Coincidence free pairs of maps

Ulrich Koschorke

Universitaet Siegen, Emmy Noether Campus, Walter-Flex-Str. 3, D-57068 Siegen, Germany


This paper centers around two basic problems of topological coincidence theory. First, try to measure (with the help of Nielsen and minimum numbers) how far a given pair of maps is from being loose, i.e. from being homotopic to a pair of coincidence free maps. Secondly, describe the set of loose pairs of homotopy classes. We give a brief (and necessarily very incomplete) survey of some old and new advances concerning the first problem. Then we attack the second problem mainly in the setting of homotopy groups. This leads also to a very natural filtration of all homotopy sets. Explicit calculations are carried out for maps into spheres and projective spaces.

AMSclassification. Primary 55M20. Secondary 55Q40, 57R22.

Keywords. Coincidence, Nielsen number, minimum number, configuration space, projective space, filtration.