A discrete d-manifold is a finite simple graph G=(V,E) where all unit spheres are (d-1)-spheres. A d-sphere is a d-manifold for which one can remove a vertex to make it contractible. A graph is contractible if one can remove a vertex with contractible unit sphere to get a contractible graph. We prove a discrete Morse-Sard theorem: if G=(V,E) is a d-manifold and f:V to R^k an arbitrary map, then for any c not in f(V), a level set { f = c } is always a (d-k)-manifold or empty. While a priori open sets in the simplicial complex of G, they are sub-manifolds in the Barycentric refinement of G. Level sets are orientable if G is orientable. Any complex-valued function psi on a discrete 4-manifold M defines so level surfaces {psi=c} which are except for c in f(V) always 2-manifolds or empty.

翻译：离散d-流形是一个有限简单图G=(V,E)，其中所有单位球面均为(d-1)-球面。d-球面是一种d-流形，可通过移除一个顶点使其可收缩。若通过移除具有可收缩单位球面的顶点能得到可收缩图，则该图是可收缩的。我们证明了一个离散Morse-Sard定理：设G=(V,E)为d-流形，f:V→R^k为任意映射，则对于任意不在f(V)中的c，水平集{f=c}总是(d-k)-流形或空集。虽然它们在G的单纯复形中是先验开集，但在G的重心细分中是子流形。若G可定向，则水平集亦可定向。离散4-流形M上的任意复值函数ψ定义了水平曲面{ψ=c}，除c在f(V)中的情形外，这些曲面总是2-流形或空集。