The Euler characteristic of a triangulated manifold in terms of even-dimensional faces
Alexey V. Gavrilov
TL;DR
This work derives a universal, dimension-free formula for the Euler characteristic of an even-dimensional triangulated closed manifold: $\chi({\mathcal M})=\sum_{n=0}^d \beta_n f_n$, where $\beta_{n-2}= (4(2^n-1)B_n)/n$ and $B_n$ are Bernoulli numbers. The proof leverages Dehn–Sommerville relations recast via $h$-numbers and a linear functional $\theta$ to show the coefficients $\beta_n$ suffice across dimensions; it also explains a boundary extension via doubling. The results illuminate how semi-Eulerian (and hence manifold) triangulations determine $\chi$ solely from even-faces, with a seamless extension to manifolds with boundary. Overall, the paper provides a dimension-independent, combinatorially grounded formula for the Euler characteristic in terms of even-dimensional faces.
Abstract
We give a formula for the Euler characteristic of a triangulated manifold of even dimension in terms of the numbers of even-dimensional faces only. The coefficients in this formula are universal (they do not depend on the dimension of the manifold).