Representing products of manifolds by edge-coloured graphs: The boundary case (Q5928271)

The aim of this paper is the study of the products of PL-manifolds by means of their representation by edge-coloured graphs. First a generalization of the construction -- introduced by Gagliardi and Grasselli in the closed case -- of a coloured-graph representing the product of two manifolds, starting by two coloured graphs representing the manifolds themselves, to the boundary case is given. This representation of manifolds allows to introduce a combinatorial invariant \(G(M^n)\), called regular genus, that in dimension 2 and 3 it coincides with the genus of a surface and the Heegaard genus of a 3-manifold respectively. Then we investigate some bounds of the regular genus of the product of 2- and 3-manifolds with a disk. Particular interest is devoted to these products because of the following conjecture which would imply the Poincaré Conjecture for all closed orientable 3-manifolds: ``For each closed, connected, orientable 3-manifold \(M^3\), it holds \(G(M^3\times S^n)\geq G(M^3\times D^n)\)''. Finally we give some bounds for the genus of the graph product of \(n\)-spheres with \(m\)-disks.