Coincidences of Fibrewise Maps Between Sphere Bundles Over the Circle

Abstract

AbstractWhen can two fibrewise maps be deformed in a fibrewise fashion until they are coincidence free? In order to get a thorough understanding of this problem (and, more generally, of minimum numbers that are closely related to it) we study the strength of natural geometric obstructions, such as ω-invariants and Nielsen numbers, as well as the related Nielsen theory.In the setting of sphere bundles, a certain degree map degB turns out to play a decisive role. In many explicit cases it also yields good descriptions of the set ℱ of fibrewise homotopy classes of fibrewise maps. We introduce an addition on ℱ, which is not always single valued but still very helpful. Furthermore, normal bordism Gysin sequences and (iterated) Freudenthal suspensions play a crucial role.