The coincidence Nielsen number for maps into real projective spaces

We give an algorithm to compute the coincidence Nielsen number N(f, g), introduced in [DJ], for pairs of maps into real projective spaces. 1. Preliminaries. Let f, g :M → N be a pair of maps between closed C-smooth connected manifolds of the same dimension. We investigate the coincidence set Φ(f, g) = {x ∈ M : fx = gx} of such a pair. The Nielsen relation (x, y ∈ Φ(f, g) are Nielsen equivalent iff there is a path ω from x to y such that fω and gω are fixed-end-homotopic) divides Φ(f, g) into Nielsen classes ([J], [M]). We will denote the quotient set by Φ′(f, g). If M and N are orientable then we use the classical coincidence index [V] to define essential and nonessential classes and the Nielsen number [M]. If the orientability assumption is dropped we use the coincidence semi-index introduced in [DJ]. We recall briefly its definition. We consider a transverse pair of maps f, g : M → N , i.e. for any x ∈ Φ(f, g) the graphs Γf , Γg ⊂ M × N are transverse at the point (x, fx = gx). Let x, y ∈ Φ(f, g) and let the path ω establish the Nielsen relation between them. Fix local orientations α0(f), α0(g) of the graphs Γf , Γg at the point (x, fx = gx). Let αt(f), αt(g) denote their translations along the paths (ω, fω), (ω, gω) in Γf and Γg respectively. Then their sum α0 = α0(f) ∧ α0(g) is an orientation of M ×N at (x, fx); let αt be its translation along (ω, fω) in M ×N . We say that x and y are R-related (reduce each other in [DJ]) iff α1 = −α1(f) ∧ α1(g) for a path ω establishing the Nielsen relation. Now we may represent a Nielsen class A as A = {a1, b1, . . . , ak, bk : c1, . . . , cs} where aiRbi but no pair {ci, cj} satisfies this relation (i 6= j). Finally, we define the semi-index of this class: |ind|(f, g : A) := s .