  Discrete normal surfaces are normal surfaces whose intersection with each
tetrahedron of a triangulation has at most one component. They are also natural
Poincar\'e duals to 1-cocycles with $\ZZ/2\ZZ$-coefficients. For a fixed
cohomology class in a simplicial poset the average Euler characteristic of the
associated discrete normal surfaces only depends on the $f$-vector of the
triangulation. As an application we determine the minimum simplicial poset
representations, also known as crystallizations, of lens spaces $L(2k,q),$
where $2k=qr+1.$ Higher dimensional analogs of discrete normal surfaces are
closely connected to the Charney-Davis conjecture for flag spheres.
