Two, more readily computable equivariant Nielsen numbers I. Nielsen theory for M-ads

In this, the first of two papers outlining a Nielsen theory for “two, more readily computable equivariant numbers”, we define and study two Nielsen type numbers N(f,k;X−{Xν}ν∈M) and N(f,k;X,{Xν}ν∈M), where f and k are M-ad maps. While a Nielsen theory of M-ads is of interest in its own right, our main motivation lies in the fact that maps of M-ads accurately mirror one of two fundamental structures of equivariant maps. Being simpler however, M-ad Nielsen numbers are easier to study and to compute than equivariant Nielsen numbers. In the sequel, we show our M-ad numbers can be used to form both upper and lower bounds on their equivariant counterparts.The numbers N(f,k;X−{Xν}ν∈M) and N(f,k;X,{Xν}ν∈M), generalize the generalizations to coincidences, of Zhao's Nielsen number on the complement N(f;X−A), respectively Schirmer's relative Nielsen number N(f;X,A). Our generalizations are from the category of pairs, to the category of M-ads. The new numbers are lower bounds for the number of coincidence points of all maps f′ and k′ which are homotopic as maps of M-ads to f, respectively k firstly on the complement of the union of the subspaces Xν in the domain M-ad X, and secondly on all of X. The second number is shown to be greater than or equal to a sum of the first of our numbers. Conditions are given which allow for both equality, and Möbius inversion. Finally we show that the fixed point case of our second number generalizes Schirmer's triad Nielsen number N(f;X1∪X2).Our work is very different from what at first sight appears to be similar partial results due to P. Wong. The differences, while in some sense subtle in terms of definition, are profound in terms of commutability. In order to work in a variety of both fixed point and coincidence points contexts, we introduce in this first paper and extend in the second, the concept of an essentiality on a topological category. This allows us to give computational theorems within this diversity. Finally we include an introduction to both papers here.