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Relative Nielsen theory for noncompact spaces and maps. (English) Zbl 1098.55001
Relative Nielsen theory deals with the estimation of the number of fixed points of maps in the relative homotopy class of a given relative map \(f\colon (X,A)\to (X,A)\) of a pair of compact polyhedra. H. Schirmer obtained two invariants, namely \(N(f;X,A)\) [Pac. J. Math. 122, 459–473 (1986; Zbl 0553.55001)] and \(N(f;\overline{X-A})\) [Topology Appl. 30, No. 3, 253–266 (1988; Zbl 0664.55003)], which are lower bounds for the number of fixed points on \(X\) and \(\overline{X-A}\), respectively, for maps in the relative homotopy class of \(f\).
The reviewer obtained an invariant \(N(f;X-A)\) [X. Zhao, Lect. Notes Math. 1411, 189-199 (1989; Zbl 0689.55008)], which is a lower bound for the number of fixed points on \(X-A\). These methods were also used to estimate the periodic points, with given least period or period, on \(X\), \(\overline{X-A}\) and \(X-A\). Corresponding Nielsen type numbers were obtained in [P. R. Heath, H. Schirmer and C. You, Topology Appl. 63, No. 2, 117–138 (1995; Zbl 0827.55002)].
The authors of the paper under review generalize these relative type Nielsen numbers to the setting of “admissible” maps of noncompact ANR-pairs. The key point is that under the admissibility assumption, the fixed point index of the given map is well-defined, and hence the generalized Nielsen type numbers share the same lower bound properties as the original ones. Such a generalization aims at the application of Nielsen theory to analysis, which was formalized by U. K. Scholz in the non-relative case [Rocky Mt. J. Math. 4, 81–87 (1974; Zbl 0275.55013)].

55M20 Fixed points and coincidences in algebraic topology
37C25 Fixed points and periodic points of dynamical systems; fixed-point index theory, local dynamics
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)
Full Text: DOI
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