Extending the concept of genus to dimension $n$

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.