Coincidences of Fibrewise Maps Between Sphere Bundles Over the Circle

Abstract

When 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 deg B 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.