Anosov theorem for coincidences on nilmanifolds

Seung Won Kim ¹   ; Jong Bum Lee ¹

¹ Department of Mathematics Sogang University Seoul 121-742, Korea

Fundamenta Mathematicae, Tome 185 (2005) no. 3, pp. 247-259

Résumé

Suppose that $L,L'$ are simply connected nilpotent Lie groups such that the groups $\gamma_i(L)$ and $\gamma_i(L')$ in their lower central series have the same dimension. We show that the Nielsen and Lefschetz coincidence numbers of maps $f,g : {\mit\Gamma}\backslash L\to {\mit\Gamma}'\backslash L'$ between nilmanifolds ${\mit\Gamma}\backslash L$ and ${\mit\Gamma}'\backslash L'$ can be computed algebraically as follows: $$ L(f,g)=\det(G_*-F_*),\quad N(f,g)=\vert L(f,g)\vert, $$ where $F_*, G_*$ are the matrices, with respect to any preferred bases on the uniform lattices ${\mit\Gamma}$ and ${\mit\Gamma}'$, of the homomorphisms between the Lie algebras $\mathfrak{L}, \mathfrak{L}'$ of $L, L'$ induced by $f,g$.

Détail

DOI : 10.4064/fm185-3-3

Keywords: suppose simply connected nilpotent lie groups groups gamma gamma their lower central series have dimension nielsen lefschetz coincidence numbers maps mit gamma backslash mit gamma backslash between nilmanifolds mit gamma backslash mit gamma backslash computed algebraically follows det * f * quad vert vert where * * matrices respect preferred bases uniform lattices mit gamma mit gamma homomorphisms between lie algebras mathfrak mathfrak induced