Klein, Bottles On The Mapping Of Coincidence Points

In this thesis,we consider two maps between topology space . A point is said to be a coincidence point of the two maps if it has the same image under the two maps. In algebraic topology ,ones pay much attention to not only the existence of the coincidence points but also the estimation of their number and classification.The Nielsen theory is a classical branch of algebraic topology . In 1927, Jacob Nielsen introduced the definition of Nielsen number, and showed that it is a lower bound of the number of the fixed points of maps in the homotopy class of given map. Later, Nielsen number of coincidence points was defined, which is a lower bound of the number of coincidence points.In this thesis, we also investigate the covering relation among the Klein bottle , the torus and the plane by the martric form, and also obtain the representation of their element. Second, we classify the maps on the Klein bottle by the different form of the lifting, and then compute the coincidence Nielsen number through the universal covering of the plane to the torus and the torus to the Klein bottle. Meanwhile,we discuss the coindition for the number of coincidence points on the Klein bottle equal to the coincidence Nielsen number.In this thesis, we also know the coincidence of two maps f and g on the closed manifolds can be represented the disjoint union of its coincidence classes and then we compute the semi-index of these coincidence classes.