Roots of Mappings from Manifolds

Assume that f : X → Y is a proper map of a connected n-manifold X into a Hausdorff, connected, locally path-connected, and semilocally simply connected space Y , and y0 ∈ Y has a neighborhood homeomorphic to Euclidean n-space. The proper Nielsen number of f at y0 and the absolute degree of f at y0 are defined in this setting. The proper Nielsen number is shown to a lower bound on the number of roots at y0 among all maps properly homotopic to f , and the absolute degree is shown to be a lower bound among maps properly homotopic to f and transverse to y0. When n > 2, these bounds are shown to be sharp. An example of a map meeting these conditions is given in which, in contrast to what is true when Y is a manifold, Nielsen root classes of the map have different multiplicities and essentialities, and the root Reidemeister number is strictly greater than the Nielsen root number, even when the latter is nonzero.