A Möbius Inversion Formula for Generalized Lefschetz Numbers

Let G be a finite group and f : U ⊂ M → M a compactly fixed G-equivariant self-map of a G-manifold M ; the aim of the paper is to give an answer to the following question: if for every subgroup H ⊂ G the restricted map f : U →M can be deformed (non-equivariantly and via a compactly fixed homotopy) to be fixed point free, is it true that f can be deformed (via a compactly fixed homotopy) equivariantly to a fixed point free map? To achieve the result, it is necessary to relate the generalized Lefschetz numbers L(f) to the obstructions of deforming f equivariantly, which are nothing but some (local) generalized Lefschetz numbers of suitable maps. Under some assumptions there is an algebraic relation, namely a Möbius inversion formula, that allows to have the full answer. This formula is a generalization of a formula of Dold [Do83] and Komiya [Ko88], with the generalized Lefschetz number in place of the Brouwer fixed point index.