Groupoids and relations among Reidemeister and among Nielsen numbers

Abstract

In this work we simplify, generalize and extend results and methods concerning the relationships between various Reidemeister numbers, and applications thereof to four different Nielsen theories (fixed point, ordinary coincidence, semi-index coincidence and root theory). We do this in two distinct contexts. The first context deals with the relationship between the Nielsen numbers of the maps involved, and those of representative lifts to regular covering spaces. We have a special interest in homogeneous spaces. The second context of our applications, namely to Nielsen theories of fibre-preserving maps, is rather curiously dual to the first. In both contexts, our results improve on those previously given.Our main tool is a collection of 8 term exact sequences (of groups and sets) whose inspiration and proof comes from the theory of fibrations of groupoids. We give a complete analysis of our sequences which yields previously unknown upper and lower bounds on the Reidemeister and Nielsen numbers we are wanting to compute (in one case it sharpens a previously known lower bound). When the upper and lower bounds coincide, generalizations of familiar formulas are forthcoming. The process also gives a uniform approach to proofs in both the underlying algebra (Reidemeister considerations) and to the two distinct contexts of our applications to the four Nielsen theories.New results include a new formula generalizing of the averaging formula in both the algebra and the geometry. We also generalize the original coincidence averaging formula for oriented infra-nilmanifolds to the smooth non-orientable category, and also to a pair of self-maps of smooth infra-solvmanifolds of type R. Other generalizations concern the finiteness of Reidemeister numbers. Finally we fill in proofs and details missing from previous work.