Euler Characteristics of Random Manifolds

2026-07-27

AI Summary

This study investigates the expected Euler characteristic of random level sets on simplicial complexes. By integrating combinatorial topology, probabilistic methods, and curvature functional analysis, the authors establish an explicit relationship between the expected face vector of a random submanifold and that of its host complex. They derive, for the first time, an analytical expression for the expected Euler characteristic of a random level set: $\mathbb{E}[\chi(H)] = 2 - 2K(G) - \chi(G)$, where $K(G)$ denotes the curvature functional of the complex $G$ and $\chi(G)$ its Euler characteristic. This formula reveals a profound connection between the expectation and intrinsic geometric-topological invariants of the underlying simplicial complex.

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.

Problem

• Euler characteristic
• random manifolds
• simplicial complex
• f-vector
• level surface

Innovation

• Euler characteristic
• random manifolds
• simplicial complex
• f-vector
• curvature functional