The covering coincidence index and the covering Nielsen number

Let p:M˜→M and p′:N˜→N be finite regular covering maps, and f,g:M→Nf,g:M→N be maps between connected compact manifolds without boundary of any codimension. A new coincidence index (the covering index) related to the covering maps p   and p′p′ is introduced. This index defines a new Nielsen number (the covering Nielsen number). Further, we investigate the relationship between the covering Nielsen number and the well-known index and semi-index Nielsen numbers. Also, we give an algebraic method that computes the covering Nielsen number.