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The geometry of maps into Euclidean space. (English. Russian original) Zbl 1011.55001

Russ. Math. Surv. 53, No. 5, 893-920 (1998); translation from Usp. Mat. Nauk. 53, No. 5, 27-56 (1998).
The far reaching scope of this interesting paper may be demonstrated by the Contents: 0. Introduction; 1. Sets in strongly general position; 2. Regularly branching maps; 3. The converse of Hurewicz’s theorem for polyhedra; 4. The problems of Hurewicz and Nöbeling; 5. The converse of the existence theorem for polyhedra; 6. Cohomological dimension 7. Historical remarks on Uryson’s formula; Bibliography (64 items).
From the introduction: In a recent paper by the author [Fundam. Prikl. Mat. 4, No. 1, 11-38 (1998; Zbl 0967.55004)] an analogue of the Lyusternik-Shnirel’man theorem is considered for the action of an arbitrary finite group. In the present article, an analogue of the Aleksandrov width is considered which, however, deals with glueing not two points, but \(k\) points having a common image, and maps into Euclidean space are used rather than maps into an arbitrary polyhedron (for the Aleksandrov width see [P. S. Aleksandrov, Fundam. Math. 20, 140-150 (1933; Zbl 0006.42602)].
We obtain some results on the converse of the existence theorem for a “sparing map” (under no requirements on the density of the set of “sparing” maps). In the case when \(k+1\) is 2, an odd prime, or a prime power, the converse of Hurewicz’s corollary on the existence of at most \(k\)-fold maps is given by theorems of E. R. van Kampen [Abh. Math. Semin. Hamb. Univ. 9, 72-78 (1932; Zbl 0005.02604)] and A. Flores [Erg. Math. Kolloqu. Wien 6, 4-7 (1935; Zbl 0011.03804)], K. S. Sarkaria [Proc. Am. Math. Soc. 111, No. 2, 559-565 (1991; Zbl 0722.57007)], and A. Yu. Volovikov [Math. Notes 59, No. 5, 477-481 (1966); translation from Mat. Zametki 59, No. 5, 663-670 (1996; Zbl 0876.57032)]. Using the Cohen-Lusk theorem [F. Cohen and E. L. Lusk, Proc. Am. Math. Soc. 56, 313-317 (1976; Zbl 0326.55002)] on the partial glueing of an orbit, we obtain a partial converse of Hurewicz’s corollary on the existence of more than \(k\)-fold maps for an arbitrary number \(k+1\), which nevertheless, for \(m=n+1\), makes it possible to obtain the full converse of this corollary:
When \(N=2n^2+5n\), for any continuous map \(f:\Delta^N_n \to\mathbb{R}^{n+1}\), where \(\Delta^N_n\) denotes the \(n\)-dimensional skeleton of an \(N\)-dimensional simplex, there exist pairwise disjoint closed simplexes \(\sigma_1,\dots, \sigma_{n+1} \subset\Delta^N_n\) such that the intersection of their images is not empty.
It remains unknown to the author whether it is possible to set \(N=n^2+3n\) in the last statement. The compactum \(\Delta^N_n\) is a polyhedron with a “sufficient” number of strongly enchained \(n\)-dimensional simplexes. However, it turns out that for \(m=n+1\) there is another class of spaces for which any map is essentially different from an embedding.
More precisely, we prove the following strengthening of Aleksandrov’s theorem on the dimensional completeness of an \(n\)-dimensional compactum in \(\mathbb{R}^{n+1}\):
For every zero-dimensional map \(f\) of an \(n\)-dimensional dimensionally incomplete compactum in \(\mathbb{R}^{n+1}\), the set of non-multiple points (in \(\mathbb{R}^{n+1}\)) is of dimension greater than or equal to \(n-1\).
We also prove that the algebraic and geometric deficiencies are equal to each other for finite-dimensional compacta, which sharpens V. G. Boltyanskij’s characterization of dimensionally complete compacta [Dokl. Akad. Nauk SSSR, n. Ser. 67, 773-776 (1949; Zbl 0037.09801)]. Our proof is based on a calculation in the Bockstein algebra that was carried out for another purpose by A. N. Dranishnikov, D. Repovš, and E. V. Shchepin [Topology Appl. 55, No. 1, 67-86 (1994; Zbl 0833.55002)].

MSC:

55M10 Dimension theory in algebraic topology
54F45 Dimension theory in general topology
54-03 History of general topology
55-03 History of algebraic topology
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