2004/08/03 by Ulrich Koschorke, Koschorke, Ulrich
Mathematics · #55M20 #55M55 #55Q25 #55Q40 #55Q55 #55Q57 #55R42 #55S35 (Primary) 55P35 #Algebraic Topology (math.AT) #FOS: Mathematics #Geometric Topology (math.GT) #math.AT #math.GT #msc:55M20 #msc:55M55 #msc:55P35 #msc:55Q25 #msc:55Q40 #msc:55Q55 #msc:55Q57 #msc:55R42 #msc:55S35
paper · pdf · doi:10.48550/arxiv.math/0408044
23 pages
arxiv created 2004/08/03 · arxiv updated 2009/12/01
Given two maps f1, f2 : Mm \longrightarrow Nn between manifolds of the indicated arbitrary dimensions, when can they be deformed away from one another? More generally: what is the minimum number MCC (f1, f2) of pathcomponents of the coincidence space of maps f'1, f'2 where f'i is homotopic to fi, i = 1, 2? Approaching this question via normal bordism theory we define a lower bound N (f1, f2) which generalizes the Nielsen number studied in classical fixed point and coincidence theory (where m = n). In at least three settings N (f1, f2) turns out to coincide with MCC (f1, f2): (i) when m < 2n - 2; (ii) when N is the unit circle; and (iii) when M and N are spheres and a certain injectivity condition involving James-Hopf invariants is satisfied. We also exhibit situations where N (f1, f2) vanishes, but MCC (f1, f2) is strictly positive.