Nielsen coincidence theory in arbitrary codimensions

Abstract

Given two maps f_1, f_2 : M^m \longrightarrow N^n between manifolds of the indicated arbitrary dimensions, when can they be deformed away from one another? More generally: what is the minimum number MCC (f_1, f_2) of pathcomponents of the coincidence space of maps f'_1, f'_2 where f'_i is homotopic to f_i, i = 1, 2? Approaching this question via normal bordism theory we define a lower bound N (f_1, f_2) which generalizes the Nielsen number studied in classical fixed point and coincidence theory (where m = n). In at least three settings N (f_1, f_2) turns out to coincide with MCC (f_1, f_2): (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 (f_1, f_2) vanishes, but MCC (f_1, f_2) is strictly positive.