Roots of mappings from manifolds

Assume that is a proper map of a connected -manifold into a Hausdorff, connected, locally path-connected, and semilocally simply connected space , and has a neighborhood homeomorphic to Euclidean -space. The proper Nielsen number of at and the absolute degree of at are defined in this setting. The proper Nielsen number is shown to a lower bound on the number of roots at among all maps properly homotopic to , and the absolute degree is shown to be a lower bound among maps properly homotopic to and transverse to . When , 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 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.