Mapping a sphere into Euclidean space

Translated from Matematicheskie Zametki, Vol. 46, No. 3, pp. 3–8, September, 1989.     

Local indexes of iterates of a holomorphic map

M. V. Lomonosov Moscow State University. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 23, No. 1, pp. 61–62, January–March, 1989.     

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

It is proved that, for all $n$ > 2, the Banach--Mazur compactum $Q(n)$ is the compactification of a $Q$ -manifold by a Euclidean point. For $n$ = 2, this was known earlier.     

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

Let be a continuous mapping between orientable closed surfaces of genus h and g and let c denote the constant map with . Let be the minimal number of roots of f' among all maps f' homotopic to f, i.e. . We prove that where and denotes the Euler characteristic. In addition, certain quadratic equations in free groups closely related to the coincidence problem are solved. Received January 26, 1999; in final form November 8, 1999 / Published online February 5, 2001     

Generic planes conjecture

In some particular cases we prove the density of the set of mappings of an n-dimensional compactum into an m-dimensional Euclidean space such that the set of all d-dimensional planes with the preimage cardinality ?? q has the dimension ?? qn - (q ? d ? 1)(m ? d).     

Obstructions to the extension of partial maps

One of the most important problems in topology is the minimization (in some sense) of obstructions to extending a partial map , i.e., of a subsetF ⊂ Z/A such thatf can be globally extended to its complement. It is shown that ifZ is a fixed metric space with dimZ ≤ n andp, q ≥−1 are fixed numbers, then obstructions to extending all partial maps can be concentrated in preselected fairly thin subsets ofZ. Translated fromMatematicheskie Zametki, Vol. 62, No. 6, pp. 803–812, December 1997 Translated by O. V. Sipacheva     

Realization of Configurations and the Löwner Ellipsoid

It is proved that any subset of an (m-1)-dimensional sphere of volume larger than l(m+ 1) of the volume of the entire sphere contains l+ 1 points forming a regular l-dimensional simplex. As a corollary, it is obtained that, if the exterior of a given m-dimensional filled ellipsoid contains no more than the 1/(m+ 1) fraction of some sphere, then the volume of the ellipsoid is no less than the volume of the corresponding ball. The existence of a pair of points a given spherical distance apart in a set of positive measure is examined.