tailieunhanh - Báo cáo hóa học: "COINCIDENCE CLASSES IN NONORIENTABLE MANIFOLDS"

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: COINCIDENCE CLASSES IN NONORIENTABLE MANIFOLDS | COINCIDENCE CLASSES IN NONORIENTABLE MANIFOLDS DANIEL VENDRUSCOLO Received 15 September 2004 Revised 20 April 2005 Accepted 21 July 2005 We study Nielsen coincidence theory for maps between manifolds of same dimension regardless of orientation. We use the definition of semi-index of a class review the definition of defective classes and study the occurrence of defective root classes. We prove a semi-index product formula for lifting maps and give conditions for the defective coincidence classes to be the only essential classes. Copyright 2006 Daniel Vendruscolo. 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 In 2 6 the Nielsen coincidence theory was extended to maps between nonorientable topological manifolds. The main idea to do this is the notion of semi-index a nonnegative integer for a coincidence set. Let f g M N be maps between closed -manifolds without boundary. If we define fi f g M N X N as usual then we may assume that h is in a transverse position that is the coincidence set Coin f g x e M I f x g x is finite and for each coincidence point x there is a chart R X R U c N X N such that U f g M n U AN n U corresponds to R X R R X 0 0 X R see 6 for details . We say that two coincidence points x y e Coin f g are Nielsen related if there is a path Y 0 1 M with y 0 x y 1 y such that f Y is homotopic to gY relative to the endpoints. In fact this is an equivalence relation whose equivalence classes are called coincidence classes of the pair f g . Let x y e Coin f g belong to the same coincidence class and let Y be a path establishing the Nielsen relation between them. We choose a local orientation p0 of M in x and denote by Pt the translation of p0 along Y t . Definition 6 Definition . We will say that two points x y e Coin f g are R-related xRy if and only if there is a path Y