Minimal Number of Preimages Under Maps of Surfaces

We compute the minimal number of roots of the equation $f(x) = c$ for a map $f$ belonging to a given homotopy class of maps between closed (not necessarily oriented) surfaces.     

Bruijning—Nagata and Hashimoto—Hattori characteristics of covering dimension revisited

New dimension functions for topological spaces are introduced in the spirit of Nagata’s approach. Expressions for the new functions in terms of covering dimension include the Bruijning—Nagata and Hashimoto—Hattori formulas.     

Realization of primitive branched coverings over closed surfaces following the hurwitz approach

Let V be a closed surface, H⊑π₁(V) a subgroup of finite index l and D=[A ₁,...,A _m ] a collection of partitions of a given number d≥2 with positive defect v(D). When does there exist a connected branched covering f:W→V of order d with branch data D and f∶W→V It has been shown by geometric arguments [4] that, for l=1 and a surface V different from the sphere and the projective plane, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D)э0 mod 2. In the case l>1, the corresponding branched covering exists if and only if v(D)э0 mod 2, the number d/l is an integer, and each partition A _i ∈D splits into the union of l partitions of the number d/l. Here we give a purely algebraic proof of this result following the approach of Hurwitz [11]. The realization problem for the projective plane and l=1 has been solved in [7,8]. The case of the sphere is treated in [1, 2, 12, 7].     

Movability relative to various classes of spaces

This article is in answer to a question posed by K. Borsuk [1]. There exists a locally connected continuum X which is movable relative to the class of all spheres, but which is not 2-movable. We shall prove that the classes of movable compacta coincide for the following : 1) all polyhedra of dimension ⩽n, 2) all compacta of dimension ⩽n, and 3) all compacta of fundamental dimension ⩽n. We shall also prove that the movability of a compactum X is equivalent to its movability relative to the class of all polyhedra. Translated from Matematicheskie Zametki, Vol. 21, No. 1, pp. 125–132, January, 1977. The authors would like to thank Yu. M. Smirnov for the great interest he showed in this article.     

Theory of retracts and infinite-dimensional manifolds

The survey is devoted to the theory of retracts and infinite-dimensional manifolds. The basic constructions and concepts are considered, and major attention is given to properties of mappings of softness type.Translated from Itogi Nauki i Tekhniki, Seriya Algebra, Topologiya, Geometriya, Vol. 24, pp. 195–270, 1986.