Nielsen fixed point theory for partially ordered sets (Q5928480)

Let \(X\) be a compact ANR and \(A_i\subseteq X\) for \(i=1, \dots,k-1\) be compact ANRs also. Partially order the \(A_i\) by setting \(A_j\leq A_i\) if and only if \(A_j\subseteq A_i\). Let \({\mathcal P}=\{X,A_1, \dots, A_{k-1}\}\) be the partially ordered set where it is assumed that if \(A_j\cap A_i\neq\emptyset\) then \(A_j\cap A_i\in {\mathcal P}\) and the subscripts are ordered so that \(A_j\leq A_i\) implies \(j\leq i\) (this so-called admissable ordering is not unique in general). A ``Nielsen type number'' \(N(f,{\mathcal P},\leq)\) of an (admissable) order-preserving map \(f:{\mathcal P}\to{\mathcal P}\) is defined by extending the inclusion-exclusion principle approach of Schirmer's original relative Nielsen number. The number \(N(f,{\mathcal P},\leq)\) is shown to have the properties one would require for a Nielsen theory, in particular it is an (order-preserving) homotopy invariant lower bound for the number of fixed points of \(f:X\to X\). The author relates this Nielsen type number, and an extension of it, to many of the existing Nielsen theories: for maps of pairs, maps of triads, fiber-preserving maps, equivariant maps, and iterates of maps. Besides serving to connect diverse Nielsen theories, these Nielsen type numbers for partially ordered sets can be used to estimate, or in some cases calculate, the corresponding Nielsen type numbers.