Semeon Antonovich Bogatyi

On the nonexistence of iterative roots

Abstract For the germ of a holomorphic mapping F : ( U , 0) → ( C , 0) of the form F ( z ) = ρz + ⋯, where ρ is a primitive root of unity of order d ⩾2, criteria for the existence of a continuous iterative root of given order and the topological linearizability of F are given. The following conditions are equivalent: (1) F d = Id; (2) the germ of the mapping F ( z ) is topologically conjugate to the germ of the mapping qz ; (3) the germ of the mapping F has a continuous iterative root of order d k for every k ⩾ 1. If F d ≠ Id, then for a given positive integer N the germ of the mapping F has a continuous iterative root of order N iff d · gcd( N , ind( F d , 0) − 1) divides ind( F d , 0) −1.

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

AbstractLet V be a closed surface, H⊑π1(V) a subgroup of finite index l and D=[A1,...,Am] 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 Ai∈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].

Realization of Primitive Branched Coverings Over Closed Surfaces

Let V be a closed surface, H ⊆ π1(V ) a subgroup of finite index and D = [A1, . . . , Am] 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#(π1(W )) = H? We show that, for a surface V different from the sphere and the projective plane and = 1, the corresponding branched covering exists (the data D is realizable) if and only if the data D fulfills the Hurwitz congruence v(D) ≡ 0 mod 2. In the case > 1, the corresponding branched covering exists if and only if v(D) ≡ 0 mod 2, the number d/ is an integer, and each partition Ai ∈ D splits into the union of partitions of the number d/ . The realization problem for the projective plane and = 1 has been solved in (Edmonds-Kulkarni-Stong, 1984). The case of the sphere is treated in (BersteinEdmonds, 1979; Berstein-Edmonds, 1984; Husemoller, 1962; Edmonds-KulkarniStong, 1984). AMS: Primary: 55M20, Secondary: 57M12, 20F99

On mappings of compact spaces into Cartesian spaces

Abstract Eilenberg proved that if a compact space X admits a zero-dimensional map f :X→Y , where Y is m -dimensional, then there exists a map h :X→I m+1 such that f×h :X→Y×I m+1 is an embedding. In this paper we prove generalizations of this result for σ -compact subsets of arbitrary spaces. An example of a compact space X and of a zero-dimensional σ -compact subset A⊂X is given such that for any continuous function f :X→ R which is one-to-one on the set A and any G δ -subset B of X with B⊃A the restriction f|B :B→ R has infinite fibers. This example is used to demonstrate that our results are sharp.

On multiplicity of mappings between surfaces

Let M and N be two closed (not necessarily orientable) surfaces, and f a continuous map from M to N. By definition, the minimal multiplicity MMR[f] of the map f denotes the minimal integer k having the following property: f can be deformed into a map g such that the number |g^{-1}(c)| of preimages of any point c in N under g is at most k. We calculate MMR[f] for any map

Schauder's fixed point and amenability of a group

f

Free equivariant extensors

of positive absolute degree A(f). The answer is formulated in terms of A(f), [pi_1(N):f_#(pi_1(M))], and the Euler characteristics of M and N. For a map f with A(f)=0, we prove the inequalities 2 <= MMR[f] <= 4.

On a question of Terasawa

A criterion for existence of a fixed point for an affine action of a given group on a compact convex space is presented. From this we derive that a discrete countable group is amenable if and only if there exists an invariant probability measure for any action of the group on a Hilbert cube. Amenable properties of the group of all isometries of the Urysohn universal homogeneous metric space are also discussed.

Banach--Mazur Compacta are Aleksandrov Compactifications of Q-manifolds

Abstract We prove for a finite group G and a compact metric G -space Y that the conditions (1) Y∈LC n−1 ∩C n−1 , and (2) Y∈G - AE(X) , for every normal n -dimensional space X endowed with a free numerable action of the group G , are equivalent. As a corollary we obtain: (A) For the space X endowed with a free action of the finite group G the conditions (1) the space X is normal, dim X⩽n and K(X;G)⩽n+1 ; (2) the space X is normal, dim X⩽n and K(X;G) ; (3) G∗⋯∗G∈G - AE(X) , are equivalent. (B) For a paracompact space X with a free action of the finite group G the inequality K(X;G)⩽ dim X+1 holds.

The minimal number of roots of surface mappings and quadratic equations in free groups

Abstract Answering a question of Terasawa, we show that for every n≥1 the spaces μ∞In and μ∞Sn are not homeomorphic.