Lefschetz coincidence numbers of solvmanifolds with Mostow conditions. (English) Zbl 1324.22003

Kasuya, Hisashi

Arch. Math., Brno 50, No. 1, 27-37 (2014).

Let \(M_1\) and \(M_2\) be two oriented compact manifolds of the same dimension and \(f,g: M_1 \to M_2\) two continuous maps. Composing the cohomology map \(H^*(f)\) with the dual of \(H^*(g)\) and using the Poincaré duality, one obtains a graded linear map \(H^*(M_1) \to H^*(M_1)\). The graded trace of this map is the so called Lefschetz coincidence number \(L(f,g)\). It is well known that for solvmanifolds \(M_1 = G_1/\Gamma_1\) and \(M_2 = G_2/\Gamma_2\) (i.e. \(G_i\) are simply connected solvable Lie groups and \(\Gamma_i\) cocompact Lie subgroups), \(L(f,g)\) yields a lower bound for the number of connected components of coincidences of \(f\) and \(g\).
The author extends (by considering the so called Mostow condition) the known class of solvmanifolds for which the Lefschetz coincidence number can be computed explicitly. The author’s approch is based on Mostow’s identification of the de Rham cohomology of \(M_1/G_1\) and the Lie algebra cohomology of \(\mathfrak{g}_1\) (the Lie algebra of \(G_1\)), see [G. D. Mostow, Ann. Math. (2) 73, 20–48 (1961; Zbl 0103.26501)].

Reviewer: Josef Šilhan (Brno)

MSC:

22E25	Nilpotent and solvable Lie groups
53C30	Differential geometry of homogeneous manifolds
54H25	Fixed-point and coincidence theorems (topological aspects)
55M20	Fixed points and coincidences in algebraic topology

Keywords:

de Rham cohomology; Lefschetz coincidence number; solvmanifold