Nielsen numbers for based maps, and for noncompact spaces

In this paper we present two (not independent) applications of the surplus Nielsen number of the complement due to the second author. For the first application we give a based Nielsen number N∗(ƒ) associated with a based map ƒ:(X,x∗) → (X,x∗) of a compact connected polyhedron X. The number N∗(ƒ) is a based homotopy invariant which is a lower bound for the number of fixed points in the based homotopy class of ƒ. Moreover N∗(ƒ) ⩾ N(ƒ;X,x∗) the relative Nielsen number of the pair (X,x∗). The inequality may be strict, in particular N∗ (ƒ) detects the two unremovable fixed points (not detected by N(ƒ; X,x∗) on Jiangʹs well known example on the figure eight. A minimum theorem is given. second part of the paper we introduce a Nielsen type number N∞(ƒ) for noncompact spaces by taking the surplus number of the complement of the point at infinity in the one point compactification of the original space. The number N∞(ƒ) is a homotopy invariant with respect to those homotopies which extend to the one point compactification. In particular it is an isotopy invariant for self-homeomorphisms. We also indicate how to extend N∞(ƒ) to the relative setting for noncompact spaces.