Extending the Concept of Genus to Dimension W

Some graph-theoretical tools are used to introduce the concept of "regular genus" §(M„), for every closed n-dimensional PL-manifold M„. Then it is proved that the regular genus of every surface equals its genus, and that the regular genus of every 3-manifold Af j equals its Heegaard genus, if M3 is orientable, and twice its Heegaard genus, if M3 is nonorientable. A geometric approach, and some applications in dimension four are exhibited. 1. Definitions and notations. Let T = (V, E) be a regular multigraph of degree n + 1, y: E -> A„ = {/ G Z|0 < i < n] an (n + l)-line-colouring of T [Ha, p. 133]. Such a pair (I\ y) is said to be an (n + l)-coloured graph. For every subset © of A„, r9 will denote the subgraph (V, y~\9>)); further, for every colour c G A„, c will denote the set A„ — {c}. A 2-cell imbedding i: |r| —> F [W, p. 40] of an (n + l)-coloured graph (I\ y) on a closed surface F is called a regular imbedding if its regions are bounded by 2-coloured cycles. Moreover, i is called a strongly-regular imbedding if there exists a cyclic permutation e = (e0, . . . , e„) of A„, so that each region is bounded by a component of one of the subgraphs r{(¡f,+i}, i being an integer modulo n + 1. It is easy to see that every strongly-regular imbedding is regular. Conversely it is proved in [G4] that every regular imbedding of a 3or 4-coloured graph is strongly regular. This is not true in general, as it is easy to check. In the present work, we only consider strongly-regular imbeddings, as they seem to apply better to the geometric situation we wish to represent. For the sake of conciseness, we shall always omit the word "strongly". So we shall simply call regular the strongly-regular imbeddings of [G4]. A pseudocomplex [HW, p. 49] K = K(T) of dimension n can be associated to every (n + l)-coloured graph, so that T becomes its dual 1-skeleton; for the construction, compare [G2]. If, for every colour c G An, the subgraph T¿ is connected, then A"(r) has exactly n + 1 vertices, and both K and T are said to be contracted. Now let M be a closed «-manifold, (I\ y) a contracted (n + l)-coloured graph, such that the space |7C(r)| of its associated complex is homeomorphic with M; then Received by the editors July 23, 1979 and, in revised form, December 19, 1979. AMS (MOS) subject classifications (1970). Primary 57C15, 57C99; Secondary 05C10, 05C15.