Normal surfaces as combinatorial slicings

We investigate slicings of combinatorial manifolds as properly embedded co-dimension 11 submanifolds. Focus is given to the case of dimension 33, where slicings are (discrete) normal surfaces. For the cases of 22-neighborly 33-manifolds as well as quadrangulated slicings, lower bounds on the number of quadrilaterals of slicings depending on its genus gg are presented. These are shown to be sharp for infinitely many values of gg. Furthermore, we classify slicings of combinatorial 33-manifolds which are weakly neighborly polyhedral maps.