Coincidence Reidemeister classes on nilmanifolds and nilpotent fibrations
The paper presents the Reidemeister classes which arise from a pair of homomorphisms between two short exact sequences of groups, the fibrations provided by the definition of a nilpotent class and the family of fibrations \({\mathcal F}C\), whose elements are certain fibre bundles over manifolds which belong to a fixed nilpotent class. The author proves as the main result Theorem 3.7. Let \(E_i\in{\mathcal F}{\mathcal C}(N)\) and \(F_i\in{\mathcal F}_0{\mathcal C} (N_1)\) for \(i=1,2\). For two fibre preserving maps \(f,g: E_1\to E_2\) where \(F_2 \to E_2\to N\) is nilpotent, \(\dim E_1=\dim E_2\), \(\pi_1(E_1) =\pi(E_2)\) and either \(f_\# (\pi_1(E_1))= \pi_1(E_2)\) or \(g_\#(\pi_1 (E_1))= \pi_1(E_2)\), the following three conditions are equivalent: (a) \(N(f,g)\neq 0\); (b) \(\text{coin} (f_\#, g_\#) =1\) and \(\Lambda(f',g')\neq 0\); (c) \(\#{\mathcal R}[f,g] <\infty\) and \(\Lambda (f',g') \neq 0\). If one of the three conditions above holds then \[ N (f,g) =N(\overline f,\overline g)N(f',g')=\#{\mathcal R}[\overline f,\overline g] \# {\mathcal R}[f',g']= \#{\mathcal R}[f,g]. \] Finally in any case, all the Nielsen classes have the same index, namely \(\pm\Lambda (f',g')/\#{\mathcal R}[f',g']\) and \(|\Lambda(f,g) | = \Lambda(f',g')|\Lambda (\overline f, \overline g)|\).
scientific article; zbMATH DE number 1308038
scientific article