The number of small covers over cubes and the product of at most three simplices up to equivariant cobordism
This paper considers the number of equivariant cobordism classes of small covers over the product of simplices. We recall that a small cover, defined by \textit{M. W. Davis} and \textit{T. Januszkiewicz} in [Duke Math. J. 62, No.2, 417--451 (1991; Zbl 0733.52006)], is a smooth closed manifold \(M^n\) with a locally standard \((\mathbb{Z}_2)^n\)-action such that its orbit space is a simple convex polytope. Denoting by \(\mathcal{M}_n\) the set of equivariant unoriented cobordism classes of all \(n\)-dimensional small covers, let \(\mathcal{M}_* = \sum_{n\geq 1}\mathcal{M}_n\), which is generated by the classes of small covers over the product of simplices. Since the equivariant cobordism class of a small cover over a simple convex polytope is determined by its tangential representation set, which can be identified with the characteristic function of the simple convex polytope, the authors using this function determine relevant results in what concerns the number of small covers over cubes and the product of at most three simplices up to equivariant cobordism.
scientific article; zbMATH DE number 5957831
scientific article