Normal surfaces as combinatorial slicings

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 1 submanifolds. Focus is given to the case of dimension 3 , where slicings are (discrete) normal surfaces. For the cases of 2 -neighborly 3 -manifolds as well as quadrangulated slicings, lower bounds on the number of quadrilaterals of slicings depending on its genus g are presented. These are shown to be sharp for infinitely many values of g . Furthermore, we classify slicings of combinatorial 3 -manifolds which are weakly neighborly polyhedral maps.