Extraordinary dimension of maps (Q2488857)

The paper deals with extensional dimension of maps, in particular with the extraordinary dimension introduced by \textit{E.~V.~Shchepin} [Russ. Math. Surv. 53, No. 5, 975--1069 (1998); translation from Usp. Mat. Nauk. 53, No. 5, 115--212 (1998; Zbl 0967.55001)] and considered in more detail by \textit{A.~Chigogidze} [Topology Appl. 138, No. 1--3, 1--20 (2004; Zbl 1046.54028 )]. If \(L\) is a CW-complex and \(X\) is a metrizable space, the extraordinary dimension \(\dim_LX\) of \(X\) generated by \(L\) is defined as the smallest integer \(n\) such that \(\Sigma^nL\) is an absolute extensor for \(X\), where \(\Sigma^nL\) is the \(n\)th iterated suspension of \(L\). For a given map \(f:X\to Y\), \(\dim_Lf \leq n\) means that \(\dim_Lf^{-1}(y)\leq n\) for every \(y\in Y\). In the paper under review the authors prove that if \(f:X\to Y\) is a perfect surjection between metrizable spaces with \(Y\) being a \(C\)-space and \(L\) is a countable CW-complex, then the property that \(\dim_Lf\leq n\) is equivalent to each of the following two conditions: (\(\ast\)) There exists a dense \(G_{\delta}\) subset \({\mathcal G}\) of the space \(C(X,{\mathbf I}^n)\) of all maps from \(X\) to \(n\)-dimensional cube \({\mathbf I}^n\) with the source limitation topology such that \(\dim_L(f\times g)=0\) for every \(g\in {\mathcal G}\); (\(\ast\ast\)) There exists a map \(g:X\to {\mathbf I}^n\) such that \(\dim_L(f\times g)=0\). Note that here and in the paper, \(f\times g\) should be understood as the map \((f,g):X\to Y\times\;{\mathbf I}^n\) defined by the formula \((f,g)(x)=(f(x),g(x))\) for all \(x\in X\). Moreover, if additionally \(X\) is compact, then \(\dim_Lf\leq n\) is equivalent to the condition: (\(\ast\ast\ast\)) There exists an \(F_{\sigma}\) subset \(A\) of \(X\) such that \(\dim_LA\leq n-1\) and the restriction map \(f| (X\setminus A)\) is of dimension \(\dim f| (X\setminus A)\leq 0\). For the covering dimension similar results were obtained by \textit{B.~A.~Pasynkov} [Sov. Math., Dokl. 16, 384-388 (1975); translation from Dokl. Akad. Nauk SSSR 221, 543--546 (1975; Zbl 0355.54027), Proc. Steklov Inst. Math. 212, 138--162 (1996); translation from Tr. Mat. Inst. Steklova 212, 147--172 (1996; Zbl 0963.54027)] and \textit{H.~Toruńczyk} [Fundam. Math. 125, 237--249 (1985; Zbl 0593.54035)], in the realm of finite-dimensional compact metric spaces and extended by \textit{M.~Tuncali} and \textit{V.~Valov} [Fundam. Math. 175, No. 1, 35--52 (2002; Zbl 1021.54027), Tsukuba J. Math. 28, No. 1, 155--167 (2004; Zbl 1069.54023)] to perfect maps between metrizable \(C\)-spaces. For the extensional dimension a similar type result was obtained by \textit{M.~Levin} and \textit{W.~Lewis} [Isr. J. Math. 133, 61--76 (2003; Zbl 1031.54019)].