Coincidence theory on the complement

In this work we generalize two aspects of Nielsen fixed point theory on the complement to Nielsen coincidence theory. The first aspect concerns the location (under relative homotopies) of coincidence points. It prepares the way for equivariant coincidence theory and the for second part. A minimum theorem is forthcoming under the condition that the subspace can be by-passed. The second aspect (the study of surplus periodic points on the complement) gives a parallel (but quite different) theory when the subspace cannot be by-passed. features of this work include a modified fundamental group approach which simplifies the exposition. Secondly in addition to the usual Jiang condition it includes an analogue of it which ensures that the Reidemeister and Nielsen numbers are the same when the Lefschetz number is nonzero.