In mathematics, the Dehn–Sommerville equations are a complete set of linear relations between the numbers of faces of different dimension of a

simplicial polytope In geometry, a simplicial polytope is a polytope whose facet_(mathematics), facets are all Simplex, simplices. For example, a ''simplicial polyhedron'' in three dimensions contains only Triangle, triangular faces Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Dehn's early life and career took place in Germany. However, he was forced to retire in 1 ...

in 1905. Their general form was established by

Duncan Sommerville Duncan MacLaren Young Sommerville (1879–1934) was a Scottish mathematician and astronomer. He compiled a bibliography on non-Euclidean geometry and also wrote a leading textbook in that field. He also wrote ''Introduction to the Geometry of N ...

in 1927. The Dehn–Sommerville equations can be restated as a symmetry condition for the ''h''-vector of the simplicial polytope and this has become the standard formulation in recent combinatorics literature. By duality, analogous equations hold for simple polytopes.

Statement

Let ''P'' be a ''d''-dimensional

f(P)=(f_0,f_1,\ldots,f_)

is called the ''f''-vector of the polytope ''P''. Additionally, set :

f_=1, f_d=1.

Then for any ''k'' = −1, 0, ..., ''d'' − 2, the following Dehn–Sommerville equation holds: :

\sum_^ (-1)^j \binom  f_j = (-1)^f_k.

When ''k'' = −1, it expresses the fact that

Euler characteristic In mathematics, and more specifically in algebraic topology and polyhedral combinatorics, the Euler characteristic (or Euler number, or Euler–Poincaré characteristic) is a topological invariant, a number that describes a topological space's ...

of a (''d'' − 1)-dimensional simplicial sphere is equal to 1 + (−1)^{''d'' − 1}. Dehn–Sommerville equations with different ''k'' are not independent. There are several ways to choose a maximal independent subset consisting of

\left frac\right /math> equations. If ''d'' is even then the equations with ''k'' = 0, 2, 4, ..., ''d'' − 2 are independent. Another independent set consists of the equations with ''k'' = −1, 1, 3, ..., ''d'' − 3. If ''d'' is odd then the equations with ''k'' = −1, 1, 3, ..., ''d'' − 2 form one independent set and the equations with ''k'' = −1, 0, 2, 4, ..., ''d'' − 3 form another.

Equivalent formulations

Sommerville found a different way to state these equations: :

\sum_^(-1)^\binom f_i = \sum_^(-1)^i \binom f_i,

where 0 ≤ k ≤ (d−1). This can be further facilitated introducing the notion of ''h''-vector of ''P''. For ''k'' = 0, 1, ..., ''d'', let :

h_k = \sum_^k (-1)^\binomf_.

The sequence :

h(P)=(h_0,h_1,\ldots,h_d)

is called the ''h''-vector of ''P''. The ''f''-vector and the ''h''-vector uniquely determine each other through the relation :

\sum_^d f_(t-1)^=\sum_^d h_k t^.

Then the Dehn–Sommerville equations can be restated simply as :

h_k = h_ \quad\text 0\leq k\leq d.

The equations with 0 ≤ k ≤ (d−1) are independent, and the others are manifestly equivalent to them. Richard Stanley gave an interpretation of the components of the ''h''-vector of a simplicial convex polytope ''P'' in terms of the projective

toric variety In algebraic geometry, a toric variety or torus embedding is an algebraic variety containing an algebraic torus as an open dense subset, such that the action of the torus on itself extends to the whole variety. Some authors also require it to be ...

''X'' associated with (the dual of) ''P''. Namely, they are the dimensions of the even

intersection cohomology In topology, a branch of mathematics, intersection homology is an analogue of singular homology especially well-suited for the study of singular spaces, discovered by Mark Goresky and Robert MacPherson in the fall of 1974 and developed by them ov ...

groups of ''X'': :

h_k=\dim_\operatorname^(X,\mathbb)

(the odd

groups of ''X'' are all zero). In this language, the last form of the Dehn–Sommerville equations, the symmetry of the ''h''-vector, is a manifestation of the

Poincaré duality In mathematics, the Poincaré duality theorem, named after Henri Poincaré, is a basic result on the structure of the homology (mathematics), homology and cohomology group (mathematics), groups of manifolds. It states that if ''M'' is an ''n''-dim ...

in the intersection cohomology of ''X''.

References

Branko Grünbaum Branko Grünbaum (; 2 October 1929 – 14 September 2018) was a Croatian-born mathematician of Jewish descentConvex Polytopes ''Convex Polytopes'' is a graduate-level mathematics textbook about convex polytopes, higher-dimensional generalizations of three-dimensional polyhedron, convex polyhedra. It was written by Branko Grünbaum, with contributions from Victor Klee, M ...

''. Second edition. Graduate Texts in Mathematics, Vol. 221, Springer, 2003 * Richard P. Stanley, ''Combinatorics and Commutative Algebra''. Second edition. Progress in Mathematics, 41. Birkhäuser Boston, Inc., Boston, MA, 1996. * D. M. Y. Sommerville (1927
The relations connecting the angle sums and volume of a polytope in space of n dimensions
Proceedings of the Royal Society Series A, 115:103–19, weblink from

JSTOR JSTOR ( ; short for ''Journal Storage'') is a digital library of academic journals, books, and primary sources founded in 1994. Originally containing digitized back issues of academic journals, it now encompasses books and other primary source ...

. * Günter M. Ziegler, ''Lectures on Polytopes''.

Springer Springer or springers may refer to: Publishers * Springer Science+Business Media, aka Springer International Publishing, a worldwide publishing group founded in 1842 in Germany formerly known as Springer-Verlag. ** Springer Nature, a multinationa ...

, 1998. {{DEFAULTSORT:Dehn-Sommerville equations Polyhedral combinatorics