Coincidence Points Of Maps Between Surfaces

Consider two maps between topological 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 in pay much attention to the estimation of the number of coincidence points as well as the existence of the coincidence points.A special and important case of coincidence point is just fixed point. In 1927, Jacob Nielsen introduced the definition of Nielsen number, and showed that it is a lower bound of the number 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 for the number of coincidence points..It is known that the computation of Nielsen number is very difficult. A natural idea is to establish the relation between the Nielsen number and other topological invariants. In this paper, we consider the coincidence points of a special kind of self homeomorphisms, namely Dehn twists. We obtain a relation between the Nielsen number of coincidence points of two Dehn twists and the minimal geometric intersection number of the simple closed curves determining these two Dehn twist. We shall show that on the torus, the coincidence Nielsen number of two Dehn twists in essential simple close curves is equal to the square of the minimal geometric intersection number of the two curves. Moreover, we try to compute the coincidence Nielsen number of Dehn twists on oriented surfaces with bigger genuses. Meanwhile, we also discuss the relation between thecoincidence points of f and g which are two homeomorphisms between oriented manifolds and the fixed points of the homeomorphism g-1 .