Extending the Concept of Genus to Dimension N

Abstract

Abstract.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.

A note about complexity of lens spaces

“…A cellular embedding of a coloured graph into a surface is said to be regular if its regions are bounded by the images of bicoloured cycles; interesting results of crystallization theory (mainly related to an n-dimensional extension of Heegaard genus, called regular genus and introduced in [20]) rely on the existence of this type of embeddings for graphs representing manifolds of arbitrary dimension.…”

An irreducible Heegaard diagram of the real projective 3‐space p3

“…Proof. Construct a crystallization Γ associated with the above Heegaard diagram via Gagliardi's method [3].…”

“…Now, we construct an extended Heegaard diagramH associated with H = (F ; u, v). The extended Heegaard diagramH contains 16 Heegaard diagrams for P 3 . At least one of them can be transformed into a Heegaard diagram H with a pair of complementary handles by a finite sequence of wave moves ( Figure 6).…”

On the genus of smooth 4-manifolds

“…The regular genus (or simply called the genus) of a closed (PL) «-manifold M is defined as the nonnegative integer (see [10]) g(M) = min{g(G)/(G, c) is a crystallization of M} .…”

“…If TIi (M) is the fundamental group of M, then g(M) > ranklli(Af) (see [1]). If M is nonorientable, then g(M) is even (see [10]). In [7] bounds were determined for the genus of any 4-manifold which is a product of Sx by a closed 3-manifold or a product of two closed surfaces.…”