MATHEMATICS

E. G. SKLYARENKO

ON THE REPRESENTATION OF INFINITE-DIMENSIONAL COMPACTA AS AN INVERSE LIMIT OF POLYHEDRA

(Presented by Academician P. S. Aleksandrov, 17 V 1960)

Freudenthal proved ((^1)) that every compactum (X) can be represented as such an inverse limit of a sequence of polyhedra ({P_i, f_i^j}), in which the mappings (f_i^j) are piecewise-affine irreducible(^*) mappings “onto.” Moreover, in order that the compactum be (n)-dimensional, it is necessary and sufficient that it be representable as an inverse limit of a sequence of (n)-dimensional polyhedra in which the mappings (f_i^j) satisfy the conditions stated above.

The aim of the present note is to give an analogous characterization for one class of infinite-dimensional compacta, namely compacta that are the sum of a countable number of their closed finite-dimensional subsets. For brevity we shall call such compacta weakly countable-dimensional.(^ {**})

Theorem 1. In order that a compactum (X) be weakly countable-dimensional, it is necessary and sufficient that it can be represented as an inverse limit of a sequence of polyhedra ({P_i, f_i^j}) such that the mappings (f_i^j) satisfy Freudenthal’s conditions and, for every thread (\xi={\xi_i}), (\xi_i \in P_i), the dimensions of the carriers (T(\xi_i)) are bounded in the aggregate.

In proving this theorem we shall, in the main outlines, follow Freudenthal.

The proof of necessity is based on the following proposition: the space (X) is weakly countable-dimensional if and only if there exists a sequence of starwise inscribed in one another, shrinking(^ {***}), finite open coverings such that at each point (x \in X) the orders of all the coverings are bounded in the aggregate ((^{3,4})). Let ({\alpha_n}) be a system of such coverings of the compactum (X). We shall construct a sequence of polyhedra (P_i) and mappings (g_i : X \to P_i), and also

(^*) A mapping is called irreducible if, after an admissible deformation, it remains a mapping “onto.” Here a deformation of a mapping is called admissible if, in the process of deformation, the image of each point does not leave its carrier. Finally, the carrier of a point (\xi) in a given complex is the closed simplex of least dimension containing this point; we shall denote it by (T(\xi)). Irreducibility of a mapping is equivalent to the fact that, for every closed simplex, its full preimage is mapped onto it essentially.

(^ {**}) Countable-dimensional spaces are those that are countable sums of their zero-dimensional (not necessarily closed) subsets. An example of a countable-dimensional but not weakly countable-dimensional compactum was constructed by Yu. M. Smirnov ((^2)).

(^ {***}) A covering (\beta) is starwise inscribed in a covering (\alpha) if the star of any element of (\beta) with respect to (\beta) is contained in some element of (\alpha). The star of a set with respect to a covering is the sum of the elements of the covering that intersect the set. A sequence of coverings is called shrinking if, for every point and every neighborhood of it, there is a covering in the sequence such that the star of the point with respect to it is contained in the given neighborhood.

maps (f_i^j:P_j\to P_i) such that the following conditions are satisfied:
1) (\dim T(g_i x)\leq k(x)) for every point (x\in X), where (k(x)) is the upper bound of the multiplicities of the coverings (\alpha_n) at the point (x); 2) the mappings (g_i) are irreducible mappings “onto”; 3) the mappings (f_i^j) are piecewise affine; 4) (T(f_i^j,g_j x)\subseteq T(g_i x)) for each point (x\in X); 5) for every simplex (T\in P_j) the diameter of its image (f_i^j T) does not exceed (1/2^{j-1}).

For the construction of the polyhedron (P_1) and the mapping (g_1) we proceed as follows: first we construct the barycentric mapping of the compactum (X) into the nerve of the covering (\alpha_1), and then, applying the sweeping-out operation ((^5)), we obtain an irreducible mapping (g_1) onto some subcomplex (P_1) of the nerve. If the polyhedra (P_1,\ldots,P_r) and the mappings (g_i, f_i^j), (i\leq j\leq r), satisfying the indicated conditions have already been constructed, then the construction of the polyhedron (P_{r+1}) and the mapping (g_{r+1}) is carried out as follows: we choose a sufficiently fine subdivision of the polyhedron (P_r) so that the diameters of the images of the simplexes of this subdivision under the mappings (f_i^r) do not exceed (1/2^r). By compactness of the space (X), in the sequence of coverings ({\alpha_n}) there is a covering, say (\alpha_{r+1}), inscribed in the covering composed of the inverse images of the principal stars of the subdivision of the polyhedron (P_r) under the mapping (g_r). Next, the polyhedron (P_{r+1}) and the mapping (g_{r+1}) are constructed from the covering (\alpha_{r+1}) in exactly the same way as the polyhedron (P_1) and the mapping (g_1) were constructed from the covering (\alpha_1); in this process, as the mapping (f_r^{r+1}) we take the simplicial mapping of the polyhedron (P_{r+1}) into the subdivision of the polyhedron (P_r), and, finally, put (f_i^{r+1}=f_i^r f_r^{r+1}). It is easy to verify that the sequence obtained in this way satisfies all the conditions formulated above.

We shall show that the inverse-limit compactum (P) of the sequence ({P_i,f_i^j}) is homeomorphic to (X). To this end we construct mappings (f_i:X\to P_i) such that (f_i=f_i^j f_j). For each mapping (f_i) we take the limit of the sequence of mappings (f_i^j g_j) ((i) fixed), which converges uniformly, since

[
\rho\bigl(f_i^{j+1}g_{j+1}x,f_i^j g_jx\bigr)
=\rho\bigl(f_i^j f_j^{j+1}g_{j+1}x,f_i^j g_jx\bigr)
\leq 1/2^{j-1},
]

which in turn follows from the fact that (g_jx) and (f_j^{j+1}g_{j+1}x) belong to one and the same closed simplex. Since the mapping (f_i) can be obtained from (g_i) by means of an admissible deformation, (f_i) is an irreducible mapping onto the polyhedron (P_i). Assigning to each point (x\in X) the thread ({f_i x}), we obtain the desired homeomorphism of the compactum (X) onto (P).

It remains to verify that for each thread (\xi={\xi_i}), (\xi_i\in P_i), the dimensions of the carriers (T(\xi_i)) are bounded in the aggregate. But this follows from the fact that every thread (\xi) has the form ({f_i x}), (x\in X), while (T(f_i x)\leq T(g_i x)) and (\dim T(g_i x)\leq k(x)).

The proof of sufficiency can also be obtained from the theorem on coverings formulated above: in the inverse-limit compactum, as the sequence of coverings one may take the coverings composed of the inverse images of the principal stars of the complexes (P_i). However, it is easy to give a direct proof as well. For any natural number (n), consider the following sequence of closed subsets (X_{i,n}) of the polyhedra (P_i): let (X_{1,n}) be the (n)-dimensional skeleton of the polyhedron (P_1), and suppose the sets (X_{i,n}) for (i\leq r) have already been constructed; put
[
X_{r+1,n}=(f_r^{r+1})^{-1}X_{r,n}\cap P_{r+1}^n,
]
where (P_{r+1}^n) denotes the (n)-dimensional skeleton of the polyhedron (P_{r+1}). The inverse limit (X_n) of the sequence ({X_{i,n}, f_i^j}) with the same mappings (f_i^j) (considered on (X_{i,n})) is naturally embedded in the compactum (X). From the condition imposed on the threads it follows directly that
[
X=\bigcup_n X_n.
]
The sets (X_n) are compact and (\dim X_n\leq n). Thus the theorem is completely proved.

Theorem 2. If a weakly countable-dimensional compactum (X) is represented as the inverse limit of a sequence of polyhedra ({Q_k, h_k^l}) satisfying Freudenthal’s conditions, then by passing to a subsequence, by an admissible deformation of the maps (h_k^l), and by subdividing the complexes (Q_k), one can pass from the original sequence to a sequence whose limit is still equal to (X), and whose nerves satisfy the condition of Theorem 1.

The proof of this theorem follows directly from the following proposition of Freudenthal: if a compactum (X) is represented as the inverse limit of two sequences of polyhedra ({P_i, f_i^j}) and ({Q_k, h_k^l}), then in them one can choose subsequences (P_{i_n}) and (Q_{k_n}) and construct maps
[
h_{2n-1}: Q_{k_n}\to P_{i_n}
\quad\text{and}\quad
h_{2n}: P_{i_{n+1}}\to Q_{k_n}
]
in such a way that these maps will be simplicial with respect to certain subdivisions of the complexes (P_{i_n}) and (Q_{k_n}), the compositions (h_{2n}h_{2n+1}) and (h_{2n-1}h_{2n}) are obtained by an admissible deformation from the maps (h_{k_n}^{k_{n+1}}) and (f_{i_n}^{i_{n+1}}), and the limit of the alternating sequence of polyhedra
[
P_{i_n}, Q_{k_n}, \qquad n=1,2,\ldots,
]
is equal to (X).

As ({P_i, f_i^j}) we take the sequence constructed in the preceding theorem, and apply Freudenthal’s proposition to the sequences ({P_i, f_i^j}) and ({Q_k, h_k^l}). From the resulting alternating sequence we choose the subsequence consisting of the polyhedra (Q_{k_n}), and take in them those same subdivisions with respect to which the maps (h_{2n}) are simplicial. This sequence will be the desired one.

In conclusion I express my sincere gratitude to Yu. M. Smirnov for valuable advice and comments.

Moscow State University
named after M. V. Lomonosov

Received
6 V 1960

REFERENCES CITED

¹ H. Freudenthal, Compositio Math., 4 (1937). ² Yu. M. Smirnov, Izv. AN SSSR, ser. matem., 23, No. 2 (1959). ³ E. Sklyarenko, DAN, 126, No. 6 (1959). ⁴ J. Nagata, Fund. Math., 48, No. 1 (1960). ⁵ P. S. Aleksandrov, Combinatorial Topology, Moscow–Leningrad, 1947.