On Nonfree Actions of Commuting Involutions on Manifolds

A new lower bound is obtained relating the rational cup-length of the base and that of the total space of branched coverings of orientable manifolds for the case in which the branched covering is a projection onto the quotient space by the action of commuting involutions on the total space. This bound is much stronger than the classical Berstein-Edmonds 1978 bound which holds for arbitrary branched coverings of orientable manifolds.In the framework of the theory of branched coverings, results are obtained that are motivated by the problems concerning n-valued topological groups. We explicitly construct m - 1 commuting involutions acting as automorphisms on the torus T-m with the orbit space RPm for any odd m = 3. By the construction thus obtained, the manifold RPm carries the structure of an 2(m-1)-valued Abelian topological group for all odd m = 3.