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On the generalization of the Nielsen number. (English) Zbl 0719.55002

Let \(f:X\to Y\) be a map, where X is a compact locally path-connected space. Suppose B is a closed subset of Y such that there is a neighborhood of B in Y of which B is a strong deformation retract. Points of \(f^{-1}(B)\) are called Nielsen equivalent if there is a path between them whose image under f is homotopic rel endpoints to a path in B. If f is deformed by a homotopy, a Nielsen equivalence class is either carried to another Nielsen class of it vanishes [compare U. Scholz, Rocky Mt. J. Math. 4, 81-87 (1974; Zbl 0275.55013)]. A Nielsen class that does not vanish under any homotopy is called essential and the Nielsen number N(f;B) of f relative to B is the number of essential Nielsen classes. The Nielsen number is a homotopy invariant lower bound for the cardinality of \(f^{-1}(B)\). If \(p_ 1,p_ 2: X\to Z\) are maps where Z is uniformly locally contractible, \(Y=Z\times Z\), \(f(x)=(p_ 1(x),p_ 2(x))\) and B is the diagonal, then N(f;B) agrees with the Nielsen coincidence number of R. B. Brooks (UCLA dissertation, 1967). When X, Y and B are compact orientable smooth closed manifolds with B a submanifold of Y such that dim Y-dim B\(=\dim X\), the authors use transversality to make \(f^{- 1}(B)\) a finite set for which an index of a Nielsen class to be the sum of the indices of its members, a class is essential if its index is nonzero. Using techniques modelled on those of Jiang Boju [Lect. Notes Math. 886, 163-170 (1981; Zbl 0482.57014)], they show that if dim \(X\geq 3\), then f is homotopic to a map g such that there are exactly N(f;B) points in \(g^{-1}(B)\). In the coincidence setting, that is when \(Y=Z\times Z\) and B is the diagonal, this proves a theorem of H. Schirmer [J. Reine Angew. Math. 194, 21-39 (1955; Zbl 0066.417)] for the case of smooth manifolds.

MSC:

55M20 Fixed points and coincidences in algebraic topology
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