MATHEMATICS

B. PASYNKOV

ON INVERSE SPECTRA AND DIMENSION

(Presented by Academician P. S. Aleksandrov on 8 II 1961)

I. A spectrum (S={P_\alpha,\delta_\alpha^\beta}) (for the definition of a spectrum see in ((^1)) or in ((^{2,3}))) is called polyhedral if the spaces (P_\alpha) are polyhedra given in some of their triangulations. If, for certain subdivisions (P_\beta) and (P_\alpha), the projections (\delta_\alpha^\beta) are simplicial and affine on the simplexes of the subdivision, then the spectrum (S) is called simplicial. A spectrum (\Sigma={K_\alpha,\delta_\alpha^\beta}) will be called combinatorial if the spaces (K_\alpha) are finite complexes and the projections (\delta_\alpha^\beta) are transitive and single-valued. These spectra will be assumed Hausdorff and essential ((^2)).

Remark. The complexes (K_\alpha) are not assumed to be complete, i.e., the faces of a simplex (t_\alpha\in K_\alpha) need not belong to (K_\alpha).

The complexes (K_\alpha) may be regarded as topological (T_0)-spaces ((^1)); in this case the projections (\delta_\alpha^\beta) are simply continuous mappings and the dimension (\dim K_\alpha) is defined for (K_\alpha). The spectrum (S={P_\alpha,\delta_\alpha^\beta}), respectively (\Sigma={K_\alpha,\delta_\alpha^\beta}), will be called (r)-dimensional, and we shall write (\dim S\le r), respectively (\dim\Sigma\le r), if for every (\alpha)
[
\dim P_\alpha\;(\dim K_\alpha)\le r.
]
It is easy to see that for the limit space (X) of an (r)-dimensional polyhedral or combinatorial spectrum one has
[
\dim X\le r.
]
One can prove:

Theorem 1. In order that a spectrum
[
\Omega={X_\alpha,\delta_\alpha^\beta}
]
with projections “onto,” where the (X_\alpha) are bicompacts (respectively, an essential spectrum of finite complexes), should give in the limit a bicompact (X) of dimension not greater than (r) in the sense of (\dim), it is necessary and sufficient that for every (X_\alpha) and its covering
[
\gamma={O_i},\quad i=1,\ldots,s,
]
there exist an (X_\beta), (\beta\ge\alpha), such that in it one can inscribe a covering of multiplicity not greater than (r+1) in the covering by the inverse images
[
(\delta_\alpha^\beta)^{-1}O_i.
]

We shall call a complex (K_\alpha) a complex of length (l) and write
[
\operatorname{ind} K_\alpha\le l,
]
if the maximal number of elements in any naturally ordered chain of its simplexes is not greater than (l+1).

Example. For a complex (K) consisting of a tetrahedron (t^3), its two-dimensional face (t^2), an edge (t^1) which is a face of (t^2), and a vertex (t^0) of the edge (t^1), we have
[
\dim K=0,\qquad \operatorname{ind}K=3.
]

We shall say that the dimension of the senior simplexes of a complex (K_\alpha) does not exceed (r) if it can be realized so that the dimension of the senior simplexes of the realization does not exceed (r). It is clear that in this case
[
\dim K\le r\quad\text{and}\quad \operatorname{ind}K\le r.
]
And, finally, a spectrum (\Sigma={K_\alpha,\delta_\alpha^\beta}) will be called a spectrum of length (l) ((\operatorname{ind}\Sigma\le l)) if for every (\alpha) we have
[
\operatorname{ind}K_\alpha\le l.
]
It is clear that
[
\dim\Sigma\le r\quad\text{and}\quad \operatorname{ind}\Sigma\le r,
]
if for every (\alpha) the dimension of the senior simplexes of (K_\alpha) does not exceed (r).

Theorem 2. If for a combinatorial spectrum (\Sigma) we have (\operatorname{ind}\Sigma\le r), then for the limit space (X) one has
[
\operatorname{Ind}X\le r.
]

Theorem 3. If a bicompact (X) is the limit of an (r)-dimensional simplicial spectrum (S) with projections “onto,” then it is the limit of a combina-

of the spectrum (\Sigma) with at most (r)-dimensional senior simplexes of the complexes of this spectrum, i.e. (\operatorname{ind}\Sigma \leqslant r) and (\dim\Sigma \leqslant r).

Further, one can construct a bicompactum (L), which is the limit of a one-dimensional combinatorial spectrum of length (1), but is not the limit of any one-dimensional polyhedral spectrum. For this bicompactum (\dim L = \operatorname{ind} L = \operatorname{Ind} L = 1), i.e. we have a strengthening of the result obtained in ((3)^*), where the absence of a one-dimensional polyhedral spectrum was derived only for a bicompactum one-dimensional in the sense of (\dim), and followed from the noncoincidence, for this bicompactum, of the dimensions (\dim) and (\operatorname{ind}) ((\operatorname{Ind})). From two copies of the bicompactum (L) (analogously to the way O. V. Lokutsievskii did this in ((5))) one can construct a bicompactum (M) for which (\dim M = 1), while (\operatorname{ind} M = 2); but then, by Theorem 2, the bicompactum (M) cannot be the limit of any combinatorial spectrum of length (1), i.e. for combinatorial spectra of the given length the sum theorem is not true.

In ((3)) it was shown (this also follows from Theorems 2 and 3) that from the existence, for a bicompactum (X), of an (r)-dimensional simplicial spectrum it follows that (\operatorname{Ind} X \leqslant r). It turns out that, for every (m = 1, 2, \ldots, \infty), one can construct a bicompactum (A(m)) such that (\dim A(m) = \operatorname{ind} A(m) = \operatorname{Ind} A(m) = 1), and (A(m)) will be the limit of a one-dimensional combinatorial spectrum of length one, but for which a simplicial spectrum will be no less than (m)-dimensional.

Construction of (L) and (A(m)). Denote on the line the point ((-1/2)) by (\alpha); the interval ([0,1]) by (I_\alpha); the Cantor perfect set, arranged in the usual way on ([0,1]), by (C_\alpha), and the point ((3/2)) by (\alpha + 1). Then the sets (\alpha \cup I_\alpha \cup \alpha + 1) and (\alpha \cup C_\alpha \cup \alpha + 1) are naturally ordered. In this case we shall say that (I_\alpha) or (C_\alpha) is placed between (\alpha) and (\alpha + 1).

We construct (L). Consider the transfinite numbers (\alpha \leqslant \omega_1). Place intervals (I_\alpha) between the pair of numbers (\alpha) and (\alpha + 1), for every (\alpha \in W(\omega_1)={\alpha < \omega_1}). The topology in the set
[
K=\bigcup_{\alpha\leqslant\omega_1}\alpha \cup \bigcup_{\alpha<\omega_1} I_\alpha
]
is given by the transitive order generated by the order of the numbers (\alpha) and the sets (\alpha \cup I_\alpha \cup \alpha + 1). Multiply the set (K) by the Cantor perfect set (C), and then glue the set (C \times \omega_1) into an interval, identifying pairwise the endpoints of the intervals adjacent to the set (C) (assuming it arranged in the usual way on the interval ([0,1])). The construction of the bicompactum (L) is finished. As is clear, it is similar to that carried out in ((5)).

We construct the bicompactum (A(m)) for (m=2). Consider all transfinite numbers (\alpha \leqslant \omega_{\tau_1}) and (\beta \leqslant \omega_{\tau_2}), where (\omega_{\tau_i}) is the first ordinal number of regular cardinality (\tau_i) (i.e. (\omega_{\tau_i}) is not a limit point of any subset of (W(\omega_{\tau_i})) of cardinality less than (\tau_i)). Place Cantor perfect sets (C_\alpha) between every pair of numbers (\alpha) and (\alpha + 1). The topology of the set
[
C_{\tau_1}=\bigcup_{\alpha\leqslant\omega_{\tau_1}}\alpha \cup \bigcup_{\alpha<\omega_{\tau_1}} C_\alpha
]
is given, as before, by the transitive order generated by the order of the numbers (\alpha) and the sets (\alpha \cup C_\alpha \cup \alpha + 1). Analogously we obtain
[
C_{\tau_2}=\bigcup_{\beta\leqslant\omega_{\tau_2}}\beta \cup \bigcup_{\beta<\omega_{\tau_2}} C_\beta .
]
Suppose (\tau_2>\tau_1). Multiply the sets (C_{\tau_1}) and (C_{\tau_2}). In this product, on the sets (C_{\tau_1}\times \omega_{\tau_2}) and (\omega_{\tau_1}\times C_{\tau_2}), glue, as was already done above, the Cantor perfect sets (C_\alpha\times \omega_{\tau_2}) and (\omega_{\tau_1}\times C_\beta) (for all (\alpha<\omega_{\tau_1}) and (\beta<\omega_{\tau_2})) into the intervals (I_\alpha) and (I_\beta). The bicompactum obtained will be the required (A(2)). (A(\infty)) is obtained from the discrete sum of the spaces (A(m)), (m=1,2,\ldots), by bicompactifying with one point.

II. One may consider spectra (S={X_\alpha,\ \delta_\alpha^\beta}), where the (X_\alpha) are Hausdorff spaces, imposing certain restrictions on the projections (\delta_\alpha^\beta) and on the dimension of (X_\alpha), and obtain estimates for the dimension of the limit space (X).

[
\underline{\hspace{2.5em}}
]

* This weaker result was also obtained by S. Mardešić in ((4)).

Theorem 4. If the projections $\omega_\alpha^\beta$ do not lower $\operatorname{ind}$ of closed subsets $X_\beta$ (i.e. $\operatorname{ind}\omega_\alpha^\beta F \geq \operatorname{ind}F$, where $F$ is closed in $X_\beta$), then from $\operatorname{ind}X_\alpha \leq r$ for every $\alpha$ it follows that $\operatorname{ind}X \leq r$.

Corollary 1. If in Theorem 4 the set $\Phi \subset X$ is a bicompactum, then $\omega_\alpha$ does not lower the dimension $\operatorname{ind}$ of the set $\Phi$ for every $\alpha$.

Corollary 2. If closedness is required of $\omega_\alpha$, then for every closed $F \subset X$ we have $\operatorname{ind}\omega_\alpha F \geq \operatorname{ind}F$ for every $\alpha$.

Corollary 3. In particular, if $\dim X=r$, the $X_\alpha$ are spaces with a countable base, $\dim X_\alpha \leq r$, and $\omega_\alpha^\beta$ are closed and zero-dimensional, then $\operatorname{ind}X=\dim X$. Hence it is clear that if for a bicompactum $X$ we have $\dim X=r<\operatorname{ind}X$, then in the spectrum $S={\Phi_\alpha,\omega_\alpha^\beta}$ for $X$, where the $\Phi_\alpha$ are $r$-dimensional compacta (such a spectrum always exists, see (4)), all projections $\omega_\alpha^\beta$ cannot be zero-dimensional.

Theorem 5. Let the bicompactum $X$ be the limit of such a spectrum $S={X_\alpha,\omega_\alpha^\beta}$ of bicompacta $X_\alpha$ that $\omega_\alpha^\beta$ do not lower $\operatorname{Ind}$ of closed subsets $X_\beta$. Then: a) from $\operatorname{Ind}X_\alpha \leq r$ for every $\alpha$ it follows that $\operatorname{Ind}X \leq r$; b) for every $\alpha$, $\operatorname{Ind}\omega_\alpha\Phi \geq \operatorname{Ind}\Phi$, where $\Phi$ is closed in $X$.

Corollary. If in the spectrum $S$ the spaces $X_\alpha$ are compacta for which $\dim X_\alpha \leq \dim X$ and the projections $\omega_\alpha^\beta$ are zero-dimensional, then $\dim X=\operatorname{ind}X=\operatorname{Ind}X$.

Example. For the transfinite line $A$ we have $\dim A=\operatorname{ind}A=\operatorname{Ind}A=1$; for it there even exists a one-dimensional simplicial spectrum, but it is not the limit of any one-dimensional spectrum of compacta with zero-dimensional projections, since every continuous function on it is finally constant.

III. Using spectra, one can prove the following assertion.

Theorem 6. Let there be a locally bicompact group $G$, its closed subgroup $H$, and the space of left (right) cosets $X=G/H$. If $\operatorname{ind}H<\infty$ or $\operatorname{ind}X<\infty$, then
$\dim X=\operatorname{ind}X=\operatorname{Ind}X=\operatorname{ind}G-\operatorname{ind}H$.

Remark. The coincidence of dimensions for locally bicompact groups was proved in (6). The equality $\operatorname{ind}X=\operatorname{ind}G-\operatorname{ind}H$, when $\operatorname{ind}X<\infty$*, follows from the results of Mostert (7).

It also turns out that Theorem 7 is valid, which is a strengthening of a result of A. Arhangel’skii (8):

Theorem 7. A homogeneous space $X=G/H$ for a locally bicompact group $G$ decomposes into a discrete sum of finally compact spaces, i.e. it is strongly paracompact, and hence normal.

The author expresses sincere gratitude to P. S. Aleksandrov for posing some of the problems solved here.

Proof-correction note. One can even construct a snake-like bicompactum $N$ for which $\dim N=\operatorname{ind}N=\operatorname{Ind}N=1$, but which will not be the limit of any one-dimensional polyhedral spectrum, in particular of a spectrum of simple arcs. From two copies of the bicompactum $N$ one can construct a snake-like bicompactum $P$ for which $\operatorname{ind}P=2$, i.e. the sum theorem for the dimension $\operatorname{ind}$ is false even for snake-like bicompacta.

Moscow State University
named after M. V. Lomonosov

Received
5 II 1961

REFERENCES

1. P. S. Aleksandrov, UMN, 2, issue 1, 5 (1947).
2. B. Pasynkov, DAN, 131, No. 2, 23 (1960).
3. B. Pasynkov, DAN, 121, No. 1, 45 (1958).
4. S. Mardešić, Illinois J. Math., 4, No. 2, 278 (1960).
5. O. V. Lokutsievskii, DAN, 67, No. 2, 217 (1949).
6. B. Pasynkov, DAN, 132, No. 5, 1035 (1960).
7. P. S. Mostert, Duke Math. J., 23, 1, 57 (1956).
8. A. Arhangel’skii, DAN, 132, No. 5, 980 (1960).

* This case includes the case of a finite-dimensional group $G$.