Fixed-point theory for homogeneous spaces

Let G be a compact connected Lie group, K a closed subgroup (not necessarily connected) and M = G/K the homogeneous space of left cosets. Assume that M is orientable and p_*: H_n(G) → H_n(M) is nonzero, where n = dim M. In this paper, we employ an equivariant version of Nielsen root theory to show that the converse of the Lefschetz fixed-point theorem holds true for all selfmaps on M. Moreover, if the Lefschetz number of a selfmap f : M → M is nonzero, then the Nielsen number of f coincides with the Reidemeister number of f, which can be computed algebraically.