ROOT PROBLEM FOR CONVENIENT MAPS

In this paper we study when the minimal number of roots of the so-called convenient maps horn two-dimensional CW complexes into closed surfaces is zero We present several necessary and sufficient conditions for such a map to be root free Among these conditions we have the existence of specific fittings for the homomorphism induced by the map on the fundamental groups, existence of the so-called mutation of a specific homomorphism also induced by the map, and existence of particular solutions of specific systems of equations on free groups over specific subgroups

Spherical space forms - Homotopy self-equivalences and homotopy types, the case of the groups Z/a x (Z/b x TL(2)(F(p)))

Let G = Z/a x(mu) (Z/b x TL(2)(F(p))) and X(n) be an n-dimensional CW-complex with the homotopy type of the n-sphere. We determine the automorphism group Aut(G) and then compute the number of distinct homotopy types of spherical space forms with respect to free and cellular G-actions on all CW-complexes X(2dn - 1), where 2d is a period of G. Next, the group E(X(2dn - 1)/alpha) of homotopy self-equivalences of spherical space forms X(2dn - 1)/alpha, associated with such G-actions alpha on X(2dn - 1) are studied. Similar results for the rest of finite periodic groups have been obtained recently and they are described in the introduction. (C) 2009 Elsevier B.V. All rights reserved.