Nielsen numbers for maps of triads

AbstractLet ƒ:(X,A1,A2) → (X,A1,A2) be a selfmap of a triad which consist of a compact connected polyhedron X and two subpolyhedra A1 and A2 so that X = A1 ∪ A2. A Nielsen type number N(ƒ;A1 ∪ A2), called the Nielsen number of the triad, is defined which is a homotopy invariant lower bound for the number of fixed points of ƒ on X. Conditions are given which ensure that N(ƒ;A1 ∪ A2) is the best possible lower bound, and the location of minimal fixed point sets is characterized. The Nielsen number of the triad is computed, e.g., in cases where X is obtained as the double of a manifold, as the suspension of a polyhedron, by attaching a handle to a manifold, or as the connected sum of two manifolds. Minimal fixed point sets and their locations are described in these four cases