数理解析研究所講究録
第 823 巻 1993 年 55-72

55

距離空間における COHOMOLOGY 次元について

横井勝弥 (KATSUYA YOKOI)
筑波大学数学系

1. INTRODUCTION AND PRELIMINARY
In the last ten years, cohomological dimension theory has striking development. A
motivation of the development is surely the Edwards-Walsh theorem, [24], as follows:
1.1. The\‘orem. Every compact metric space $X$ of cohomological dimension $c-\dim_{Z}X\leq$(in teger coefficient) is the image of a cell-like map $f:Zarrow X$ from a compact metricspace of $\dim Z\leq n$ .
$n$

$Z$

Not only the result but also techniques of the proof gave an important influence to thedevelopment. After them, L. R. Rubin and P. J. Schapiro [22] showed the noncompact
version of the Edwards-Walsh theorem and S. Marde\v{s}i\v{c} and L. R. Rubin [17] gave thenonmetrizable version. On the other hand, A. N. Dranishnikov, [5] and [6], characterized
by the Edwards-Walsh’s way and showedcohomological dimension with respect tothe Edwards-Walsh-like theorem:$Z_{p}$

1.2. Theorem. Every compact metric space $X$ of cohomological dimension with re$\prime c-\dim_{Z_{p}}X\leq n$ , is the image of a map $f:Zarrow X$ from a compact metric
spect tospace of $\dim Z\leq n$ whose fibers are acyclic modulo .
$Z_{p},$$Z$$p$

Motivated above results and Marde\v{s}i\v{c}’s characterization of c- $\dim_{Z}X\leq n$ , we willshow a characterization of c- $dimz_{p}X\leq n$ for noncompact case. Using the characterization, we will give the existence of an acyclic resolution modulo . In fact, our characterization suggests a dimension-like function, called approximable dimension, and canobtain the following more general results.
$p$1.3. Theorem. Let $X$ be a metriz able space havin approximable dimension withresp $ect$ to an arbitrary coefFcients $G\leq n$ . Then there exists a map $f:Zarrow X$ from ametriza $ble$ spa ce of $\dim Z\leq n$ and $w(Z)\leq w(X)$ onto $X$ such that $H^{*}(f^{-1}(x);G)=0$
for all $x\in X$ .
$g$

$Z$

As its consequence, we have noncompact versions of Theorems 1.1 and 1.2. We may
call such a mapping an acyclic resolution of $X$ (with respect to ), specially, in the
$f$

$G$

56
case of $G=Z_{p}$ , an acyclic resolution of $X$ modulo . Finally we will note that there
exists a compact metric space $X$ of c- $\dim_{Q}X=1$ which does not admit an acyclic$[11,12]$ . Thereby we can see that approximable dimension
resolution with respect tois different from cohomological dimension and Theorem 1.3 is a good property obtained
from approximable dimension.
In this paper, we mean the definition of cohomological dimension as follows: thecohomological dimension of a space $X$ with resp $ect$ to a coefficient groupis less thanand equal to , denoted by c- $\dim_{G}X\leq n$ , provided that every map $f:Aarrow K(G, n)$ of
a closed subset of $X$ into an Eilenberg-MacLane space $K(G, n)$ of type $(G, n)$ admits acontinuous extension over $X$ (c.f. [10]). The dimension of a space $X$ means the covering
dimension of $X$ and denotes by $\dim$ X.
is the additive group of all integers and foreach prime number
is the cyclic group of order .
By a polyhedron we mean the space $|K|$ of a simplicial complex $K$ with the Whitehead
topology. In section 5, the topology of $|K|$ may be generated by a uniformity [Appendix,
$p$

$Q$

$G$

$n$

$A$

$Z$

$p,$

$Z_{p}$$p$22].
If is a vertex of a simplicial complex $K$ , let $st(v, K)$ be the open star of in $|K|$and $s^{-}t(v, K)$ be the closed star of in $|K|$ . If $A\subseteq|K|$ , then we define $st(A, K)=$
and $s-t(A, K)=\cup\{\sigma : \sigma\in K, \sigma\cap A\neq\phi\}$ . The symbol
and
Sdj $K$ means the j-th barycentric subdivision of $K$ . We define the symbols
for a simplicial complex
with an index to be the cover $\{st(v, K_{i}) : v\in K_{i}^{(0)}\}$ and the, respectively.
cover
We use the $symbol\prec both$ to mean ‘refine’ for covers and ‘subdivides’ for subdivisionsof a complex. The $symbol\prec*is$ used for star refines.
Let be an open cover of a space $X$ . Then for
,
$v$

$v$

$v$

$\cup\{Int\sigma :\sigma\in K, \sigma\cap A\neq\emptyset\}$$S_{i}$$\overline{S}_{i}$$K_{i}$$\{s\overline{t}(v, K_{i}):v\in K_{i}^{(0)}\}$$U\in \mathcal{U}$$\mathcal{U}$$st(U,\mathcal{U})=st^{1}(U,\mathcal{U})=\cup\{U’ : U’\in \mathcal{U}, U’\cap U\neq\emptyset\}$$st^{j+1}(U,\mathcal{U})=\cup\{U’ : U’\in \mathcal{U}, U’\cap st^{j}(U,\mathcal{U})\neq\emptyset\}$

,

.

By st
we mean the cover. If and are maps from a space toa space $X,$
means that for each $z\in Z$ , there exists
with $f(z),$ $g(z)\in U$ .
If $X$ is a metric space with a metric , we write, whereinstead of$X$
is the cover whose consists of all-neighborhoods in . By the symbol, the symbol
we mean the nerve of the cover U. For covers
is used for thefollowing cover $\{U\cap V, U, V : U\in \mathcal{U}, V\in \mathcal{V}\}$ .
$\{st^{j}(U,\mathcal{U}) :$j(\mathcal{U})$U\in \mathcal{U}\}$

$f$

$Z$

$g$

$(f, g)\leq \mathcal{U}$$U\in \mathcal{U}$$d$$(f, g)\leq\epsilon$$(f, g)\leq \mathcal{U}_{\epsilon}$$\epsilon/2$$\mathcal{U}_{\epsilon}$$\mathcal{N}(\mathcal{U})$$\mathcal{U},$$\mathcal{U}\wedge \mathcal{V}$$\mathcal{V}$2. EDWARDS-WALSH COMPLEXESIn the latter section, we need Edwards-Walsh complexes for arbitrary simplicial complexes.
2.1. Lemma. Let
be a simplicial complex with th Whitehea topology, be aprime um $ber$ and be a natural number. Then there exists a combin atorial map $(i.e$ .
$|L|$

$n$

$n$

$e$

$d$

$p$

57
is a subcomplex of $EW_{Z_{p}}(L, n)$ if $L’$ is a subcomplex of )such th at$L$$\pi_{L}^{-1}(L’)$$|L|$$\psi_{L}$: $EW_{Z_{p}}(L, n)arrow$(i) for, wherewith $\dim\sigma\geq n+1,$,$\dim$
w漁(ii) for,a(iii) $EW_{Z_{p}}(L, n)$ is a CW-complex,(iv)
is a subcomplex of $EW_{Z_{p}}(L,.n)$ with respect to the triangulation in (3),(v)
is 五 nite CW-complex for $k\geq n$ ,(vi) for any sub complex $L’$ of and map $f:|L’|arrow K(Z_{p}, n)$ , there exis an $ext$ ension of.
$\sigma\in L$$r_{\sigma}=rank\pi_{n}(\sigma^{(n)})$$\psi_{L}^{-1}(\sigma)\in K(\oplus_{1}^{r_{\sigma}}Z_{p}, n)$$\sigma\in L$$\leq n,$$\psi_{L}^{-1}(\sigma)=\sigma$$\psi_{L}^{-1}(\sigma)$$\psi_{L}^{-1}(\sigma)^{(k)}$

$a$

$L$

$ts$

$fo\psi_{L}|_{\psi_{L}^{-1}(|L|)}$

Sketch of Proof. We give its proof by using Edwards-Walsh’s modification by Dranishnikov [6]. By the induction on $\dim L,$
is constructed to satisfy the following:$\psi_{L}$(1) $\psi_{L}^{-1}(L^{(n)})=L^{(n)}$ is a subcomplex of $EW_{Z_{p}}(L, n)$ and$\psi_{L}|_{|L^{(n)}|}=id_{|L^{(n)}|}$.

Let be a simplex of with $\dim$ a $=n+1$ . Let
be an Eilenberg-MacLane
space of type $(Z_{p}, n)$ obtained fromby attaching an (n+l)-cell by a map of degree. Hence$L$$\sigma$$K(\sigma)$$\partial\sigma$

$p$

(2)

$K(\sigma)^{(n)}=\partial\sigma$and$K( \sigma)^{(n+1)}=\partial\sigma\bigcup_{\alpha}B^{n+1}$, where$\alpha:\partial B^{n+1}arrow\partial\sigma$

is a mapof degree .
$p$

If $\dim a\geq n+2$ and $n\geq 2$ , then $K(\sigma)=K_{1}(\sigma)\cup K_{2}(\sigma)\cup\ldots$ such that

, where the union is taken over all proper faces of ,(3)
is obtained from
(4) for $i=2,3,$
by attaching to $K_{i-1}(\sigma)^{(n+i-1)}$
a finite collection of $(n+i)$ -cells killing the $(n+i-1)$ -th homotopy group.
$\tau$$K_{1}( \sigma)=\bigcup_{\tau_{\neq}\sigma}\prec K(\tau)$$\ldots,$$\sigma$$K_{i-1}(\sigma)$$K_{i}(\sigma)$

If $\dim\sigma\geq n+2$ and $n=1$ , then $K(\sigma)=K_{1}(\sigma)\cup K_{2}(\sigma)$ U. . . such that, by attaching finite collection of 2-cells abelizis obtained froming the fundamental group,
is obtained fromby attaching to $K_{i-1}(\sigma)^{(n+i-1)}$(6) for $i=2,3,$
a finite collection of $(n+i)$ -cells killing the $(n+i-1)$ -th homotopy group.
(5)$K_{1}(\sigma)$$\bigcup_{\tau_{\neq}\sigma}\prec K(\tau)$$\ldots,$$K_{i-1}(\sigma)$$K_{i}(\sigma)$

Then we construct as(7)(8)$\psi_{L}^{-1}(\sigma)$$\psi_{L}|_{M_{\sigma}}$,is the mapping cylinderof the embedding :is the cone ofis the barycentre of .
such that$M_{\sigma}$$j_{\sigma}$$\psi_{L}(K(\sigma))$$\psi_{L}|_{\psi_{L}^{-1}(\partial\sigma)}$Hence for each simplex$\sigma$$\psi_{L}^{-1}(\partial\sigma)arrow+K(\sigma)$$\sigma$of $\dim\sigma\geq n+1$ , we have the property:(9) if $n\geq 2$ ,,$\psi_{L}^{-1}(\sigma)^{(n+1)}=\sigma^{(n)}\cross[0,1]\bigcup_{\alpha_{1}}B^{n+1}\bigcup_{\alpha_{2}}\cdots\bigcup_{\alpha_{r_{\sigma}}}B^{\dot{n}+1}$

where for each (n+l)-dimensional faceof degree ,$p$$\tau_{i}$of$\sigma,$$\alpha_{i}$:$\partial B^{n+1}arrow\partial\tau_{i}\cross\{1\}$is a map

58(10) if $n=1$ ,$\psi_{L}^{-1}(\sigma)^{(2)}=\sigma^{(1)}\cross[0,1]\bigcup_{\alpha_{1}}B^{2}\bigcup_{\alpha_{2}}\cdots\bigcup_{\alpha_{r_{\sigma}}}B^{2}\bigcup_{\beta_{1}}B^{2}\bigcup_{\beta_{2}}\cdots\bigcup_{\beta_{k_{\sigma}}}B^{2}$

where for each 2-dimensional facedegree and the collectionofof$\sigma,$$\{[\beta_{1}], \ldots[\beta_{k_{\sigma}}\cdot]\}$$p$$\pi_{1}$$\tau_{i}$$( \sigma^{(1)}\cross[0,1]\bigcup_{\alpha_{1}}B^{2}\bigcup_{\alpha_{2}} ...\bigcup_{\alpha_{r_{\sigma}}}B^{2})$,:is a map ofgenerates the commutator ‘subgroup$\partial B^{2}arrow\partial\tau_{i}\cross\{1\}$$\alpha_{i}$.

$\square$

3. CHARACTERIZATIONS FOR METRIZABLE SPACES
Let us establish definitions. Let $K$ be a simplicial complex and $f,g:Xarrow|K|$ be$f(x)\in\sigma$
maps. We say that is a K-modification of if for each $x\in X$ andis called
implies $g(x)\in\sigma$ . Let be an open cover of $X$ . Then a map$b^{-1}(st(U,\mathcal{U}))=U$
for eachand is essential on each simplex-normal map if). Note that if:
is a essential map for each(i.e.
ofis a locally finite, then -normal map exists.
$\sigma\in K,$

$f$

$g$

$b:Xarrow|\mathcal{N}(\mathcal{U})|$$\mathcal{U}$$b$$U\in \mathcal{U}$$\mathcal{U}$$b|_{b^{-1}(\sigma)}$$\mathcal{N}(\mathcal{U})$$b^{-1}(\sigma)arrow\sigma$$\sigma\in \mathcal{N}(\mathcal{U})$$\mathcal{U}$$\mathcal{U}$3.1. Definition. Let $Q,$ $P$ be polyhedra, be an abelian group, be an open cover of$P$ andbe a natural number. We say that a map $\psi:Qarrow P$ is-approximable
if there exists a triangulation of such that for any triangulation $M$ of there is a
PL-map: $|M^{(n)}|arrow|L^{(n)}|$ satisfying the following conditions:$G$$\mathcal{U}$$(G, n,\mathcal{U})$

$n$

$L$

$P$

$Q$

$\psi’$,(i)$\alpha:|L^{(n)}|arrow
K(G, n)$ , there exists an extension $\beta:|M^{(n+1)}|arrow$(ii) for any map$K(G, n)$ of.
$(\psi’, \psi|_{|M^{(n)}|})\leq \mathcal{U}$$\alpha 0\psi’$be an abelian group and be a natural number. A map
Deflnition. Let
is called ( , n)-cohomological if$f:Xarrow P$ of a metrizable space $X$ to a polyhedron
for any open cover of there exist a polyhedron and maps $\varphi:Xarrow Q,$ $\psi:Qarrow P$
such that
3.2.

$G$

$n$

$P$

$\mathcal{U}$

(i)
(ii)

$P$

$G$

$Q$

,
is ( , n,U)-approximable.

$(\psi 0\varphi, f)\leq \mathcal{U}$$\psi$$G$

3.3. Theorem. Let $X$ be a metriza $ble$ space, be a prim $en$ um er and be a naturalof 1 $ess$ than and equal
number. Then $X$ has cohomological dimension $wi$ th respect to$P$$X$to if and only if every map of to a polyhedron is ( , n)-cohomological.
$b$

$p$

$n$

$Z_{p}$$f$$n$$Z_{p}$

Proof of necessity. Suppose that c- $dimz_{p}X\leq n$ . Let $f:Xarrow P$ be a map of

$X$

to a

polyhedron and be an open cover of . Then take a star refinementof .
First, we show that there exist a simplicial complex $K$ and maps $\varphi:Xarrow|K|$ ,$\psi:|K|arrow P$ such that$P$

(1) if$\sigma\in K$$P$$\mathcal{U}$, there exists$U\in \mathcal{U}_{0}$with$\psi(\sigma)\subseteq U$$\mathcal{U}_{0}$,$\mathcal{U}$59(2) for each $x\in X$ if, there exists
with,(3) there exist a triangulation of $P$ and a PL-map : $|K^{(n)}|arrow|L^{(n)}|$ such that(i)(ii) for any map $\alpha:|L^{(n)}|arrow K(G, n)$ there is an extension $\beta:|K^{(n+1)}|arrow$$K(G, n)$ of.
$\varphi(x)\in Int\sigma,$ $\sigma\in K$$\psi(\sigma)\cup\{f(x)\}\subseteq$$U\in \mathcal{U}_{0}$

$U$

$L$

$\psi’$$(\psi’, \psi|_{|K^{(n)}|})\leq \mathcal{U}_{0}$$\alpha 0\psi’$By J. H. C. Whitehead’s theorem [25], take a triangulation(4) st$\{s\overline{t}(v, L):v\in L^{(0)}\}\prec \mathcal{U}_{0}$of $P$ such that$L$.

We will construct a map $c:Xarrow EW_{Z_{p}}(L, n)$ such that
(5) $c|_{f^{-1}(|L^{(n)}|)}=f|_{f^{-1}(|L^{(n)}|)}$ ,(6)

for
structed in Lemma 2.1.
$c(f^{-1}(\sigma))\subseteq\psi_{L}^{-1}(\sigma)$

We define the map$\sigma\in L$, where$\psi_{L}$: $EW_{Z_{p}}(L, n)arrow L$ is the map con-

: $f^{-1}(|L^{(n)}|)arrow|L^{(n)}|\subseteq EW_{Z_{p}}(L, n)$. Inductively,$c_{n}\equiv f|_{f^{-1}(|L^{(n)}|)}$

suppose that for $n\leq k$ we have defined the function : $f^{-1}(|L^{(k)}|)arrow EW_{Z_{P}}(L, n)$:such thatis continuous andonfor. Now, let a $\in L$ with $\dim\sigma=k+1$ . By
the construction of and $EW_{Z_{p}}(L, n),$:is continuous. Henceby c- $\dim_{Z_{p}}f^{-1}(\sigma)\leq c-$ $\dim_{Z_{p}}X\leq n$ and (i) in Lemna 2.1, we have an continuous
extension :of. Define
to be
on
for
with $\dim$ a $=k+1$ . Finally, we define to be. Then since $X$ is compactly
generated, the function is continuous.
in the following way:
We define an open cover$c_{k}$$f^{-1}(\sigma)arrow\psi_{L}^{-1}(\sigma)\subseteq EW_{Z_{p}}(L, n)$$C_{k}|_{f^{-1}(\sigma)}$$f^{-1}(\sigma)\cap f^{-1}(\tau)$$C_{k}|_{f^{-1}(\tau)}$$\sigma,$$\tau\in L^{(k)}$$c_{k}|_{f^{-1}(\partial\sigma)}$$c_{k}$$c_{\sigma}$$f^{-1}(\sigma)arrow\psi_{L}^{-1}(\sigma)$$C_{k}|_{f^{-1}(\sigma)}=$$\partial\sigmaarrow\psi_{L}^{-1}(\sigma)$$C_{k}|_{f^{-1}(\partial\sigma)}$$c_{k+1}$$f^{-1}(\sigma)$$c_{\sigma}$$\sigma\in L$$\bigcup_{k=n}^{\infty}c_{k}$

$c$

$c$

$\mathcal{B}=\{B_{\sigma} :\sigma\in L\}$

$B_{\sigma}\equiv EW_{Z_{p}}(L, n)\backslash \cup\{\psi_{L}^{-1}(\tau) :\sigma\cap\tau=\emptyset\}$.

Then note that we have(7)(8) if$\psi_{L}^{-1}(\sigma)\subseteq B_{\sigma}$$x\in B_{\sigma}$and$x\in\psi_{L}^{-1}(\tau),$$\sigma\cap\tau\neq\emptyset$.

, for a star refinementof , there exists an open refinement
Since $EW_{Z_{p}}(L, n)$ is$K$ofsuch that if
is a simplicial complex of $\dim K\leq n+1$ , then every partial$K$
in $EW_{Z_{p}}(L, n)$ relative torealization ofextended to a full realization relative to. Select a star refinementof .
Then by [21, Lemma 9.6], there exist an open cover of $X$ refining
and mapssuch that$LC^{n}$$\mathcal{B}_{2}$$\mathcal{B}_{1}$$\mathcal{B}$$\mathcal{B}_{1}$$\mathcal{B}_{2}$$\mathcal{B}_{1}[2]$$\mathcal{B}_{2}$$\mathcal{B}_{3}$$\mathcal{V}$$\varphi:Xarrow|\mathcal{N}(\mathcal{V})|,$$f^{-1}(\mathcal{U}_{0})\wedge c^{-1}(\mathcal{B}_{3})$$\psi:|\mathcal{N}(\mathcal{V})|arrow P$

is V-normal,(9)
is L-modification of ,(10)
, there exists(11) if$\varphi$$\psi 0\varphi$$\sigma\in \mathcal{N}(\mathcal{V})$$f$$U\in \mathcal{U}_{0}$with$f(\varphi^{-1}(\sigma))\cup\psi(\sigma)\subseteq U$.

60Then theseand satisfy the conditions (1) $-(3)$ .
It is easily seen that (11) implies (1) and (2). It remain to prove that (3) holds.
:
We shall construct a map
in the following way: note
, there existsthat ifwith $U\subseteq c^{-1}(B_{U})$ .
onfor each \langle ). Let
is defined by an element, we have. Then by$\mathcal{N}(\mathcal{V}),$$\psi$$\varphi$$|\mathcal{N}(\mathcal{V})^{(n+1)}|arrow EW_{Z_{p}}(L, n)$$\psi_{0}$$\langle U\rangle\in \mathcal{N}(\mathcal{V})^{(n+1)}$$B_{U}\in \mathcal{B}_{3}$$U$$\psi_{0}(\langle U))\in B_{U}$$|\mathcal{N}(\mathcal{V})^{(0)}|$$\psi_{0}$$\in \mathcal{N}(\mathcal{V})^{(0)}$$\langle U_{0}, \ldots, U_{m}\rangle\in$$\emptyset\neq U_{0}\cap\cdots\cap U_{m}\subseteq c^{-1}(B_{U_{0}})\cap\cdots\cap c^{-1}(B_{U_{m}})$$\mathcal{N}(\mathcal{V})^{(n+1)}$$\psi_{0}(\{\langle U_{0}\rangle, \ldots, \langle U_{m}\cdot\rangle\})\subseteq st(B_{U_{0}}, \mathcal{B}_{3})\subseteq B$

for some$B\in \mathcal{B}_{2}$.

is a partial realization ofin $EW_{Z_{p}}(L, n)$ relative to
It show that.
Therefore, by the construction of , we may define
to be a full realization relative. Then by the same way in [21, p245 (8)] we can show thatto$\mathcal{N}(\mathcal{V})^{(n+1)}$$\psi_{0}$$\mathcal{B}_{2}$$\psi_{0}$$\mathcal{B}_{2}$$\mathcal{B}_{1}$(12) ifexist

withsuch that$t\in|\mathcal{N}(\mathcal{V})^{(n+1)}|$$\sigma,$$\lambda\in L$$\psi(t)\in Int\delta$$\delta\prec\sigma$

and

and

for

$\psi_{0}(t)\in\psi_{L}^{-1}(\tau)$$\sigma\cap\lambda\neq\emptyset\neq\lambda\cap\tau$$\delta,$$\tau\in L$, then there.

Now, by the property (v) in Lemma 2.1, we can choose(13) a cellular map, if$|\mathcal{N}(\mathcal{V})^{(n+1)}|$$\psi_{1}$: $|\mathcal{N}(\mathcal{V})^{(n+1)}|arrow EW_{Z_{p}}(L, n)^{(n+1)}$ such that for each $t\in$, then.
$\psi_{1}(t)\in\psi_{L}^{-1}(\tau)^{(n+1)}$$\psi_{0}(t)\in\psi_{L}^{-1}(\tau)$

By the simplicial approximation theorem, we assume thatis PL.
If $n\geq 2$ , by the properties (9) and (1) in Lemma 2.1, we have$\psi_{1}$$EW_{Z_{p}}(L, n)^{(n+1)}=|L^{(n)}|\cup\cup\{\partial\sigma\cross[0,1]\bigcup_{\alpha_{\sigma}}B_{\sigma}^{n+1} :

where
point

$\alpha_{\sigma}$:$\partial B_{\sigma}^{n+1}arrow\partial\sigma$\sigma\in L, \dim\sigma=n+1\}$

is a map of degree . For each (n+l)-simplex, and take the retraction$p$$\sigma$,of , choose a$L$$z_{\sigma}\in B_{\sigma}^{n+1}\backslash \partial B_{\sigma}^{n+1}$$r:EW_{Z_{p}}(L, n)^{(n+1)}\backslash \{z_{\sigma} :\sigma\in L, \dim\sigma=n+1\}arrow|L^{(n)}|$induced by the compositions of the radial projection of
the natural projection of
onto
If $n=1$ , for every simplex of $\dim$ aform (10) in Lemma 2.1:

$B_{\sigma}^{n+1}\backslash \{z_{\sigma}\}$$\partial\sigma\cross\{0\}\subseteq|L^{(n)}|$$\partial\sigma\cross[0,1]$$\geq 2,$$\sigma$$\psi_{L}^{-1}(\sigma^{(2)})$

onto$\partial\sigma\cross\{1\}$.
may be represented as the$\psi_{L}^{-1}(\sigma)^{(2)}=\sigma^{(1)}\cross[0,1]\bigcup_{\alpha_{1}}B^{2}\bigcup_{\alpha_{2}}\cdots\bigcup_{\alpha_{r_{\sigma}}}B^{2}\cup\rho_{1}B^{2}\bigcup_{\beta_{2}}\cdots\bigcup_{\beta_{k_{\sigma}}}B^{2}$ofThen choose points
the retraction
induced by the compositions of the radial projections ofonto
and the natural projection of
In both cases, we put$u_{1}^{\sigma},$$\ldots,$$u_{r_{\sigma}}^{\sigma},$$v_{1}^{\sigma},$$\ldots,$$\psi_{L}^{-1}(\sigma^{(1)})^{(2)}\backslash \sigma^{(1)}\cross[0,1]$$v_{k}^{\sigma_{\sigma}}$$r:EW_{Z_{p}}(L, n)^{(2)}\backslash \{u_{1}^{\sigma}, \ldots, u_{r_{\sigma}}^{\sigma}, v_{1}^{\sigma}, \ldots, v_{k_{\sigma}}^{\sigma} :$\sigma^{(1)}\cross[0,1]$$\psi’\equiv ro\psi_{1}|_{|\mathcal{N}(\mathcal{V})^{(n)}|}$or$B^{2}\backslash \{u_{i}^{\sigma}\}$$\sigma^{(1)}\cross\{0\}\subseteq|L^{(1)}|$:and$|\mathcal{N}(\mathcal{V})^{(n)}|arrow|L^{(n)}|$

.

.

.

for each$B^{2}$

and\sigma\in L, \dim\sigma\geq 2\}arrow|L^{(1)}|$$B^{2}\backslash \{v_{j}^{\sigma}\}$onto$S^{1}$

61
Then the map

holds the conditions (i),(ii). First, we show the condition (i). Let. By (12), there exist
such that,and. Then since, we have $\psi’(t)\in\tau$ . Hence,is an element of
we havefor some(see (4)). Next, we must show thecondition (ii). But, it is easy to show that. Hence, we omitted it here.
Now, we shall show that is ( , n)-cohomological. By (2), we can easily see that
. So, we show that is-approximable.
Let $M$ be a triangulation of $|K|$ . Note that for a simplicial approximation of: $|M|=|K|arrow|K|$ with respect to $K$ , we have that$\psi’$$t\in|\mathcal{N}(\mathcal{V})^{(n)}|$$\sigma,$$\psi_{0}(t)\in\psi_{L}^{-1}(\tau)$$\lambda,$$\tau\in L$$\psi_{L}^{-1}(\tau)^{(n)}$$\psi_{1}(t)$$\psi(t),$ $\psi’(t)\in s-t(\lambda, L)\subseteq U$$f$$(\psi 0\varphi, f)\leq \mathcal{U}$$\psi(t)\in\sigma$$\sigma\cap\lambda\neq\emptyset\neq\lambda\cap\tau$$U\in \mathcal{U}_{0}$$Z_{p}$$(Z_{p}, n,\mathcal{U})$$\psi$$j$$id_{|M|}$

and $j(|M^{(n)}|)\subseteq|K^{(n)}|$ .

$j(|M^{(n+1)}|)\subseteq|K^{(n+1)}|$

Then by (1) and (3), we can easily see that the map$\psi’’\equiv\psi’oj:|M^{(n)}|arrow|L^{(n)}|$holds the conditions.

$\square$

The reverse implication is proved by the standard way [21]. First, we need somenotations.
We may assume that the Eilenberg-MacLane space $K(Z_{p}, n)$ is a metrizable, locallycompact separable space. Then by the Kuratowski-Wojdyslawski’s theorem, we canconsider that $K(Z_{p}, n)$ is a closed subset of a convex subset of a normed linear space$E$ . Note thatis AR(metrizable spaces). Since $K(Z_{p}, n)$ is ANR, there exist a closed
and a retraction $r:Farrow K(Z_{p}, n)$ . Further, we can choose anneighborhood $F$ in
open coverof $Int_{C}F$ such that
$C$

$C$

$C$

$\mathcal{W}_{0}$(1) for any space$ro\alpha,$$\beta:Zarrow F$ withand any maps
$ro\beta:Zarrow K(Z_{p}, n)$ are homotopic in $K(Z_{p}, n)$ .
$Z$$\alpha,$Then we take an open, convex cover(2) if$W\in \mathcal{W}$$\mathcal{W}$

of

$C$

$(\alpha, \beta)\leq \mathcal{W}_{0}$, the maps

such thatwith $W\cap K(Z_{p}, n)\neq\emptyset$ , there exists$U\in \mathcal{W}_{0}$with$st(W, \mathcal{W})\subseteq U$.

Select a star refinement V of .
be a Kuratowski’s map with respect to V and define a map:
Leton
is defined by an element:in the following way: a map. Next, by using the convexity of , we extend
for each \langle ),. Let. Then by)linearly on each simplex,$\mathcal{W}$$h_{0}$$h_{1}$$Carrow|\mathcal{N}(\mathcal{V})|$$|\mathcal{N}(\mathcal{V})|arrow C$$|\mathcal{N}(\mathcal{V})^{(0)}|$$h_{1}$$V$$h_{1}(\langle V\rangle)\in V$$C$$\in|\mathcal{N}(\mathcal{V})^{(0)}|$$|\mathcal{N}(\mathcal{V})|$$\sigma=\langle V_{0},$$\ldots$$V_{m}$$V_{0}\cap\cdots\cap V_{m}\neq$$\in|\mathcal{N}(\mathcal{V})|$$\emptyset$$h_{1}(\{\langle V_{0}),$$\ldots,$$(V_{m})$})Thus, by the construction of$\subseteq st(V_{o}, \mathcal{V})\subseteq W_{\sigma}$$h_{1}$, we havefor some$h_{1}(\sigma)\subseteq W_{\sigma}$.

$W_{\sigma}\in \mathcal{W}$$h_{1}$.

62Let
be a subcomplex
simplicial neighborhood ofsuch that if (inwith
. Then we can easily see the followings:$\mathcal{N}(\{V\in \mathcal{V} :$\mathcal{N}_{1}$$\mathcal{N}_{1}$V\cap K(Z_{p}, n)\neq\emptyset\})$$\mathcal{N}(\mathcal{V})$$\mathcal{N}(\mathcal{V})$$\in \mathcal{N}_{0}$$V_{0}\rangle$

of
. Let
, there exists

be a$\mathcal{N}_{0}$$(V_{1}\rangle$$\in \mathcal{N}_{1}$$V_{0}\cap V_{1}\neq\emptyset$(3) for each $x\in K(Z_{p}, n)$ , there exists
,
)
(4)
.
(5)
$h_{1}(|\mathcal{N}_{0}|)\subseteq st(K(Z_{p}, n),$$\mathcal{W}$$W\in \mathcal{W}$with $x,$ $h_{1}oh_{0}(x)\in W$ ,$\subseteq F$$h_{0}(K(Z_{p}, n))\subseteq|\mathcal{N}_{1}|\subseteq|\mathcal{N}_{0}|$Proof of sufficiency. Let

be a closed subset of $X$ and $h:Aarrow K(Z_{p}, n)$ be a map.
and maps. We take an open cover
We consider the above-mentioned nerveofsuch that$A$$h_{0},$$\mathcal{N}(\mathcal{V})$$\mathcal{U}$$h_{1}$$|\mathcal{N}(\mathcal{V})|$

(6)
(7)

$st^{3}(|\mathcal{N}_{1}|,\mathcal{U})\subseteq|\mathcal{N}_{0}|$$st^{3}(\mathcal{U})\prec h_{1}^{-1}(\mathcal{W})$,,and choose a subdivisionofsuch that if
there exists
with.
Since is AE, there is an extension $H:Xarrow C$ of . Then by the assumption, themapis $(Z_{p}, n)$ -cohomological. Hence, there exist a polyhedronand maps $\varphi:Xarrow Q,$
such that$\mathcal{N}$$\sigma\in \mathcal{N}$$\mathcal{N}(\mathcal{V})$$C$$U\in \mathcal{U}$$\sigma\subseteq U$$h$$h_{0}oH:Xarrow|\mathcal{N}(\mathcal{V})|$$Q$$\psi:Qarrow|\mathcal{N}(\mathcal{V})|$

(8)
(9)

,
-approximable.

$(\psi 0\varphi, h_{0}oH)\leq \mathcal{U}$$\psi$

is$(Z_{p}, n,\mathcal{U})$By using the simplicial approximation theorem, we obtain a triangulation $M$ ofa simplicial approximationof . Then by (8),(9), we have$\psi^{*}:$(10)(11) $\psi*is$ ($Marrow \mathcal{N}$

$Q$

and

$\psi$,, st )-approximable.

$(\psi^{*}0\varphi, h_{0}oH)\leq st\mathcal{U}$$Z_{p},$ $n$$\mathcal{U}$Now, by (11) with respect to $M$ , there exist a triangulationsuch that$L$

and a PL-map$\psi’$: $|M^{(n)}|arrow$$|L^{(n)}|$,(12)(13) for any map $\alpha:|L^{(n)}|arrow K(Z_{p}, n)$ , there exists an extension $\beta:|M^{(n+1)}|arrow$$K(Z_{p}, n)$ of.
$(\psi’, \psi^{*}|_{|M^{(n)}|})\leq st\mathcal{U}$$\alpha 0\psi’$

Claim. There exists a map $\xi:Qarrow K(Z_{p}, n)$ such th at$\xi|_{\psi^{*-1}(|\mathcal{N}_{0}|)}=roh_{1}o\psi^{*}|_{\psi^{*-1}(|\mathcal{N}_{0}|}$Construction of . First, we shall see that$\xi$(14) for each $x\in D\equiv\psi^{*-1}(|\mathcal{N}_{0}|)\cap|M^{(n)}|$ , there exists$h_{1}o\psi’(x)\in U$ .

$U\in \mathcal{W}_{0}$such that$h_{1}o$$\psi^{*}(x),$By (12), there exist,
such thatand$\psi’(x)\in U_{3}$ . Then by (7), we havewith
. Since$W‘\cap
K(Z_{p}, n)\neq\emptyset$ .
$h_{1}o\psi^{*}(x)\inW$, by (4), there exists
such that
andHence by (2), we obtainsuch that $h_{1}o\psi^{*}(x),$ $h_{1}o\psi’(x)\in st(W’, \mathcal{W})\subseteq U$ .
Therefore by (14) and (1), we see the followings:$U_{1},$ $U_{2},$ $U_{3}\in \mathcal{U}$$W\in \mathcal{W}$$|\mathcal{N}_{0}|$$W’\in \mathcal{W}$$U\in \mathcal{W}_{0}$(15) $h_{1}o\psi’(D)\subseteq F$ ,$U_{1}\cap U_{2}\neq\emptyset\neq U_{2}\cap U_{3}$$h_{1}(U_{1}\cup U_{2}\cap U_{3})\subseteq W$$\psi^{*}(x)\in U_{1}$$\psi^{*}(x)\in$63(16) $roh_{1}o\psi^{*}|_{D}\simeq roh_{1}o\psi’|_{D}$ in $K(Z_{p}, n)$ .

Since is a subpolyhedron of $|M^{(n)}|$ and is PL, $\psi’(D)$ is subpolyhedron ofHence, from $\pi_{q}(K(Z_{p}, n))=0$ for $q<n$ (if $n=1$ , the path-connectedness ofthere exists an extension$D$$\psi’$$|L^{(n)}|$$K(Z_{p’\prime}n$.

)),$\alpha:|L^{(n)}|arrow K(Z_{p}, n)$of $roh_{1}|_{\psi’(D)}$ : $\psi’(D)arrow K(Z_{p}, n)$ .
Then by (13), we have an extension$\beta:|M^{(n+1)}|arrow K(Z_{p}, n)$of$\alpha 0\psi’$.

Now, put$R\equiv|M^{(n+1)}|\backslash \cup\{Int\sigma :\sigma\in M, \dim\sigma=n+1, \sigma\subseteq\psi^{*-1}(|\mathcal{N}_{0}|)\}$.

Then since. for each $x\in D\subseteq R$ we have $\beta(x)=\alpha 0\psi’(x)=roh_{1}o\psi’(x)$ ,(17)

in $K(Z_{p}, n)$ .

$\beta|_{D}\simeq roh_{1}o\psi’(x)|_{D}\simeq roh_{1}o\psi^{*}|_{D}$By the homotopy extension theorem, there exists an extension$roh_{1}o\psi^{*}|_{D}$$\xi_{R}$: $Rarrow K(Z_{p}, n)$ of.

, we haveSince for a $\in M$ with $\dim\sigma=n+1$ and, there exists an extension: $|M^{(n+1)}|arrow K(Z_{p}, n)$ ofsuch that$\sigma\subseteq\psi^{*-1}(|\mathcal{N}_{0}|)$$roh_{1}o\psi^{*}|_{\partial\sigma}$$\xi_{n+1}$$\xi_{R}|_{\partial\sigma}=$$\xi_{R}$.

$\xi_{\mathcal{R}+1}|_{\psi^{*-1}(|\mathcal{N}_{0}|)\cap||}M(n+1)=roh_{1}o\psi^{*}|_{\psi^{*-1}(|\mathcal{N}_{0}|)\cap|M(n+1)}|$Hence, we can define a map$\xi’$: $\psi^{*-1}(|\mathcal{N}_{0}|)\cup|M^{(n+1)}|arrow K(Z_{p}, n)$ by the following:$\xi’\equiv(roh_{1}o\psi^{*}|_{\psi^{*-1}(|\mathcal{N}_{0}|)})\cup\xi_{n+1}$.

Therefore from $\pi_{q}(K(Z_{p}, n))=0$ for $q>n$ , we obtain an extension $\xi:Qarrow K(Z_{p}, n)$of such that
. It completes the construction.
$\xi|_{\psi^{*-1}(|\mathcal{N}_{0}|)}=roh_{1}o\psi^{*}|_{\psi^{*-1}(|\mathcal{N}_{0}|)}$$\xi’$

Now, we put$h’\equiv\xi 0\varphi:Xarrow K(Z_{p}, n)$.

Then to complete the proof it suffices to prove(18)$h’|_{A}\simeq h$

in $K(Z_{p}, n)$ .

First , we shall see that$\psi^{*}0\varphi(A)\subseteq|\mathcal{N}_{0}|$Let $a\in A$ . By (10), there exist(19)$U_{1}\cap U_{2}\neq\emptyset\neq U_{2}\cap U_{3}$$U_{1},$

and$U_{2},$ $U_{3}\in \mathcal{U}$.

such that$\psi^{*}0\varphi(a)\in U_{\dot{1}},$$h_{0}oH(a)\in U_{3}$.

64Then since $h_{0}oH(a)=h_{0}oh(a)\in h_{0}(K(Z_{p}, n))\subseteq|\mathcal{N}_{1}|$ , we haveby
(6).
Hence, by Claim, we have for each $a\in A$ $h’(a)=\xi 0\varphi(a)=roh_{1}o\psi^{*}0\varphi(a)$ .
Therefore, by (1), it suffices to see that$\psi^{*}o(\backslash \rho(a)\in|\mathcal{N}_{0}|$(20) there exists$U\in \mathcal{W}_{0}$such that $h_{1}o\psi^{*}0\varphi(a),$ $h(a)\in U$ .

with the property (19). By (7), there exists
such that. By (3) we choosesuch that $h(a),$ $h_{1}oh_{0}oh(a)\in W’$ .
Therefore, since $h(a)\in K(Z_{p}, n)$ , there exists
such thatLet$W\in \mathcal{W}$$U_{1},$ $U_{2},$ $U_{3}\in \mathcal{U}$$U_{1}\cup U_{2}\cup U_{3}\subseteq h_{1}^{-1}(W)$$W’\in \mathcal{W}$$U\in \mathcal{W}_{0}$.

$h_{1}o\psi^{*}0\varphi(a),$ $h(a)\in st(W’, \mathcal{W})\subseteq U$

It completes the proof.

$\square$

4. APPROXIMABLE DIMENSION
4.1. Deflnition. A space $X$ has approximable dimension with respect to a coefficientgroupof less than and equal to (abbreviated, $a-\dim_{G}X\leq n$ ) provided that forevery polyhedron $P$ , map $f:Xarrow P$ and open cover , there exist a polyhedron andmaps $\varphi:Xarrow Q,$ $\psi:Qarrow P$ such that
$G$

$n$

$Q$

$\mathcal{U}$

(i)
(ii)

$(\psi 0\varphi, f)\leq \mathcal{U}$$\psi$

is$(G, n,\mathcal{U})$,-approximable.

First, we state fundamental inequalities of a- $\dim_{G}$ .

4.2. Theorem. For a metriz a $ble$ space $X$ and an arbi trary abelian groupthe following inequalities:$c-\dim_{G}X\leq a-\dim_{G}X\leq\dim X$$G$, we A old.

Proof. The second inequality is trivial. We can see the first inequality by the strategy
similar to the proof of the sufficiency in Theorem 33.

口As we will show in latter sections, our approach of agives useful applications.
$a-\dim_{G}$In general,
is different from c. However, in special cases of coefficient$G,$acoincides with c.
group$\dim_{G}$$\dim_{G}$$\dim_{G}$4.3. Theorem. If $G=Z$ or$X$ , we have$\dim_{G}$$Z_{p}$, where is a prime number, for every metrizable space$p$$a-\dim_{G}X=c-\dim_{G}X$ .

Proof. From Theorem 3.3, 4.2, we see the fact.

$\square$65
5. RESOLUTIONS FOR METRIZABLE SPACES
By a polyhedron we mean the space $|K|$ of a simplicial complex $K$ with the Whitehead). We may define a topology for $|K|$ by means of a uniformity
topology (denoted by).
in [Appendix, 22] (denoted by$|K|_{w}$$|K|_{u}$5.1. Theorem. Let $X$ be a metrizable space having approximable dimension with$respect$ to an abelian groupof less than and equal to . Then there exist an ndimensional metrizable space and a perfect $UV^{n-1}$ -surjection $\pi:Zarrow Xsu$ ch thatfor $x\in X$ , the set $[\pi^{-1}(x), K(G, n)]$ of homotopy classes is trivial.
$G$

$n$

$Z$

Proof. The strategy is like the construction of Walsh-Rubin $[24,22]$ .
Let be a metric for $X$ and let
be a sequence of open covers of$X$ where eachconsists of all $1/(i+1)$ -neighborhoods.
First, we shall construct the followings:
open covers
of $X$ whose nerves
are locally finite dimensional, maps : $Xarrow$for $i\geq 0,$:
for $i\geq 1$ and sequencesof subdivisions offor $i\geq 0$ such that$d$$\{\mathcal{U}_{i} :i\in N\cup\{0\}\}$$\mathcal{U}_{i}$$\mathcal{N}(\mathcal{V}_{i})$$\mathcal{V}_{i}$$|\mathcal{N}(\mathcal{V}_{i})|$$f_{i^{*}},$$f_{i}$$b_{i}$$\mathcal{N}_{i}^{j},j\in NU\{0\}$$|N(\mathcal{V}_{i})|arrow|\mathcal{N}(\mathcal{V}_{i-1})|$$\mathcal{N}(\mathcal{V}_{i})$for $j\geq 0$ ,is normal with respect tofor $j\geq 0$ ,and:is simplicial for $i\geq 1$ ,is-modification of $b_{i-1},0\leq j\leq 3$ for $i\geq 1$ ,onto a compact set inmaps each compact set in,contained in a finite union of simplexes offor $i\geq 1$ ,(6)
for $k\geq 1$ andfor $k\geq 4$ ,
(7)
,
(8)
A
A
(1)
(2)
(3)
(4)
(5)

$\overline{S}_{i}^{j+1}\prec^{*}S_{i}^{j}$$b_{i}^{-1}(S_{i}^{j})$$b_{i}$$f_{i}$$\mathcal{N}_{i}^{j}$$\mathcal{N}_{i}^{0}arrow \mathcal{N}_{i-1}^{3}$$\mathcal{N}_{i-1}^{j}$$f_{i}ob_{i}$$|\mathcal{N}_{i}|_{u}$$f_{i}$$|\mathcal{N}_{i-1}|_{u}$which is$\mathcal{N}_{i-1}$$S_{i}^{0}\prec f_{i}^{-1}(S_{i-1}^{3})$$\overline{S}_{i}^{k}\prec f_{i}^{*-1}(S_{i-1}^{k+3})$$\overline{S}_{i}^{k}\prec f_{i}^{-1}(S_{i-1}^{k+3})$$b_{i-1}^{-1}(S_{i-1}^{3})$$\mathcal{V}_{i}\prec \mathcal{U}_{i}$$b_{i-2}^{-1}(S_{i-2}^{6})\wedge\cdots\wedge b_{0}^{-1}(S_{0}^{3i})$as the uniform space with the uniform topology induced by thewhere we regard.
uniform base
Further, we shall construct continuous (w.r. . the Whitehead topology), uniformly
suchcontinuous (w.r. . the uniform topology) PL-maps :that$|\mathcal{N}_{i}|_{u}$$\{S_{i}^{j}\}_{j0}^{\infty_{=}}$

$t$

$t$

(9) for each$g_{i}$$t\in|(\mathcal{N}_{i}^{3})^{(n)}|$,(10) for any map, there exist$\sigma,$$\tau\in \mathcal{N}_{i-1}^{2}$$|(\mathcal{N}_{i}^{3})^{(n)}|arrow|(\mathcal{N}_{i-1}^{3})^{(n)}|$such that$f_{i}(t)\in\sigma,$$g_{i}(t)\in\tau$and$\sigma\cap\tau\neq\emptyset$$\alpha:|(\mathcal{N}_{i-1}^{3})^{(n)}|_{w}arrow K(G, n)$$arrow K(G, n)$ of $\alpha og_{i}$ :(11) for eachpolyhedral subset of$x\in|\mathcal{N}_{i}|,$, there exists an extension,$|(\mathcal{N}_{i}^{3})^{(n)}|_{w}arrow|(\mathcal{N}_{i-1}^{3})^{(n)}|_{w}arrow K(G, n)$$g_{i}(st(x,\overline{S}_{i}^{2})\cap|(\mathcal{N}_{i}^{3})^{(n)}|)$$|\mathcal{N}_{i-1}|$

is a Whitehead (i.e. finite) compact.

Let us start the construction. We take an open refinement
is locally finite dimensional and -normal map
nerve$\mathcal{N}(\mathcal{V}_{0})$$\beta:|(\mathcal{N}_{i}^{3})^{(n+1)}|$$\mathcal{V}_{0}$$\mathcal{V}_{0}$$b_{0}$:

of$\mathcal{U}_{0}$in $X$ whose. We$Xarrow|\mathcal{N}(\mathcal{V}_{0})|$66
defineto be a subdivision of[22, Proposition A.3], for the cover$\mathcal{N}_{0}^{j}$

cover
that

of

$\mathcal{B}_{0}$

(12)
(13)

for $j=0,1,2$ with

$Sd_{2j}\mathcal{N}(\mathcal{V}_{0})$$|\mathcal{N}(\mathcal{V}_{0})|$and a PL,$\mathcal{N}_{0}^{2}$. By using, we obtain an openof the identity such$\overline{S}_{0}^{j}\prec S_{0}^{j-1}$$\mathcal{E}_{0}\equiv\{st(x,\overline{S}_{0}^{2}):x\in|\mathcal{N}(\mathcal{V}_{0})|\}$-modification,(Cl $B$ ) is compact for$B\cupr_{0}(C1B)\subseteq
E$for someCl$r_{0}$:$|\mathcal{N}_{0}^{2}|arrow|\mathcal{N}_{0}^{2}|$$B\in \mathcal{B}_{0}$$r_{0}$$E\in \mathcal{E}_{0}$.

is ( , n)-cohomological, from the similar argument to the proof of the neSince
cessity in Theorem 3.3 we can take the followings:, locally finite open coverofof $X$ and maps : $Xarrow$subdivisionsuch that$G$$b_{0}$$Sd_{2}\mathcal{N}_{0}^{2}$$\mathcal{N}_{0}^{3}$$|\mathcal{N}(\mathcal{V}_{1})|,$$f_{1^{*}}:$$b_{1}$$\mathcal{V}_{1}$$|\mathcal{N}(\mathcal{V}_{1})|arrow|\mathcal{N}_{0}^{3}|$

,
(14)
,
(15)
A
is -normal,
(16)
(17)
is
-modification of ,, there exists,(18) for eachsuch that$M$, there exists a PL-map :of(19) for any triangulationsuch that,(i), there exists an extension $\beta:|M^{(n+1)}|$(ii) for any map$arrow K(G, n)$ of.
$\overline{S}_{0}^{3}\prec^{*}S_{0}^{2}\wedge \mathcal{B}_{0}$$b_{0}^{-1}(S_{0}^{3})$$\mathcal{V}_{1}\prec^{*}\mathcal{U}_{1}$$b_{1}$$\mathcal{V}_{1}$$f_{1^{*}}ob_{1}$$b_{0}$$\mathcal{N}_{0}^{3}$$U\in stS_{0}^{3}$$\sigma\in \mathcal{N}(\mathcal{V}_{1})$$b_{0}(b_{1}^{-1}(\sigma))\cup f_{1^{*}}(\sigma)\subseteq U$$p’$$|\mathcal{N}(\mathcal{V}_{1})|$$(p’, f_{1^{*}}|_{1|}M(n))\leq\{s^{-}t(\lambda,\mathcal{N}_{0}^{3}) :$|M^{(n)}|arrow|(\mathcal{N}_{0}^{3})^{(n)}|$\lambda\in \mathcal{N}_{0}^{3}\}$$\alpha:|(\mathcal{N}_{0}^{3})^{(n)}|arrow K(G, n)$$\alpha op’$Let
denote a subdivision ofwith
for $j\geq 3$ .
Now, letdenotewith the metric topology [19, p301]. Then there isa-modification :of the identity function [19, p302]. By thesimplicial approximation theorem, we obtain a subdivisionofand a simplicial. Then by the simpliciality
approximation :of. Letdenoteofand (17) , we have$\overline{S}_{0}^{j+1}\prec^{*}S_{0}^{j}$$Sd_{2}\mathcal{N}_{0}^{j}$$\mathcal{N}_{0^{+1}}^{j}$$|\mathcal{N}_{0}^{3}|_{m}$$|\mathcal{N}_{0}^{3}|$$|\mathcal{N}_{0}^{3}|_{m}arrow|\mathcal{N}_{0}^{3}|_{w}$$j_{0}$$\mathcal{N}_{0}^{3}$$\mathcal{N}_{1}$$f_{1}$$\mathcal{N}_{1}arrow \mathcal{N}_{0}^{3}$$j_{0}of_{1^{*}}$$\mathcal{N}(\mathcal{V}_{1})$$\mathcal{N}_{1}$$\mathcal{N}_{1}^{0}$$f_{1}$

(20)
(21)

,
-modification of$S_{1}^{0}\prec f_{1}^{-1}(S_{0}^{3})$$f_{1}ob_{1}$is$\mathcal{N}_{0}^{3}$

We take a subdivisions
(22)
(23)
(24)

$\overline{S}_{1}^{j+1}\prec^{*}S_{1}^{j}$$\mathcal{N}_{1}^{j+1}$of$\mathcal{N}_{1}^{0}$$b_{0}$.

for $j=0,1$ such thatfor $j=0,1$ ,for $j=1,2$ ,for $j=1,2$ .

$\overline{S}_{1}^{j}\prec f_{1}^{-1}(S_{0}^{j+3})$$\mathcal{N}_{1}^{j}\prec Sd_{2j}\mathcal{N}_{1}^{0}$

By using Lemma [22, Proposition A.3], for the coverofobtain an open coveridentity map such that$\mathcal{B}_{1}$

(12)
(13)

$|\mathcal{N}(\mathcal{V}_{0})|$and a PL,,
(Cl $B$ ) is compact forfor some(Cl $B$ )Cl$B\in \mathcal{B}_{1}$$r_{1}$$B\cup r_{1}$$\subseteq E$$E\in \mathcal{E}_{1}$.

$\mathcal{N}_{1}^{2}$$\mathcal{E}_{1}\equiv\{st(x,\overline{S}_{1}^{2})$-modification$r_{1}$::$x\in|\mathcal{N}_{1}|\}$$|\mathcal{N}_{1}^{2}|arrow|\mathcal{N}_{1}^{2}|$, weof the67
is ( , n)-cohomological, from the similar argument to the proof of the neSince
cessity in Theorem 3.3 we can take the followings:, locally finite open coverofsubdivisionof $X$ and maps
: $Xarrow$$G$$b_{1}$$Sd_{2}\mathcal{N}_{1}^{2}$$\mathcal{N}_{1}^{3}$$|\mathcal{N}(\mathcal{V}_{2})|,$$|\mathcal{N}(\mathcal{V}_{2})|arrow|\mathcal{N}_{1}^{3}|$$f_{2^{*}}:$$(,14)_{2}\overline{S}_{1}^{3}\prec^{*}S_{1}^{2}$A$\mathcal{B}_{1}$A$b_{2}$$\mathcal{V}_{2}$such that$f_{1}^{-1}(S_{0}^{6})$

,

,
(15)
A
A
is -normal,
(16)
is
(17)
-modification of ,, there exists
such that,(18) for each$M$, there exists a PL-map :(19) for any triangulationofsuch that,(i), there exists an extension $\beta:|M^{(n+1)}|$(ii) for any map$arrow K(G, n)$ of.
$b_{1}^{-1}(S_{1}^{3})$$\mathcal{V}_{2}\prec^{*}\mathcal{U}_{2}$$b_{2}$$b_{0}^{-1}(S_{0}^{6})$$\mathcal{V}_{2}$$f_{2^{*}}ob_{2}$$\mathcal{N}_{1}^{3}$$b_{1}$$U\in stS_{1}^{3}$$\sigma\in \mathcal{N}(\mathcal{V}_{2})$$b_{1}(b_{2}^{-1}(\sigma))\cup f_{2^{*}}(\sigma)\subseteq U$$p’$$|\mathcal{N}(\mathcal{V}_{2})|$$(p’, f_{2^{*}}|_{|M^{(n)}|})\leq\{s-t(\lambda,\mathcal{N}_{1}^{3}) :$|M^{(n)}|arrow|(\mathcal{N}_{1}^{3})^{(n)}|$\lambda\in \mathcal{N}_{0}^{3}\}$$\alpha:|(\mathcal{N}_{1}^{3})^{(n)}|arrow K(G, n)$$\alpha op’$Now, by. using (19) about the triangulationsuch that$g_{1}^{*}:$$\mathcal{N}_{1}^{3}$of$|\mathcal{N}(\mathcal{V}_{1})|$, we obtain a PL-map$|(\mathcal{N}_{1}^{3})^{(n)}|arrow|(\mathcal{N}_{0}^{3})^{(n)}|$(25)(26) for any map$K(G, n)$ of,, there exists an extension$(g_{1}^{*}, f_{1^{*}}|_{|(\mathcal{N}_{1}^{3})^{(n)}|})\leq\{s-t(\lambda,\mathcal{N}_{0}^{3}):\lambda\in \mathcal{N}_{0}^{3}\}$$\alpha:|(\mathcal{N}_{0}^{3})^{(n)}|arrow K(G, n)$$\alpha og_{1}^{*}$$\beta:|(\mathcal{N}_{1}^{3})^{(n+1)}|arrow$.

Consider the inclusion map$i_{0}$:$|(\mathcal{N}_{0}^{3})^{(n)}|^{c}arrow|\mathcal{N}_{0}^{3}|$and the composition$|(\mathcal{N}_{1}^{3})^{(n)}|arrow|(\mathcal{N}_{0}^{3})^{(n)}|arrow|\mathcal{N}_{0}^{3}|=|\mathcal{N}(\mathcal{V}_{0})|arrow|\mathcal{N}(\mathcal{V}_{0})|$$r_{0}oi_{0}og_{1}^{*}:$.

of the PL-map
The imagehas dimension $\leq n$ . Then we can take aof the inclusion map-modification :. Let :
denote the composition map $s_{0}or_{0}oi_{0}og1$ .
Then this has the following properties:$A$$r_{0}oi_{0}og_{1}^{*}$$\mathcal{N}_{0}^{3}$$s_{0}$$Aarrow|(\mathcal{N}_{0}^{3})^{(n)}|$$Aarrow|\mathcal{N}_{0}^{3}|$$g_{1}$$|(\mathcal{N}_{1}^{3})^{(n)}|arrow$$|(\mathcal{N}_{0}^{3})^{(n)}|$Claim 1.
(9) for each$t\in|(\mathcal{N}_{1}^{3})^{(n)}|$,(10) for any map$K(G, n)$ of,(11) for eachpolyhedral $su$ bset

, there exist$\sigma,$$\tau\in \mathcal{N}_{0}^{2}$such that$f_{1}(t)\in\sigma,$$g_{1}(t)\in\tau$and$\sigma\cap\tau\neq\emptyset$$\alpha:|(\mathcal{N}_{0}^{3})^{(n)}|arrow K(G, n)$, there exist an extension$\beta:|(\mathcal{N}_{1}^{3})^{(n+1)}|arrow$$\alpha og_{1}$$x\in|\mathcal{N}_{1}|,$$g_{1}(st(x,\overline{S}_{1}^{2})\cap|(\mathcal{N}_{1}^{3})^{(n)}|)$$of|\mathcal{N}_{0}|$.

Proof of Claim 1. We show the property (9) . Let$\sigma,$$\lambda,$$\tau\in \mathcal{N}_{0}^{3}$$\lambda=|v_{0},$$v_{1}|,$

such thatand$f_{1^{*}}(t)\in\sigma,$$v_{0}\in\sigma$$v_{1}\in\tau$.

is a Whitehead (i.e. finit e) compact$g_{1}^{*}(t)\in\tau$

and$t\in|(\mathcal{N}_{1}^{3})^{(n)}|$$\sigma\cap\lambda\neq\emptyset\neq\lambda\cap\tau$. By (25) , there exist. We may assume that

68
is-modification of the identity function, we have
. Since, we haveis simplicial approximation of.
with. Since is -modification of the identity map, we have
Select, we. Further since is -modification ofandhave $g_{1}(t)=s_{0}or_{0}oi_{0}og_{1}^{*}(t)\in\tilde{\tau}$.
(i.e.
).
Case 1.
, we have
By. Hence, there exists
such that, we have. Then if
andwith. Therefore wehave A
and.
.
Case 2.
, the proof is similar to Case 1. Let
If
,
. By
there exist
such thatand. Thenor, we haveif. Similarly, we have
. Therefore we have
with
.
and
, we can see the property (10) by the homotopy extension theorem andBy(26) .
We show the property (11) . First, we shall see thatSince

$j_{0}of_{1^{*}}(t)\in\sigma$$\mathcal{N}_{0}^{3}$$j_{0}$$f_{1}(t)\in\sigma$$j_{0}of_{1^{*}}$$\tilde{\tau}\in \mathcal{N}_{0}^{2}$$\tau\subseteq\tilde{\tau}$$v_{0}\not\in(\mathcal{N}_{0}^{2})^{(0)}$$v_{0}\in Int\gamma$$\tilde{\tau}\neq\emptyset,$$\mathcal{N}_{0}^{3}\prec \mathcal{N}_{0}^{2}$$v_{1}\in\tilde{\tau}^{(0)}$$\mathcal{N}_{0}^{3}\prec Sd_{2}\mathcal{N}_{0}^{2}$$\tilde{\sigma}$$A=*|\mathcal{N}_{0}^{3}|$$\mathcal{N}_{0}^{3}$$s_{0}$$v_{1}\in(\mathcal{N}_{0}^{2})^{(0)}$$|v_{0},$ $v_{1}|\subseteq\gamma$$\mathcal{N}_{0}^{2}$$r_{0}$$r_{0}oi_{0}og_{1}^{*}(t)\in\tilde{\tau}$$f_{1}$$\gamma\in \mathcal{N}_{0}^{2}$$\tilde{\sigma}\in \mathcal{N}_{0}^{2}$$f_{1}(t)\in\tilde{\sigma}$$\sigma\subseteq\tilde{\sigma}$$\gamma\prec\tilde{\sigma}$$g_{1}(t)\in\tilde{\tau}$$v_{1}\not\in(\mathcal{N}_{0}^{2})^{(0)}$$v_{0}\in(\mathcal{N}_{0}^{2})^{(0)}$$v_{0}\not\in(\mathcal{N}_{0}^{2})^{(0)}$$v_{0}\in Int\gamma_{0},$$\gamma_{1}\in \mathcal{N}_{0}^{2}$$\gamma 0,$$\tilde{\sigma}\in \mathcal{N}_{0}^{2}$$\sigma\subseteq\tilde{\sigma}$$f_{1}(t)\in\tilde{\sigma}$$\tilde{\sigma}\cap\tilde{\tau}\neq\emptyset,$$v_{1}\in Int\gamma_{1}$$\mathcal{N}_{0}^{3}\prec Sd_{2}\mathcal{N}_{0}^{2}$$\gamma 0\prec\gamma_{1}$$\gamma_{1}\prec\gamma_{0}$$\gamma_{1}\prec\tilde{\tau}$$\gamma 0\prec\tilde{\sigma}$$g_{1}(t)\in\tilde{\tau}$$g_{1}^{*}\simeq g_{1}$(27)$g_{1}^{*}(st(x,\overline{S}_{1}^{2})\cap|(\mathcal{N}_{1}^{3})^{(n)}|)\subseteq B$Let$st(x,\overline{S}_{1}^{2})$$x\in Int\sigma_{x}$be represented byfor some$\cup\{s^{-}t(v_{\alpha},\mathcal{N}_{1}^{2}) :\alpha\in A\}$

$B\in \mathcal{B}_{0}$.

. There exists

with$\sigma_{x}\in \mathcal{N}_{1}^{2}$.

, we choose
For eachand. Further we selectsuch that
respectively
withminimum and maximal dimensional simplexes
andsuch that.
, we have. Then there exists a vertex
Iffrom. Since $fi$ is the simplicial map from,such thatto. By the nearness between andwe have(see proof of (9) ) and (14) , we obtain$\alpha\in A$$\sigma_{\alpha}\in \mathcal{N}_{1}^{2}$$\sigma_{x}\neg\prec\sigma_{\alpha}$$\tau_{x},$$\sigma_{x}\subseteq\tau_{x}$$v\in \mathcal{N}_{1}^{2}$$\tau_{\alpha}\in \mathcal{N}_{1}^{0}$$\tau_{x}\neg\prec\tau_{\alpha}$$\sigma_{\alpha}\subseteq\tau_{\alpha}$$s^{-}t(v_{\alpha},\mathcal{N}_{1}^{2})\subseteq\tau_{\alpha}$$\sigma_{x}\subseteq Int\tau_{x}$$v_{\alpha}\in\sigma_{\alpha}$$v_{\alpha}\in Int\tau_{\alpha}$$\bigcup_{\alpha}\tau_{\alpha}\subseteq s^{-}t(v,\mathcal{N}_{1}^{0})$$\mathcal{N}_{1}^{0}$$f_{1}( \bigcup_{\alpha}\tau_{\alpha})\subseteq f_{1}(s^{-}t(v,\mathcal{N}_{1}^{0}))\subseteq s-t(f_{1}(v),\mathcal{N}_{0}^{3})$$\mathcal{N}_{0}^{3}$$f_{1}$$g_{1}^{*}$(28)$g_{1}^{*}(st(x,\overline{S}_{1}^{2})\cap|(\mathcal{N}_{1}^{3})^{(n)}|)\subseteq st(s-t(fi(v),\mathcal{N}_{0}^{3}),\overline{S}_{0}^{3})\subseteq B$

for some$B\in \mathcal{B}_{0}$.

, we choose a faceandwithsuch that. Then there exists a vertex
such that.
Hence we have (28) in the same way.
is a subpolyhedron ofis a PL-map, we see
Since
and
. Then by (27) and (12) ,that
is a subpolyhedron ofIf$\sigma_{x}\cap\partial\tau_{x}\neq\emptyset$$\sigma_{x}\cap Int\tau_{x}\neq\emptyset$$\sigma_{x}\cap\partial\tau_{x}\subseteq\tilde{\tau}_{x}$$v\in\tilde{\tau}_{x}$$st(x,\overline{S}_{1}^{2})\cap|(\mathcal{N}_{1}^{3})^{(n)}|$$g_{1}^{*}(st(x,\overline{S}_{1}^{2})\cap|(\mathcal{N}_{1}^{3})^{(n)}|)$$|\mathcal{N}_{0}|_{w}$. Since$s_{0}$$\bigcup_{\alpha}s-t(v_{\alpha},\mathcal{N}_{1}^{2})\subseteq s-t(v,\mathcal{N}_{1}^{0},)$$|\mathcal{N}_{1}|$$g_{1}^{*}$$|\mathcal{N}_{0}|$is a subpolyhedron ofis a PL-map, we have see the property (11) .

$r_{0}oi_{0}og_{1}^{*}(st(x,\overline{S}_{1}^{2})\cap|(\mathcal{N}_{1}^{3})^{(n)}|)$$\tilde{\tau}_{x}\neq\prec\tau_{x}$$\tilde{\tau}_{x}$$|\mathcal{N}_{0}|$and a compact set of

69
Now, we shall take a base for a uniformity for
for $j\geq 4$ ofsuch that$|\mathcal{N}_{1}|$. We choose a subdivisions$\mathcal{N}_{1}^{j}$$\mathcal{N}_{1}$(29) A(30)(31)$\overline{S}_{1}^{j+1}\prec^{*}S_{1}^{j}$$\cdot$where

for $j\geq 3$ ,for $j\geq 3$ ,$j+11\prec Sd_{2}\mathcal{N}_{1}^{j}$$\overline{S}_{1}^{j+1}\prec f_{1}^{-1}(S_{0}^{j+4})\wedge f_{1}^{*-1}(S_{0}^{j+4})\wedge \mathcal{F}_{1}^{j+4}$$\mathcal{F}_{1}^{j+4}$

is defined as follows.

for $j\geq 3$ ,$g_{1}^{-1}(S_{0}^{j+4}\cap|(\mathcal{N}_{0}^{3})^{(n)}|)$

is the open cover$of|(\mathcal{N}_{1}^{3})^{(n)}|_{w}$of. Then clearly the uniformity make
Extend it to an open coveruniformly continuous.
andWe shall show that $fi$ holds the property (5). First, note that the composition$\mathcal{F}_{1}^{j+4}$$f_{1},$$|\mathcal{N}_{1}|_{w}$.

$f_{1^{*}}$$g_{1}$$j_{0}oidof_{1}^{*}:$$|\mathcal{N}_{1}|_{u}arrow|\mathcal{N}_{0}|_{u}arrow|\mathcal{N}_{0}|_{m}arrow|\mathcal{N}_{0}|_{w}$,is the identity map, is continuous.
where. There exist
such that $j_{0}of_{1^{*}}(K)=$Let $K$ be a compact set of, we have. Since is a simplicial approximation of. By the continuity of $fi,$ $fi(K)$ is a compact set of.
As we proceed in this work, we have
and with the properties (1) $-(11)$ .
From now on, we consider $X$ to be the uniform space with the uniformity generated
to be the uniform space with theby the sequenceof open covers of $X$ anduniformity generated by the sequence. Then by the construction, the topology
and the original metric topology are identical.
induced by
We shall construct the resolution of $X$ . The construction essentially depends on
Rubin’s way [22]. Hence, the detail is omitted here.
and letdenote the composition
For $j\geq 0$ , letdenote the identity on$f_{j+1}o\cdots of_{i}$ :for $i>j$ .
The functions:$id:|\mathcal{N}_{0}|_{u}arrow|\mathcal{N}_{0}|_{m}$$|\mathcal{N}_{1}|_{\tau\iota}$$\sigma_{1},$$j_{0}oidof_{1^{*}}(K)\subseteq\sigma_{1}\cup\cdots\cup\sigma_{l}$$\sigma_{l}\in \mathcal{N}_{0}$$\ldots,$$j_{0}of_{1^{*}}$$f_{1}$$fi(K)\subseteq\sigma_{1}\cup\cdots\cup\sigma\iota$$|\mathcal{N}_{0}|_{u}$$\mathcal{V}_{i},$$f_{i^{*}},$$f_{i},$$\mathcal{N}_{i}^{j}$$g_{i}$$|\mathcal{N}_{i}|$$\{\mathcal{V}_{i}\}_{i0}^{\infty_{=}}$$\{S_{i}^{j}\}_{j0}^{\infty_{=}}$$\{\mathcal{V}_{i}\}_{i0}^{\infty_{=}}$$f_{j,j}$$\mathcal{N}_{j}$$f_{i,j}$$|\mathcal{N}_{i}|arrow|\mathcal{N}_{j}|$$b_{i}$$(X, \{\mathcal{V}_{i}\}_{i0}^{\infty_{=}})arrow(|\mathcal{N}_{i}|, \{S_{i}^{j}\}_{j0}^{\infty_{=}})$

and$f_{i+1,i}$:$(|\mathcal{N}_{i+1}|, \{S_{i+1}^{j}\}_{j0}^{\infty_{=}})arrow(|\mathcal{N}_{i}|, \{S_{i}^{j}\}_{j0}^{\infty_{=}})$

is Cauchy inare uniformly continuous for $i\geq 0$ . Then since the sequence
the uniform space $C(X, \mathcal{N}_{j}|.)$ with the uniformity of uniform convergence, we have auniformly continuous, limit map$\{f_{i,j}ob_{i}\}_{i=j}^{\infty}$$f_{\infty,J} \equiv\lim_{qarrow\infty}f_{q,j}ob_{q}$:$(X, \{\mathcal{V}_{i}\}_{i0}^{\infty_{=}})arrow(|\mathcal{N}_{i}|, \{S_{j}^{i}\}_{i0}^{\infty_{=}})$,

such thatis-modification of ,
(32)
,
(33)
is a topological irreducible (i.e. surjective) map relative to(34)(35) $f_{i+1,i}of_{\infty,i+1}=f_{\infty,i}$ for $i\geq 0$ .
$f_{\infty,J}$$\mathcal{N}_{j}^{3}$$b_{j}$$(f_{\infty,j}, b_{j})\leq S_{j}^{1}$$f_{\infty,j}$$\mathcal{N}_{j}^{3}$,

70
to be the uniform space by the product uniformity. Note
We consider
is a non-empty subspace by the property (34).
that
Then by (35), there exist a uniformly continuous map :withis a uniformly embedding onto a dense subset
and especially the map, where:is the natural projection.
in. Then we consider
Let denote the limit of the inverse sequence. Note thatto be the sub-uniform space of the uniform spacehas.
dimension
We begin with a description of the map . For $j\geq 0$ , a uniformly continuous map:
is defined by$\prod_{i0}^{\infty_{=}}|\mathcal{N}_{i}|_{c\iota}$$\lim_{arrow}\{|\mathcal{N}_{j}|_{u}, f_{i+1,i}\}$$f_{\omega}$$pr_{i}of_{\omega}$$f_{\infty,i}=$$X arrow\lim_{arrow}|\mathcal{N}_{i}|_{\tau\iota}$$f_{\omega}(X)$$f_{\omega}$$\prod_{j0}^{\infty_{=}}|\mathcal{N}_{j}|_{u}arrow|\mathcal{N}_{i}|_{u}$$pr_{i}$$\lim_{arrow}|\mathcal{N}_{i}|_{u}$$Z$$\{|(\mathcal{N}_{i}^{3})^{(n)}|_{u}, g_{i+1,i}\}$

$Z$

$Z$

$\prod_{i0}^{\infty_{=}}|\mathcal{N}_{i}|_{u}$$\leq n$$\pi$

$Z arrow\prod_{i=0}^{\infty}|\mathcal{N}_{i}|_{u}$$\pi_{j}$$\pi_{j}(z)\equiv(f_{j,0}(z_{j}), f_{j,1}(z_{j}),$$f_{j,j-1}(z_{j}),$ $z_{j},$ $z_{J+1},$$\ldots,$$\ldots$)

be the inclusion map. Then since the sequence
for $z=(z_{j})\in Z$ and let), there is a uniformly continuous, limit map : $Zarrow$is Cauchy in ( , 垣巽 0
. Then the map
is proper fromonto
. We must$\{\pi_{j}\}_{j0}^{\infty_{=}}$$\pi_{0}$

$C$

$Z$

$|\mathcal{N}_{i}|_{u}$$\prod_{i0}^{\infty_{=}}|\mathcal{N}_{i}|_{u}$

$s$

$\pi$

$Z$

$\pi$

$\lim_{arrow}\{|\mathcal{N}_{i}|_{u}, f_{i+1,i}\}$how that $\pi^{-1}(x)$ is a $UV^{n-1}$ -set and the set $[\pi^{-1}(x), K(G, n)]$ is trivial for $x\in$$\lim_{arrow}\{|\mathcal{N}_{i}|_{u}, f_{i+1,i}\}$For.

$x=(x_{i})\in\lim_{arrow}\{|\mathcal{N}_{i}|_{u}, f_{i+1,i}\}$, let$\delta N(x_{i})$and$\epsilon N(x_{i})$denote$st(x_{i},\overline{S}_{i}^{0})$and, respectively. Then we have the following properties [22]: for $x=(x_{i})\in$,$st(x_{i},\overline{S}_{i}^{2})$$\lim_{arrow}\{|\mathcal{N}_{i}|_{u}, f_{i+1,i}\}$(36) $gi,i-1(\delta N(x_{i})$ 寡(37){$\lim_{arrow}$By$\epsilon N(x_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|,gi,i-1$$\overline{S}_{i}^{2}\prec*S_{i}^{1}$

there is a$|(\mathcal{N}_{i}^{3})^{(n)}|)\subseteq\epsilon N(x_{i-1})$, there exists
such that$S\in S_{i}^{0}$$= \pi^{-1}(x)=\lim_{arrow}\{\delta N(x_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|,g_{i,i-1}|\ldots\}$,such that. Further, by. Hence we have the contractible set such that$st(x_{i},\overline{S}_{i}^{2})\subseteq F_{i}$$F_{i}\in S_{i}^{1}$$F_{i}\subseteq S$

(38)

沖.}

,

$F_{i}$$\epsilon N(x_{i})\subseteq F_{i}\subseteq\delta N(x_{i})$Claim 2.

$\pi^{-1}(x)$is a $UV^{n-1}$ -set for for$S_{i}^{1}\prec S_{i}^{0}$.

$x=(x_{i})\in\lim_{arrow}\{|\mathcal{N}_{i}|_{u}, f_{i+1,i}\}$.

Proof of Claim 2. It suffices to show that the map$g_{i+1,i}|\ldots$:$\delta N(x_{i+1})\cap|(\mathcal{N}_{i+1}^{3})^{(n)}|arrow\delta N(x_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|$induces a zero homomorphism of homotopy group of dimension less thanand (38), we have$n$

. By (36)$g_{i+1,i}(\delta N(x_{i+1})\cap|(\mathcal{N}_{i}^{s_{+1}})^{(n)}|)\subseteq F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n)}|\subseteq\delta N(x_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|$Since$F_{i}$

is contractible, we have$\pi_{k}(F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n)}|)=0$

Therefore
than .
$n$.

$g_{i+1,i}|\ldots$

for $k<n$ .

induces a zero homomorphism of homotopy group of dimension less

71
Claim 3. $[\pi^{-1}(x), K(G, n)]\approx\check{H}^{n}(\pi^{-1}(x);G)$ is trivial for$x\in\lim_{arrow}\{|\mathcal{N}_{i}|_{u}, f_{i+1,i}\}$.

Proof of Claim 3. By (11),(36),(37) and the continuity of \v{C}ech cohomology, we have.

$\check{H}^{n}(\pi^{-1}(x);G)\approx\lim_{arrow}\{H^{n}(g_{i,i-1}(\epsilon N(x_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|_{u});G),g_{i,i-1}|:.\}$

Hence it suffices to show that$g_{i,i-1}|^{*}..$: $H^{n}(g_{i,i-1}(\epsilon N(x_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|);G)arrow H^{n}(g_{i+1,i}(\epsilon N(x_{i+1})\cap|(\mathcal{N}_{i+1}^{3})^{(n)}|);G)$is the zero homomorphism.
Letdenotes. Then by (11) the subspaceofand the subspaceofis identical. Hence from now on,
we may consider thatis the subspace of.
Let $[\alpha]\in[G_{i,i-1}, K(G, n)]$ . Then from $\pi_{q}(K(G, n))=0$ for $q<n$ , there exist an extensionof . By (10), we have an extensionof.
is the contractible set,
is contractible in.
Since
Hence, there exists a homotopy
such thatis the inclusion map andis a constant map. Since, we can define the following compositions:$g_{i,i-1}(\epsilon N(x_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|_{u})$$G_{i,i-1}$$|(\mathcal{N}_{i-1}^{3})^{(n)}|_{u}$$G_{i,i-1}$$|(\mathcal{N}_{i-1}^{3})^{(n)}|_{w}$$G_{i,i-1}$$\tilde{\alpha}:|(\mathcal{N}_{i-1}^{3})^{(n)}|_{w}arrow K(G, n)$$s$$G_{i,i-1}$$|(\mathcal{N}_{i-1}^{3})^{(n)}|_{w}$$\beta:|(\mathcal{N}_{i}^{3})^{(n+1)}|_{w}arrow K(G, n)$$\alpha$$\tilde{\alpha}og_{i,i-1}|_{G_{i+1,i}}$$F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n+1)}|_{w}$$F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n)}|_{w}$$F_{i}$$H:(F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n)}|_{w})\cross Iarrow F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n+1)}|_{w}$$G_{i+1,i}\subseteq\epsilon N(\dot{x}_{i})\cap|(\mathcal{N}_{i}^{3})^{(n)}|_{w}\subseteq$$H_{1}$$H_{0}$$F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n)}|_{w}$$\tilde{H}\equiv\beta oi_{2}oHoi_{1}$:$G_{i+1,i}\cross I\llcorner_{arrow}(F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n)}|_{w})\cross Iarrow F_{i}\cap|(\mathcal{N}_{i}^{3})^{(n+1)}|_{w}$$arrow|(\mathcal{N}_{i}^{3})^{(n+1)}|_{w}arrow K(G, n)$where and are the inclusion maps.
Then we have $\tilde{H}_{0}=\beta|_{G_{i+1,i}}=\alpha og_{i,i-1}|c_{:+1,:}$ andthe proof of Claim 3. Then the map,$i_{2}$$i_{1}$$\pi_{X}\equiv\pi|_{\pi^{-1}(X)}$is a desired one for Theorem.

$\tilde{H}_{1}=$a constant. It completes: $\pi^{-1}(X)arrow X$$\square$5.2. Corollary. Let $X$ be a metriza $ble$ space having cohomological dimension withof less than and equal to . Then there exist an n-dimensional metrizable
respect to
space
an a perfect $UV^{n-1}$ -surjection $\pi:Zarrow X$ such th at for $x\in X$ , the set$[\pi^{-1}(x), K(Z_{p}, n)]$ of homotopy classes is trivial. .
$Z_{p}$

$Z$

$d$

$n$

72
REFERENCES
1. P. S. Alexsandrov and P. S. Pasynkov, Introduction to Dimension Theory, Moscow, 1973.
2. K. Borsuk, Theory of retract, PWN, Warszaw, 1967.
3. M. Brown, Some applications of an approximation theorem for inverse limits, Proc. Amer. Math.
Soc. 11 (1960), 478-483.
4. C. H. Dowker, Mapping theorems for non-compact spaces, Amer. J. Math. 69 (1947), 200-242.
5. A. N. Dranishnikov, On homological dimension modulo , Math. USSR-Sb. 60(2) (1988), 413-425.
6. –, Homological dimension theory, Russian Math. Surveys 43(4) (1988), 11-63.
7. –and E. V. Shchepin, Cell-hke maps. The problem of raising dimension, Russian Math.
Surveys 41 (6) (1986), 59-111.
8. J. Dydak, Cohomological dimension and metrizable spaces, preprint.
9. –and J. J. Walsh, Aspects of cohomological dimension for prin cipal ideal domains, preprint.
10. Y. Kodama, Appendix to K. Nagami, Dimension theory, Academic Press, New York, 1970.
11. A. Koyama, Approximable dimension and acychc resolutions, preprint (1989).
12. –, Approximable dimension and factorization theorems, preprint (1989).
13. –and T. Watanabe, Notes on cohomological dimension modulo –nonmetrizable version,
Kyoto Mathematical Science Research Institute Kokyuroku 659 (1988), 1-24.
14. V. I. Kuz’minov, Homological dimension theory, Russian Math. Surveys 23 (1968), 1-45.
15. S. Marde\v{s}i\v{c}, On covering dimension and inverse limits of compact spaces, Illinois J. Math. 4(1960), 278-291.
16. –, Factorization theorems for cohomological dimension, Topology Appl. 30 (1988), 291-306.
dimension, Pacific17. –and L. R. Rubin, Approximate inverse systems of compacta and coveJ. Math. 183 (1989), 129-144.
18. –, Cell-hke maps and non-metrizable compacta of finite covering dimension, Trans. Amer.
Math. Soc. 313 (1989), 53-79.
19. S. MardeSi\v{c} and J. Segal, Shape Theory, North-Holland, Amsterdam, 1982.
20. L. R. Rubin, Irreducible representations of normal spaces, Proc. Amer. Math. Soc. 107(1) (1989),
277-283.
21. –, Characterizing cohomological dimension: The cohomological dimension of $A\cup B$ , Topology Appl. 40 (1991), 233-263.
22. –and P. J. Schapiro, Cell-like maps onto non-compact spaces of finite cohomological dimension, Topology Appl. 27 (1987), 221-244.
23. E. Spanier, Algebraic topology, Springer-Verlag, New York, 1989.
24. J. J. Walsh, Dimension, cohomological dimension, and cell-like mappings, Lecture Note in Math.,
vol. 870, 1981, pp. 105-118.
25. J. H. C. Whitehead, Simplicial spaces, nuclei, and m-gropes, Proc. London Math. Soc. 45 (1939),
243-327.
$p$$p$$r^{l}ing$

INSTITUTE OF MATHEMATICS, UNIVERSITY OF TSUKUBA, TSUKUBA-SHI, IBARAKI, 305, JAPANE-mail address: yokoi@sakura.cc.tsukuba.ac.jp