9
Euler Characteristic, Dehn–Sommerville
Characteristics, and Their Applications
V.M. Buchstaber
Summary.In this chapter we present several classical results centring on the no-
tion of Euler characteristic of simplicial complexes and manifolds. We also consider
some results that are not discussed in the textbooks of topology. These results are
concerned with construction of certain combinatorial invariants of manifold trian-
gulations, which we call theDehn–Sommerville characteristics.
11.1 Introduction
In May 2002 the author read a mini-course of lectures in algebraic topology
at the Max Planck Institute, Dresden. As the audience consisted mostly of
physicists and biologists, the course aimed at introducing several fundamental
concepts, requiring only basic mathematical knowledge. The notes from the
lecture course have grown up into this text.
We give several classical results centring on the notion of Euler character-
istic of simplicial complexes and manifolds. The chapter also contains some
results that have not yet found their way into algebraic topology textbooks,
but are of considerable interest due to their applications in several fields,
including discrete mathematical physics. These results are connected with
construction and applications of certain combinatorial invariants of manifold
triangulations, which we call theDehn–Sommerville characteristics.
All the key properties are illustrated on the triangulations of two-
dimensional surfaces. Detailed proofs and further developments for most of
the results of this study can be found in [1, 2].
9.2 Simplicial Complexes and Maps
DenoteRnas ann-dim Eucledian space. Ann-dim simplexσnis the convex
hull inRnof any (n+ 1) pointsα 0 ,...,αnnot contained in an (n−1)-dim
hyperplane.