2013/05/07 by Ulrich Koschorke, Koschorke, Ulrich
Mathematics · #55M20 #55S35 #Algebraic Topology (math.AT) #FOS: Mathematics #Primary 54H25 #Secondary 55R10 #math.AT #msc:54H25 #msc:55M20 #msc:55R10 #msc:55S35
paper · pdf · doi:10.48550/arxiv.1305.1654
fiberwise maps, Nielsen number, Reidemeister set, orbit structure, torus bundles
arxiv created 2013/05/07 · arxiv updated 2013/05/09
Given two fiberwise maps f1, f2 between smooth fiber bundles over a base manifold B, we develop techniques for calculating their Nielsen coincidence number. In certain settings we can describe the Reidemeister set of (f1,f2) as the orbit set of a group operation of π1(B). The size and number of orbits captures crucial extra information. E.g. for torus bundles of arbitrary dimensions over the circle this determines the minimum coincidence numbers of the pair (f1,f2) completely. In particular we can decide when f1 and f2 can be deformed away from one another or when a fiberwise selfmap can be made fixed point free by a suitable homotopy. In two concrete examples we calculate the minimum and Nielsen numbers for all pairs of fiberwise maps explicitly. Odd order orbits turn out to play a special role.