Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Minimal Nielsen Root Classes and Roots of Liftings | Hindawi Publishing Corporation Fixed Point Theory and Applications Volume 2009 Article ID 346519 16 pages doi 2009 346519 Research Article Minimal Nielsen Root Classes and Roots of Liftings Marcio Colombo Fenille and Oziride Manzoli Neto Instituto de Ciencias Matematicas e de Computacao Universidade de Sao Paulo Avenida Trabalhador Sao-Carlense 400 Centro Caixa Postal 668 13560-970 Sao Carlos SP Brazil Correspondence should be addressed to Marcio Colombo Fenille fenille@ Received 24 April 2009 Accepted 26 May 2009 Recommended by Robert Brown Given a continuous map f K M from a 2-dimensional CW complex into a closed surface the Nielsen root number N f and the minimal number of roots p f of f satisfy N f p f . But there is a number pc f associated to each Nielsen root class of f and an important problem is to know when p f pc f N f . In addition to investigate this problem we determine a relationship between p f and p f when f is a lifting of f through a covering space and we find a connection between this problems with which we answer several questions related to them when the range of the maps is the projective plane. Copyright 2009 M. C. Fenille and O. M. Neto. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. 1. Introduction Let f X Y be a continuous map between Hausdorff normal connected locally path connected and semilocally simply connected spaces and let a e Y be a given base point. A root of f at a is a point x e X such that f x a. In root theory we are interested in finding a lower bound for the number of roots of f at a. We define the minimal number of roots of f at a to be the number Af a minị F -1 a such that p is homotopic to f . When the range Y of f is a manifold it is easy to prove that this number is independent of the selected point a e Y and from 1 Propositions
