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