Nielsen coincidence theory on infra-solvmanifolds of Sol
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https://doi.org/10.1007/s11425-020-1767-x
Nielsen coincidence theory on infra-solvmanifolds of Sol Jong Bum Lee1,∗ & Karen Regina Panzarin2 1Department 2Departamento
of Mathematics, Sogang University, Seoul 121-742, Korea; de Matem´ atica, Universidade Federal de S˜ ao Carlos, S˜ ao Carlos, SP 13565-905, Brazil Email: [email protected], [email protected] Received February 19, 2020; accepted August 20, 2020
Abstract
We derive averaging formulas for the Lefschetz coincidence numbers, the Nielsen coincidence num-
bers and the Reidemeister coincidence numbers of maps on infra-solvmanifolds modeled on a connected and simply connected solvable Lie group of type (R). As an application, we compare our formula for the Nielsen coincidence numbers with a result of Jezierski (1992) for pairs of maps on some infra-solvmanifolds of Sol. For all pairs of self-maps of a nonorientable infra-solvmanifold of Sol, we determine the sets of all possible values of the Nielsen coincidence numbers and the Reidemeister coincidence numbers. Keywords
averaging formula, infra-solvmanifold, Lefschetz coincidence number, Nielsen coincidence number,
Reidemeister coincidence number, Sol-geometry MSC(2010)
55M20, 57S30
Citation: Lee J B, Panzarin K R. Nielsen coincidence theory on infra-solvmanifolds of Sol. Sci China Math, 2021, 64, https://doi.org/10.1007/s11425-020-1767-x
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Introduction
Let M and N be closed manifolds of the same dimension and let f, g : M → N be continuous maps. In order to study the coincidence points of f and g, the Lefschetz coincidence number L(f, g), the Nielsen coincidence number N (f, g) and the Reidemeister coincidence number R(f, g) are associated with f and g. These numbers are homotopy invariants. The Nielsen coincidence number gives more precise information concerning the existence of coincidence points than the Lefschetz coincidence number, but its computation when compared with that of the Lefschetz coincidence number is in general much more difficult. The Reidemeister coincidence number is an upper bound of the Nielsen coincidence number. ¯ and N ¯ , respectively, and f, g : M → N can be lifted to When M and N are finitely covered by M ¯ ¯ ¯ f , g¯ : M → N , the relation between the Nielsen number N (f, g) and the Nielsen number N (f¯, g¯) is studied in [8,10,16,19]. It is shown that the averaging inequality formula always holds, and under certain conditions the averaging equality formula holds (see [16, Theorem 4.6]). In practice, in order to use the averaging equality formula for N (f, g), one should (1) lift f, g to f¯, g¯, (2) understand the Nielsen number N (f¯, g¯) and (3) check the conditions for the equality. It is shown in [9, 24] that if M and N are special solvmanifolds of type (R), the Lefschetz number L(f, g) and the Nielsen number N (f, g) can be computed * Corresponding author c Science China Press and Springer-Verlag GmbH Germany, part of Springer Nature 2020 ⃝
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Lee J B et al.
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Sci China Math
algebraically as N (f, g) = |L(f,
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