Euler Characteristics of Random Manifolds
2026-07-27 • Discrete Mathematics
Discrete Mathematics
AI summaryⓘ
The authors show that when you randomly pick a 'level surface' inside a shape made from simple building blocks (a simplicial complex), the average shape characteristic called the Euler characteristic follows a specific formula involving another quantity called curvature. They relate this average Euler characteristic of the smaller surface to the Euler characteristic and curvature of the entire shape. More broadly, they connect how the counts of different building block types (faces) in a smaller shape relate to those in the bigger shape it comes from.
Euler characteristicsimplicial complexlevel surfacecurvature functionalf-vectorsubmanifoldexpectationtopologycombinatorial topology
Authors
Oliver Knill
Abstract
We prove that the expectation of the Euler characteristic X(H) of random level surface H in a given simplicial complex G is E[X(H)] =2-2K(G)-X(G), where K(G)=1-f_0/2+f_1/3- ... is the curvature functional of G and X(G)=f_0-f_1+f_2-... is the Euler characteristics. More generally, the expectation of the f-vector of a submanifold is explicitly linked to the f-vector of the host manifold.