Real Projective Plane Mapping Nielsen Number

Topological fixed point theory deals with the estimation of the number of fixed points of maps.The number of essential fixed point classes of self-map f of a compact polyhedron is called the Nielsen number of f,denoted N(f).It is a lower bound for the number of fixed points of f.One naturally wonder the expression of any given self map.But,the computation of N(f) has a deep relationship with the fundamental group,there is no general formula.In this thesis,we obtain the relation for Nielsen numbers amongst given self map on a non-orientable manifold and its liftings on the orientation covering map according to the induced endomorphism on fundamental group.For the real projection plane,we compute out the all Nielsen numbers concretely by using the homotopy classification of self-maps of real projection plane.Moreover,we discuss the reduciblity and irreduciblity of periodic point classes.Therefore we compute the other two Nielsen type numbers NP_n(f) and Nφ_n(f),which estimate the number of periodic points and fixed points of the iterates of map f.