MATHEMATICS

P. S. SOLTAN

ON THE DIMENSION OF PREIMAGES UNDER MAPPINGS OF COMPACTA INTO POLYHEDRA

(Presented by Academician P. S. Aleksandrov on 12 VIII 1959)

Let \(f\) be a continuous mapping of a compactum \(X\) into a Euclidean space \(Y\), \(\dim Y \leq \dim X\). Then \({}^{(1)}\), by an arbitrarily small displacement of the mapping \(f\), one can ensure that the complete preimage of any point \(y \in Y\) has dimension \(\leq \dim X-\dim Y\). If, however, \(Y\) is an arbitrary polyhedron, the matter becomes more complicated. Suppose, for example, that \(Y\) consists of two \(n\)-dimensional simplexes with one common vertex \(y\), and \(X\) is a sphere \((\dim X \geq n)\). Then, if \(f:X \to Y\) is such a continuous mapping that the set \(f(X)\) contains interior points of both simplexes, the preimage of the point \(y\) has dimension \(\geq \dim X-1\) (since it separates the sphere), and by no small “shaking” of the mapping \(f\) can this dimension be lowered. Thus, in the given case, to the number \(\dim X-\dim Y\) one adds the number \(\dim X-1\). In general, to the difference \(\dim X-\dim Y\) one adds a certain number \(\chi_0(y)\), depending on the structure of the polyhedron \(Y\) near the point \(y\) (see the formulation of the main theorem).

Let \(P\) be some polyhedron and let \(\chi\) be a nonnegative integer-valued function defined on it with values \(\leq \dim P\). Denote by \(M_i\), \(i=0,1,2,\ldots,\dim P\), the set of all points \(x \in P\) for which \(\chi(x) \geq i\). We shall call the function \(\chi(x)\) defective if: 1) all \(M_i\) are closed and 2) for any mapping \(\varphi:Q \to P\), where \(Q\) is a polyhedron of dimension \(\geq \dim P\), and for any \(\eta>0\) there exists a mapping \(\psi:Q \to P\) such that \(\rho(\varphi,\psi)<\eta\) and

\[ \dim\bigl(\psi^{-1}(x)\bigr) \leq \dim Q-\dim P+\chi(x) \]

for any point \(x \in N_i(\eta)=M_i \setminus U(M_{i+1},\eta)\), where \(U(M_{i+1},\eta)\) is the \(\eta\)-neighborhood of the set \(M_{i+1}\) in the polyhedron \(P\).

Theorem 1. Let \(\chi(x)\) be a defective function defined on a polyhedron \(P\); let \(X\) be some compactum of dimension \(n \geq \dim P\). Then, whatever the mapping \(f:X \to P\) and the number \(\varepsilon>0\) may be, there exists a mapping \(g:X \to P\) such that \(\rho(f,g)<\varepsilon\) and

\[ \dim\bigl(g^{-1}(x)\bigr) \leq \dim X-\dim P+\chi(x) \]

for every point \(x \in P\).

Proof. Construct a sequence of \(n\)-dimensional simplicial partitions \(\{K_i\}\) in \(E^{2n+1}\), which are nerves of increasingly fine open coverings of the compactum \(X\), and a sequence of continuous mappings \(\psi_i:V_i \to P\), where \(\{V_i\}\) is a decreasing sequence of open sets of the space \(E^{2n+1}\), with \(V_i \supset |K_i|\). The barycentric mappings (see \({}^{(2)}\), p. 208) \(X \to |K_i|\) will be denoted by \(\varphi_i\), \(i=1,2,\ldots\). We choose the elements \(K_1,V_1,\psi_1\) arbitrarily, subjecting them to the sole condition: \(\rho(f_1,\psi_1\circ\varphi_1)<\varepsilon/2\). Suppose that \(K_1,\ldots,K_\nu; V_1,\ldots,V_\nu; \psi_1,\ldots,\psi_\nu\) have already been constructed \((\nu \geq 1)\). Suppose, moreover, that certain positive numbers \(\omega_1,\omega_2,\ldots,\omega_{\nu-1}\) have been defined inductively. Put

\[ \eta_\nu=\min\left(\frac{\varepsilon}{2^{\nu+2}},\frac{\omega_1}{2^{\nu+1}},\frac{\omega_2}{2^\nu},\ldots,\frac{\omega_{\nu-1}}{2^3}\right) \]

and choose so small a positive--

some \(\gamma_\nu\) such that \(\rho(\psi_\nu(y),\psi_\nu(y'))<\eta_\nu\) when \(\rho(y,y')<\gamma_\nu\) \((y,y'\in \overline{V}_\nu)\). Now choose a sufficiently fine open covering of the compactum \(X\) so that its nerve, which we take for \(K_{\nu+1}\), can be realized in \(V_\nu\cap (|K_\nu|,\eta_\nu)\), and so that at the same time \(\rho(\varphi_\nu,\varphi_{\nu+1})<\gamma_\nu\) (for the construction of such nerves see the proof of Theorem 3 in \((^3)\)). Since \(\chi(x)\) is a defect function, there exists a mapping \(\xi_{\nu+1}:|K_{\nu+1}|\to P\) satisfying the condition \(\rho(\psi_\nu(y'),\xi_{\nu+1}(y'))<\eta_\nu,\ y'\in |K_{\nu+1}|\), and having the property that
\[ \dim\bigl(\xi_{\nu+1}^{-1}(x)\bigr)\leq \dim X-\dim P+\chi(x) \]
for every point \(x\in N_j(\eta_\nu)\), \(j=0,1,2,\ldots,\dim P\).

Extend \(\xi_{\nu+1}\) to a mapping \(\psi_{\nu+1}:\overline{W}_{\nu+1}\to P\) of the closure of some neighborhood \(W_{\nu+1}\subset E^{2n+1}\) of the polyhedron \(|K_{\nu+1}|\). Then there exists a neighborhood \(W'_{\nu+1}\subset W_{\nu+1}\) of the polyhedron \(|K_{\nu+1}|\) and a number \(\omega_\nu>0\) such that the complete preimage of the neighborhood \(U(x,\omega_\nu)\) of any point \(x\in N_j(\eta_\nu)\) under the mapping \(\psi_{\nu+1}:W'_{\nu+1}\to P\) admits a \(1/2^\nu\)-covering of multiplicity \(\leq \dim X-\dim P+\chi(x)+1\). There exists, further, a positive number \(\delta_\nu<\gamma_\nu\) such that, when \(\rho(y,y')<\delta_\nu\), where \(y,y'\in W'_{\nu+1}\), we shall have \(\rho(\psi_{\nu+1}(y),\psi_{\nu+1}(y'))<\eta_\nu\). Now take for \(V_{\nu+1}\) the neighborhood \(U(|K_{\nu+1}|,\delta_\nu)\cap W'_{\nu+1}\cap V_\nu\) of the polyhedron \(|K_{\nu+1}|\), and consider the mapping \(\psi_{\nu+1}\) only on this set.

Thus the sequences \(\{K_i\}\), \(\{V_i\}\), \(\{\psi_i\}\), \(\{\omega_i\}\) have been constructed inductively. In doing so we may assume that the conditions (a)—(i) used in the proof of Theorem 3 in \((^3)\) are satisfied. Then the polyhedra \(|K_i|\) converge to some compactum \(X'\), and the mappings \(\varphi_i\) to a homeomorphic mapping \(\varphi\) of the compactum \(X\) onto \(X'\).

It is not difficult to see, further, that for any point \(x\in X'\) and for any \(i<\nu\) the inequality
\[ \rho(\psi_{\nu+1}(x),\psi_\nu(x))<\omega_i/2^{\nu+1} \]
holds; from this it follows that the mappings \(\psi_i:X'\to P\) converge to some continuous mapping \(\psi:X'\to P\) satisfying the condition
\[ \rho(\psi_{\nu+1}(x),\psi(x))<\omega_\nu\quad (x\in X'). \tag{*} \]

We now show that the mapping \(g=\psi\circ\varphi:X\to P\) is the required one. We have: \(\lim(\psi_i\circ\varphi_i)=g\). Further, for \(x\in X\),
\[ \rho(\psi_{\nu+1}(\varphi_{\nu+1}(x)),\psi_\nu(\varphi_\nu(x)))\leq \rho(\psi_{\nu+1}(\varphi_{\nu+1}(x)),\psi_\nu(\varphi_{\nu+1}(x)))+ \rho(\psi_\nu(\varphi_{\nu+1}(x)),\psi_\nu(\varphi_\nu(x)))\leq \eta_\nu+\eta_\nu=\varepsilon/2^{\nu+1}. \]
Thus,
\[ \rho(\psi_{\nu+1}\circ\varphi_{\nu+1},\psi_\nu\circ\varphi_\nu)\leq \frac{\varepsilon}{2^{\nu+1}},\quad \nu=1,2,\ldots;\qquad \rho(\psi_1\circ\varphi_1,f)<\frac{\varepsilon}{2}, \]
whence, by passage to the limit, we obtain \(\rho(f,g)<\varepsilon\).

Let \(x\in P\) be an arbitrary point. Since \(\rho(x,M_{\chi(x)+1})>0\), and the numbers \(\eta_\nu\) decrease without bound, for all sufficiently large \(\nu\) we have \(x\in N_{\chi(x)}(\eta_\nu)\). Therefore the complete preimage of the neighborhood \(U(x,\omega_\nu)\) under the mapping \(\psi_{\nu+1}:V_{\nu+1}\to P\) admits a \(\frac{1}{2^\nu}\)-covering of multiplicity \(\leq \dim X-\dim P+\chi(x)+1\). The set \(\psi^{-1}(x)\), contained (see (*)) in the indicated complete preimage, admits the same covering. Hence \(\dim\psi^{-1}(x)\leq \dim X-\dim P+\chi(x)\). It remains to note that the sets \(g^{-1}(x)\) and \(\psi^{-1}(x)\) are homeomorphic, for \(g^{-1}(x)=\varphi^{-1}(\psi^{-1}(x))\), and \(\varphi\) is a homeomorphism. Thus Theorem 1 is proved.

We proceed to the construction of a certain defect function connected with the homotopy properties of the polyhedron \(P\). Let \(P\) be a polyhedron, piecewise linearly situated in a Euclidean space \(E\), and let \(x\) be an arbitrary point of it. Consider in \(E\) the sphere \(\Sigma_x\) with center at the point \(x\) and of so small a radius that the sphere \(\Sigma_x\) intersects only those simplices of \(P\) whose closures contain the point \(x\). The intersection

$S_x(P)=P\cap\Sigma_x$ will be called the spherical representative of the point $x$ in $P$. The polyhedron $S_x(P)$ is isomorphic to the boundary of the open star of the point $x$. By $r(x)$ we shall denote the maximal one of those numbers $r$ such that, for all points $y\in P$ sufficiently close to $x$, the polyhedron $S_y(P)$ is aspherical in dimensions $<r$ (by asphericity in dimension 0 we mean connectedness).

Lemma. Let $P$ and $Q$ be two simplicial decompositions, of dimensions $s$ and $n$, respectively, and let the mesh of the decomposition $P$ be less than $\delta$. Denote by $M_0,M_1,\ldots,M_{s-1}$ the subsets of the polyhedron $P$ constructed for the function $\chi_0(x)$ as indicated above. Put, further, $\Sigma_i^0=M_i$ $(i=0,1,\ldots,s-1)$, and denote by $\Sigma_i^\nu$ the union of all closed simplices of the decomposition $P$ that do not meet $\Sigma_i^{\nu-1}$ $(\nu=1,2,\ldots)$. Finally, let $f:Q\to P$ be an arbitrary continuous mapping; let $i,k$ be integers satisfying the conditions $0\leq k\leq s$, $0\leq i\leq k+1$. Then there exist a simplicial subdivision $Q_k^{(i)}$ of the decomposition $Q$ and a simplicial mapping $f_k^{(i)}:Q_k^{(i)}\to P$ such that the following conditions are satisfied:

1) $\rho(f,f_k^{(i)})<2(ks+s+i)\delta$;

2) $\dim f_k^{(i)}(T)\geq \dim T-n+j$, if $T$ is such an (open) simplex of the decomposition $Q_k^{(i)}$ that

\[ f_k^{(i)}(T)\subset M_{s-j}\setminus \Sigma_{s-j-i}^{ks+s+i}\quad (j=1,2,\ldots,k); \]

3) if $T$ is such an (open) simplex of the decomposition $Q_k^{(i)}$ that

\[ f_k^{(i)}(T)\subset P\setminus \Sigma_{s-k}^{ks+s+i}, \]

then

\[ \dim f_k^{(i)}(T)\geq \begin{cases} \dim T-n+k+1 & \text{if } \dim T<n-k+i-1,\\ \dim T-n+k & \text{otherwise.} \end{cases} \]

If, moreover, the mapping $f$ itself is simplicial, then one may assume that $f_k^{(i)}=f$ on all simplices satisfying conditions 2) and 3) (i.e. in passing from the decomposition $Q$ to $Q_k^{(i)}$ such simplices are not subdivided and on them the mapping $f_k^{(i)}$ coincides with $f$).

We shall carry out the construction of the mappings $f_k^{(i)}$ successively, increasing the indices $i$ and $k$. For $i=k=0$ conditions 2) and 3) impose no requirements, and therefore for $f_0^{(0)}$ one may take the corresponding simplicial approximation to the mapping $f$ (or the identity mapping, if it is simplicial). Further, if the mapping $f_k^{(k+1)}$ has already been constructed, then we may put $f_{k+1}^{(0)}=f_k^{(k+1)}$, since in this case all the conditions, as is easy to see, are satisfied. It remains to carry out the construction of the mapping $f_k^{(i+1)}$, assuming that the mapping $f_k^{(i)}$ has been constructed $(0\leq i\leq k)$.

Let $T$ be an open simplex of dimension $n-k+i$, satisfying the condition

\[ f_k^{(i)}(T)\subset P\setminus \Sigma_{s-k}^{ks+s+i+1}. \]

For every $(n-k+i-1)$-dimensional face of the simplex $T$ condition 3) is fulfilled, and therefore $\dim f_k^{(i)}(\dot T)\geq i$. If $\dim f_k^{(i)}(T)\geq i+1$, then we may assume that the simplex $T$ is not subdivided in passing from $Q_k^{(i)}$ to $Q_k^{(i+1)}$ and that $f_k^{(i)}=f_k^{(i+1)}$ on $T$. Consider the case $\dim f_k^{(i)}(T)=i$ and denote the simplex $f_k^{(i)}(T)$ by $\tau$. Denote by $З(T)$ and $З(\tau)$ the closures of the open stars of the simplices $T$ and $\tau$, respectively, in the decompositions $Q_k^{(i)}$ and $P$, by $\dot З(T)$ and $\dot З(\tau)$ the boundaries of these open stars, and by $\Pi(T)$ and $\Pi(\tau)$ their representatives (i.e. $\Pi(T)*T=З(T)$, $\Pi(\tau)*\tau=З(\tau)$, where $*$ is the sign of the combinatorial join). It is easy to see that $f_k^{(i)}(\Pi(T))\subset \Pi(\tau)$ and that the mapping $f_k^{(i)}:\Pi(T)\to \Pi(\tau)$ is homeomorphic (see condition 3) in the formulation of the lemma).

Let $b$ be an arbitrary interior point of the simplex $T$. Then the pyramid $b*\Pi(T)$ is a simplicial decomposition of dimension $\leq k-i$. Since the polyhedron $\Pi(\tau)$ is aspherical in dimensions $<k-i$ (this follows easily from the relation $\tau\subset M_{s-k-1}\setminus M_{s-k}\subset P\setminus \Sigma_{s-k}^{ks+s+i+1}$ on the basis of Hurewicz’s theorem and theorem (3), p. 51 of paper (4)), the mapping

\(f_k^{(i)}:\Pi(T)\to\Pi(\tau)\) can be extended to a continuous mapping \(\varphi:b*\Pi(T)\to\Pi(\tau)\). We may assume that \(\varphi\) is a simplicial mapping of some simplicial subdivision \(B\) of the polyhedron \(b*\Pi(T)\) into the carrier \(\Pi(\tau)\), and all simplices of the carrier \(\Pi(T)\) enter the subdivision \(B\) unsubdivided. Moreover, we may assume that the simplicial mapping \(\varphi:B\to\Pi(T)\) is a homeomorphism; this follows from the lemma being proved (as applied to the mapping \(\varphi:B\to\Pi(\tau)\)) for those values of \(i,k\) for which, by induction, we assume the lemma proved.

Let us now denote the boundary of the simplex \(T\) by \(\dot T\). Then the combinatorial join \(\dot T*B\) is a subdivision of the star \(\mathfrak Z(T)\), and the boundary \(\dot{\mathfrak Z}(T)\) of this star is not subdivided. Define the mapping \(f_k^{(i+1)}\) on the star \(\mathfrak Z(T)\), taking it on \(\dot T\) to coincide with \(f_k^{(i)}\), and on \(B\) to coincide with \(\varphi\), and extending it to the simplices of the subdivision \(T*B\) by linearity. It is then easy to see that condition 3) for the mapping \(f_k^{(i+1)}\), considered on \(\mathfrak Z(T)\), is fulfilled, and moreover \(f_k^{(i)}=f_k^{(i+1)}\) on \(\dot{\mathfrak Z}(T)\). Carrying out this construction for all \((n-k+i)\)-dimensional simplices \(T\) for which
\(\dim\bigl(f_k^{(i)}(T)\bigr)\subset P\setminus\Sigma_{s-k}^{ks+s-i+1}\), we put \(f_k^{(i)}=f_k^{(i+1)}\) on all remaining simplices. The constructed mapping \(f_k^{(i+1)}\) satisfies conditions 2), 3). Condition 1) follows from the fact that \(\rho(f_k^{(i)},f_k^{(i+1)})<2\delta\), for both the mapping \(f_k^{(i)}\) and the mapping \(f_k^{(i+1)}\) carry the star \(\mathfrak Z(T)\) into the star \(\mathfrak Z(\tau)\). The induction just carried out proves the lemma. The final assertion of the lemma is easily verified.

Theorem 2. The function \(\chi_0(x)=\dim P-r(x)-1\) is deficient.
This theorem follows immediately from the lemma proved, for \(k=s-1,\ i=s\), if \(\delta\) is chosen so small that \(\rho(f,f_{s-1}^s)<\eta\) and \(\Sigma_i^{s^2+s}\subset U(M_i,\eta)\).

Comparing Theorems 1 and 2, we obtain the following result:

Main theorem. The function \(\chi_0(x)=\dim P-r(x)-1\) has the following property. Whatever finite-dimensional compactum \(X\), continuous mapping \(f:X\to P\), and positive number \(\varepsilon\) may be, there exists a mapping \(g:X\to P\) such that \(\rho(f,g)<\varepsilon\) and

\[ \dim\bigl(g^{-1}(x)\bigr)\leq \dim X-\dim P+\chi_0(x)=\dim X-r(x)-1 \]

for every point \(x\in P\).

In conclusion, we note that for a number of simple polyhedra \(P\) the function \(\chi_0(x)\) is the least of the functions \(\chi(x)\) for which the assertion of the main theorem is valid (cf. the example given at the beginning of the note). Whether the function \(\chi_0\) is the least in all cases is unknown to the author.

I take this opportunity to express my deep gratitude to V. G. Boltyanskii, who suggested the subject of the present work and supervised it.

Kishinev State
Pedagogical Institute
named after I. Creangă

Received
1 VIII 1959

CITED LITERATURE

1. Hurewicz, Sitzungsber. Preuss. Akad. d. Wiss., phys.-math. Kl., H. 24/25, 754 (1933).
2. P. S. Aleksandrov, Combinatorial Topology, Moscow, 1947.
3. V. G. Boltyanskii, Izv. AN SSSR, ser. matem., 23, No. 6 (1959).
4. V. G. Boltyanskii, Tr. Matem. inst. im. V. A. Steklova AN SSSR, 47 (1955).