MATHEMATICS

B. PASYNKOV

ON POLYHEDRAL SPECTRA AND THE DIMENSION OF BICOMPACTA, IN PARTICULAR BICOMPACT GROUPS

(Presented by Academician P. S. Aleksandrov on 6 V 1958)

Introduction. Below, unless otherwise stated, we consider spectra (for the definition see ((^1)), § 6) with projections “onto.”

We shall call a spectrum (S={X_\alpha,\ \omega_\alpha^\beta}) polyhedral if the (X_\alpha) are polyhedra; if, in addition, the projections (\omega_\alpha^\beta) are, under certain subdivisions of (X_\beta) and (X_\alpha), simplicial and affine on the simplexes of the subdivision, then the spectrum (S) will be called simplicial. A spectrum (S={X_\alpha,\ \omega_\alpha^\beta}) will be called (n)-dimensional in the sense of (\dim) if (\dim X_\alpha \le n) for all (\alpha).

The main results of this paper are the following propositions:

Theorem 1. a) Every bicompactum is the space of some polyhedral spectrum.

b) For zero-dimensional bicompacta this spectrum may be taken to be zero-dimensional, simplicial, and with projections “onto.”

Theorem 2. a) If a bicompactum (X) is the space of an (n)-dimensional in the sense of (\dim) spectrum (S={X_\alpha,\ \omega_\alpha^\beta}), then (\dim X \le n).

b) If a bicompactum (X) is the space of an (n)-dimensional polyhedral spectrum, then for any closed set (\Phi \subseteq X) there exist arbitrarily close neighborhoods whose boundary belongs to the space of some ((n-1))-dimensional spectrum of compacta.

Along the way it is proved that:

c) If (\dim X=n), where (n=0) or (1), and (X) is the space of an (n)-dimensional polyhedral spectrum, then

[
n=\dim X=\operatorname{ind} X=\operatorname{Ind} X.
]

It follows from this that the bicompacta of Lunc ((^2)) and Lokucievskii ((^3)), for which (\dim X=1), (\operatorname{ind} X=2), are not spaces of any one-dimensional polyhedral spectrum; i.e., generally speaking, the dimension (\dim X) of a bicompactum (X) cannot be defined as the least of the dimensions of the polyhedra approximating this bicompactum.

Theorem 3. If a bicompactum (X) is the space of an (n)-dimensional simplicial spectrum, then (n \ge \operatorname{Ind} X).

Hence it follows: if (\dim X=n) and (X) is the space of an (n)-dimensional simplicial spectrum, then

[
n=\dim X=\operatorname{ind} X=\operatorname{Ind} X.
]

In conclusion it is proved that:

Theorem 4. For a bicompact topological group (X)

[
\dim X=\operatorname{ind} X=\operatorname{Ind} X.
]

§ 1. Proof of Theorem 1. a) Let (\tau) be the weight of the bicompactum (X); then (X) may be regarded as embedded in the Tychonoff cube (J^\tau) of weight (\tau). Let us supply all one-dimensional faces (J^\tau) with indices from some set (A={\alpha}). By (J^{n(\alpha_1\ldots\alpha_n)}) we shall denote the face of (J^\tau) equal to (\prod_{i=1}^{n} J^{1(\alpha_i)}), and by (X^{n(\alpha_1\ldots\alpha_n)}) the projection of the set (X) onto this face. Consider such subdivisions of the (n)-dimensional faces (J^\tau) into cubes that the length of the one-dimensional faces of the cubes of the subdivision is equal to (1/2^n). From (J^{n(\alpha_1\ldots\alpha_n)}) discard successively (in order of dimension) the open senior cubes that contain no points of (X^{n(\alpha_1\ldots\alpha_n)}). Taking the bordering polyhedra that remain after the operation of discarding, with the natural order and projections, we obtain the required spectrum.

b) If the bicompactum (X) is zero-dimensional, then in it one can choose a base of open-closed sets (\gamma={\Gamma_\alpha}), and define an embedding system of functions as follows: (f_\alpha=0) on (\Gamma_\alpha), (f_\alpha=1) on (X\setminus\Gamma_\alpha). In this case the (X^{n(\alpha_1\ldots\alpha_n)}) will consist simply of a finite number of points and will give us the required spectrum.

Proof of Theorem 2. By definition, a base (\mathfrak B) in the space (X) of the spectrum (S={X_\alpha,\mathfrak D_\alpha^\beta}) is defined as follows. A finite number of indices (\alpha_1,\ldots,\alpha_s) is fixed; arbitrary open sets (V_{\alpha_1},\ldots,V_{\alpha_s}) in (X_{\alpha_1},\ldots,X_{\alpha_s}) are chosen, and the set

[
O=\mathcal E{x,\ x={x_\alpha},\quad x_{\alpha_i}\in V_{\alpha_i},\quad i=1,\ldots,s}
]

is declared to be an element of the base of the space (X), induced by the collection (V_{\alpha_1},\ldots,V_{\alpha_s}). We first prove two lemmas.

Lemma 1. Each element (O) of the base (\mathfrak B) of the space (X) may be assumed to be induced by only one (V_\alpha).

Indeed, let (O) be defined by the collection (V_{\alpha_1},\ldots,V_{\alpha_s}). Taking in (X_\alpha), where (\alpha>\alpha_i,\ i=1,\ldots,s), the set (\bigcap(\mathfrak D_{\alpha_i}^{\alpha})^{-1}V_{\alpha_i}), we obtain the required (V_\alpha).

Lemma 2. For any finite covering (\gamma={O_1,\ldots,O_s}), consisting of elements of the base (\mathfrak B), of a closed set (\Phi\subseteq X), one can find (X_\alpha) and in it a system of open sets (\gamma_\alpha={V_{\alpha_1},\ldots,V_{\alpha_s}}) inducing in (X) the system (\gamma).

Indeed, we may assume that (O_i) is induced by the set (V_{\alpha_i}), (i=1,\ldots,s). Taking in (X_\alpha), (\alpha>\alpha_i), (V_{ai}=(\mathfrak D_{\alpha_i}^{\alpha})^{-1}V_{\alpha_i}), (i=1,\ldots,s), we obtain the desired system (\gamma_\alpha).

Remark to Lemma 2. It is clear that (V_\alpha=\bigcup V_{\alpha i}) induces in (X) the element of the base (O=\bigcup O_i), containing the set (\Phi).

Let us now prove assertion a) of Theorem 2. Take an arbitrary covering of the bicompactum (X). Inscribe in it a finite covering (\gamma) from elements of the base (\mathfrak B). By Lemma 2 we find the corresponding (X_\alpha) and its covering (\gamma_\alpha). Since (\dim X_\alpha\le n), we inscribe in (\gamma_\alpha) a covering (\omega_\alpha) of multiplicity (\le n+1), which induces in (X) a covering (\omega) of multiplicity (\le n+1), inscribed in (\gamma). Item a) is proved.

For the proof of item b) we preface two further lemmas.

Lemma 3. A bicompactum (X) that is the space of a zero-dimensional spectrum of bicompacta is zero-dimensional.

This follows from item a) of Theorem 2.

Lemma 4. Let (X) be the space of the spectrum (S={X_\alpha,\mathfrak D_\alpha^\beta}); then (X) is the space of the spectrum (S'={X_\alpha,\mathfrak D_\alpha^\beta}), where (\alpha\ge \alpha_0).

The proof of this lemma presents no difficulties.

We now prove assertion b) of Theorem 2. (X) is the space of an (n)-dimensional polyhedral spectrum (S={X_\alpha,\wp^\beta_\alpha}). Since (\Phi\subset X) is a bicompactum, by the remark to Lemma 2 one may assume that an arbitrary neighborhood (O\Phi) is induced by an element (V_{\alpha_0}). By Lemma 4, we now pass to the consideration of the spectrum (S'), (\alpha\geqslant\alpha_0).

For each (\alpha) take the sets (F_\alpha=[V_\alpha]\setminus V_\alpha), where (V_\alpha=(\wp^\alpha_{\alpha_0})^{-1}V_{\alpha_0}). The sets (F_\alpha) form a spectrum of compacta, if they are taken with the natural projections and order. It is enough to show that (\wp^\beta_\alpha F_\beta\subset F_\alpha). But this follows easily from the transitivity of the spectrum. Take an arbitrary subdivision (X_{\alpha'}); from the fact that (F_\alpha) is nowhere dense in each (n)-dimensional simplex of the subdivision, it follows that (\dim F_\alpha\leqslant n-1).

We denote the space of the spectrum (s={F_\alpha,\wp^\beta_\alpha}) by (F). We shall show that ([O\Phi]\setminus O\Phi\subseteq F). Let (x={x_\alpha}\in[O\Phi]\setminus O\Phi). Every neighborhood (Ox) contains points of (O\Phi), in particular also when (Ox) is a basic neighborhood induced by an arbitrary (V_{x_\alpha}\subseteq X_\alpha), i.e. (V_\alpha\cap V_{x_\alpha}\ne\Lambda), i.e. (x_\alpha\in F_\alpha) for every (\alpha). Assertion b) is proved.

We prove part c). For bicompacta it is known that (\dim X\leqslant \operatorname{ind}X\leqslant \operatorname{Ind}X), and for zero-dimensional bicompacta all these three dimensions coincide.

For (n=0) assertion c) is obvious. Let (n=1). Then every closed set (\Phi\subset X) has an arbitrarily close neighborhood whose boundary belongs to a zero-dimensional bicompactum, i.e. is itself zero-dimensional; hence, (\operatorname{Ind}X\leqslant1).

§ 2. Proof of Theorem 3. Let the bicompactum (X) be the space of an (n)-dimensional simplicial spectrum (S={X_\alpha,\wp^\beta_\alpha}). We shall prove that (\operatorname{Ind}X\leqslant n). Take an arbitrary closed set (\Phi\subset X). Let (O\Phi) be a neighborhood of (\Phi); one may assume that it is induced by some set (V_{\alpha_0}). Consider now the spectrum (S'={X_\alpha,\wp^\beta_\alpha}), (\alpha\geqslant\alpha_0), and let (\Phi_\alpha) be the projection of (\Phi) in (X_\alpha). Then (\Phi_{\alpha_0}\subseteq V_{\alpha_0}). One can find a polyhedral neighborhood (W_{\alpha_0}) of the set (\Phi_{\alpha_0}), contained in (V_{\alpha_0}). Consider the sets (P_\alpha=[W_\alpha]\setminus W_\alpha), where (W_\alpha=(\wp^\alpha_{\alpha_0})W_{\alpha_0}). Repeating the arguments of part b) of the preceding theorem, we obtain that (P_\alpha) form an ((n-1))-dimensional simplicial spectrum, whose space contains the boundary of the basic neighborhood of the set (\Phi) induced by (W_{\alpha_0}).

For (n=0) Theorem 3 is true; by induction it extends to the case of arbitrary (n).

§ 3. The proof of Theorem 4 rests on some auxiliary considerations. Suppose two spectra are given: (S_X={X_\alpha,\wp^{\alpha'}\alpha}) and (S_Y={Y\beta,\wp^{\beta'}\beta}), whose sets of indices we denote respectively by (A={\alpha}) and (B={\beta}), and whose limiting spaces by (X) and (Y). Partially order the set (C) of all pairs (\gamma=(\alpha,\beta)) by putting ((\alpha',\beta')>(\alpha,\beta)) if simultaneously (\alpha'>\alpha) and (\beta'>\beta). To the index (\gamma=(\alpha,\beta)) we put in correspondence the space (Z\gamma=X_\alpha\times Y_\beta), and for (\gamma'>\gamma) define the projection (\wp^{\gamma'}_\gamma) by the formula

[
\wp^{\gamma'}\gamma(x)},y_{\beta'
=
(\wp^{\alpha'}\alpha x},\wp^{\beta'\beta y).
]

Thus we obtain a spectrum (S_Z={z_\gamma,\wp^{\gamma'}_\gamma}), which we call the product of the spectra (S_X) and (S_Y). It is easy to verify that the space (Z) of the spectrum (S_Z) is homeomorphic to the product (X\times Y).

§ 4. Proof of Theorem 4. It is known that every connected finite-dimensional bicompact group is simply a compactum ((4), p. 107), i.e. is the space of a simplicial spectrum of the corresponding dimension ((5) Ch. IV, p. 183). From Mostert’s results ((6)) it follows that the space of every bicompact group (G) decomposes into a to-

topological product of the space of the connected component (K) of its identity and the space of the quotient group by this component. This quotient group is zero-dimensional, i.e. it has a zero-dimensional simplicial spectrum, and the spaces that constitute it consist of a finite number of points. Multiplying the simplicial spectra for (K) and (G/K), we obtain a simplicial spectrum for (G) of the same dimension as for (K), i.e.
[
\dim G = \operatorname{ind} G = \operatorname{Ind} G = \dim K,
]
which was to be proved.

This work was carried out under the supervision of Acad. P. S. Aleksandrov, to whom the author expresses his deep gratitude.

Moscow State University
named after M. V. Lomonosov

Received
25 IV 1958

References

(^{1}) P. S. Aleksandrov, Uspekhi Mat. Nauk, 2, no. 1 (1947).
(^{2}) A. L. Lunts, DAN, 66, no. 5 (1949).
(^{3}) O. V. Lokutsievskii, DAN, 67, no. 2 (1949).
(^{4}) A. Weil, Integration in Topological Groups and Its Applications, IL, 1950.
(^{5}) H. Freudenthal, Compositio Math., 4 (1937).
(^{6}) P. S. Mostert, Duke Math. J., 23, 57 (1956).