborhood V'm of Vm in N andan extension gm : V'm-+U of gm＼Vm: Vm-^U. Since Vm+1, VmEiLCn, by Lemma2 and (3) there is an open neighborhood V'm+1 of Vm+i in M such that(8) ~k{i＼V'm+l): 7rk(V'm+1)―> 7tk(V'm) is a zero-homomorphismfor each O^k^n.Let W be an open neighborhood of rm1(FOT+2) in Q such that U'cU and there is anextension rm: U'-+V/m+1 of rm＼r£(Vm+2): r^＼Vm+2)-*Vm+i. Since gnirm＼r^{Vm+2)=gmirm＼r-m＼Vm+a), by (6) there is an open neighborhood U"(ZU' of rmKVm+2) inQ such that(9) d{gmirm＼U", iu>)<e, where iv.＼ U"-*U is the inclusion.By (9), we have(10) gmirm＼U"2liv, in U .By (8) and (10), jck{iu")'-nk{U")^-n:k{U) is a zero-homomorphism. Note that86 Hisao Kator-＼y)(Zr7n＼Vm+i)(ZU". Hence r＼y)^AC＼If Y is a compact ANR, the proof is similar. This completes the proofRemark 1. Note that if n=0, Theorem implies the result of J. Ford andJ.W. Rogers.By Theorem and [3, Theorem 8.5], we have the following.Corollary 1. // a map r: X^Y between compacta is refinable and Y^LCn(w^l), for any compactum BaY and xe.r~＼B),pro-itk(r＼r~＼B)):pro-izk{r~＼B),x)-^pro-itk{B, r(x))is an isomorphism of pro-groups for l^k^n and an epimorphismof pro-groups for k=n-＼-l.Corollary 2. Let X and Y be compacta and r: X―+Y be a refinable map.If Y^LCn and Fd(F)^n (see [2]), then r induces a shape equivalence.Proof. By [5, Theorem 1.8],Fd (X)=Fd (Y)^?i. By Theorem and the resultof [3, Theorem 8.14], [6] or [8], r induces a shape equivalence.Corollary 3. // a map r: X―>Y between compacta is refinable and Y is afinite-dimensional ANR, then r induces a hereditary shape equivalence,i.e.,forany compactum B, r＼r~l(B):r~1(B)―*B induces a shape equivalence.Corollary 4. Let r be a map from a (S^SzV ■･･VSn)-like continuum onto5iVS2V ･■･V5n, where SyVS^W ■■■VSn denotes the one point union of n circles.Then the fallowings are equivalent.(1) r is refinahle.(2) r is a CE-map.(3) r is monotone.Proof.implies (2).By [5, Theorem 3.2],(1) and (3) are equivalent. By Theorem, (1)Obviously (2) implies (3).Remark 2. In the statement of Theorem, we cannot replace ACn by Cn(n-connected).Remark 3. By [4, p. 264], there is a refinable map r: X-≫Y such that X,Y are 1-dimensional continua and r~1(y0)^AC° for some ^eF (cf.[5, Example2.7]). In [5, Example 2.6], for each n=l, 2, 3, ･･･, we constructed a refinablemap r: X―*F such that X and Y are n-dimensional continua, YgLC71"1 andSh(X)^Sh(F). In fact, for some yoe Y, r'Ky,)^ ACn. Thus those show thatRefinablemaps onto locallyn-connectedcompacta 87in the statement of Theorem we cannot replace LCn by LCn~l. Moreover, in[5, Example 2.8], we constructed a near homeomorphism h: X―>X such that Xis a n-dimensionalcontinuum, X^LC71'1 and r does not induce a shape equi-valence. In fact,for some yo^X, r'1(y0)^ACn.References[ 1 ] Borsuk, K., Concerning homotopy properties of compacta, Fund. Math. 62 (1968)pp. 223-254.[2] Borsuk, K., Theory of shape, PWN. Warszawa 1975.[ 3 ] Dydak, J., The Whitehead and Smale theorems in shape theory, DissertationesMath. 1979.[4] Ford, J. and Rogers, J.W., Refinable maps, Collq. Math. 39 (1978) pp. 263-269.|"5 ] Kato, H., Refinable maps in the theory of shape, to appear in Fund. Math.[6] Kodama, Y., Decomposition spaces and the shape in the sense of Fox, Fund.Math. 97 (1977) pp. 200-208.j"7 ] Loncar, I. and Mardesic, S., A note on inverse sequences of ANRs, Glasnik Math.3 (23) (1968) pp. 41-48.[8] Sher, R.B., Realizing cell-likemaps in Euclidean space, Gen. Top. and its appl.2 (1972) pp. 75-89.Institute of MathematicsUniversity of TsukubaIbaraki, Japan

Refinable maps onto locally n-connected compacta

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