We determine bounds for the regular genus of any 4-manifold, which is a product of $1 by a closed 3-manifold, or a product of two closed surfaces. This is done by an explicit construction of a graph representing the manifold, and by finding a minimal regular imbedding of it. S UNTO, Determiniamo una limitazione del genere regolare di una 4-varieta prodotto di Sa per una 3-varieta chiusa, o prodotto di due superficie chiuse, costruendo esplicitamente un grafo chela rappresenta, e trovandone un'immersione regolare minimale.
1. PRELIMINARIES
Throughout this work, all spaces and maps are piecewise-linear in the sense of [ 14]. For graph theory, we refer the reader to [ 10].
We recall the main concepts and definitions used in the work. In order to represent n-dimensional polyhedra, we use (n + 1)-coloured 9raphs, i.e. multigraphs (no loops are allowed) F = (V, E), regular, of degree n + 1, together with edge-colorations 7 :E ~ A = {ieZ [0 ~< i ~< n} (Y(et) ~ 7(%) for any pair of adjacent edges et, e2). The pseudocomplex K(F), 2 attached to (F, 7), is built by taking a set of n-simplexes in bijection with the set V of vertices of F, by injectively labelling their vertices in A and by glueing two of them along their faces opposite the i-labelled vertex when the corresponding vertices of F are joined by an edge e with y(e) = i (so that equally labelled vertices are identified).
Let ~ denote the set A -{c} for every colour c~A. For any subset of colours, set F~ = (V, 7-1(~)). If for every cEA the partial graph F c is connected, then K(F) has exactly n + 1 vertices; in this case, both (F, 7) and K(F) are said to be contracted. A crystallization of an n-manifold M is a contracted (n + 1)-coloured graph (F, 7) such that IK(F) I ~-M.
Most of the theorems about crystallizations are contained in [3], and others can be found in [2], [7] and [8].
A 2-cell imbedding [17, p. 40] ,'[FIEF of an (n+ 1)-coloured graph (F, 7) into a closed surface F is said to be regular if there exists a cyclic permutation e = (Co, ..., e) of the colour set such that each region of t is bounded by an (imbedded) cycle, whose edges are alternatively coloured by e~, This work was performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy.
2 K, and p defined later, depend also on the coloration 7. The notation K(F) is adopted for brevity.