Nielsen Coincidence Theory in Arbitrary Codimensions

Given two maps f1, f2 : M −→ 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(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 JamesHopf invariants is satisfied. We also exhibit situations where N(f1, f2) vanishes, but MCC(f1, f2) is strictly positive.