Self-closeness numbers of finite cell complexes

We reformulate the inequalities among self-closeness numbers of spaces in cofibrations making use of homology dimension and show that the self-closeness number of a space is less than or equal to the homology dimension of the space. Then we prove a relation of self-closeness numbers and the connectivity for manifolds satisfying Poincar\'{e} duality. On the other hand we determine the self-closeness numbers of the real projective spaces, lens spaces and a cell complex defined by Mimura and Toda. Moreover, making use of the models of Sullivan and Quillen, we show several properties of self-closeness number for finite cell complexes, and rational examples are udied to obtain some precise results. Finally, we prove relations among self-closeness numbers defined by homotopy groups, homology groups and cohomology groups.