Coincidence theory on the complement (Q1304877)

The authors study relative Nielsen coincidence theory and Nielsen coincidence theory ``on the complement'', i.e., they consider compact oriented manifolds \(X\) and \(Y\) of the same dimension with locally flat submanifolds \(A\subset X\) and \(B\subset Y\) of the same dimension and relative maps \(f,g:(X,A)\to(Y,B)\). The aim of the paper is to determine the minimal number of coincidence points under relative homotopies of \(f\) and \(g\) on the complement \(X\smallsetminus A\). The paper offers several approaches: first, the authors define a Nielsen type number \(N(f,g;X\smallsetminus A)\) that will detect which of the \(N(f,g)\) Nielsen classes lie entirely in \(X\smallsetminus A\) and none of whose representatives can be moved to \(A\) by homotopies of \(f\) and \(g\). Second, the authors define a kind of finer coincidence class defined on the subspace \(X\smallsetminus A\). This leads to a new Nielsen number \(SN(f,g;X\smallsetminus A)\geq N(f,g;X\smallsetminus A)\) which reduces to \(N(f,g;X\smallsetminus A)\) when \(A\) ``can be by-passed''. The authors particularly emphasize their use of a ``modified fundamental group'' [introduced by the second author in Pac. J. Math. 117, 267-289 (1985; Zbl 0571.55002)] in dealing with Reidemeister classes. Contrary to \textit{Boju Jiang}'s statement in his ``Lectures on Nielsen fixed point theory'' [Contemp. Math. 14 (1983; Zbl 0512.55003)] they claim that this approach has all the advantages of the covering space approach without using the machinery of that theory. \{A crucial notion in this article is that of a subspace which can be by-passed. Awkwardly enough, this notion is never defined. Maybe, the authors intended to give a precise definition after their Lemma 3.19 but the words ``can be by-passed'' have been lost in the production process\}.