Generalized Dehn-Sommerville relations for polytopes, spheres and Eulerian partially ordered sets

This paper generalizes the Dehn-Sommerville equations for simplicial spheres to related classes of objects. The underlying motivation is to understand the combinatorial structure of arbitrary polytopes, that is, polytopes that are not necessarily simplicial. Towards this end we determine the affine span of the extended f-vectors of d-polytopes. A polytope is the convex hull of finitely many points in R e . We will generally consider a polytope of affine dimension d to be a subset of Re; this is referred to as a d-polytope. A face of a polytope is the intersection of a supporting hyperplane with the polytope. For the most part we identify a polytope P with the abstract cell complex (or lattice) realized by the boundary of P, and write a face of P as the set of vertices of P it contains. That is, we will shorten F = c o n v { v 0 . . . . . Vk} to F={Vo,.. . ,Vk} when there is no risk of confusion. By convention, the empty set is considered a ( -1)-dimensional face, and the polytope itself is a d-dimentional face; these faces will be called improper faces of P. A polytope P is called simplicial if each of its faces, except possibly P itself, is a simplex (the convex hull of affinely independent points). We will write Y (respectively, ~ ) for the set of all (respectively, all simplicial) d-polytopes. The number of/-dimensional faces (or/-faces) of a polytope P is written f , and f(P)=(fo,fl . . . . . fd-1) is called the f-vector of P. The set off -vectors of all (simplicial) polytopes is written f ( ~ e ) (f(~d)). A certain transformation on the f-vectors of simplicial polytopes has arisen in a number of different contexts, and will play an important part here. For a d-polytope P define the h-vector h(P)=(ho, hl,...,he) b y h i = i ( 1 ) i s ( d j ) s= o d i f j 1 (here we use the convention

[1]  P. McMullen,et al.  AN INTRODUCTION TO CONVEX POLYTOPES (Graduate Texts in Mathematics, 90) , 1984 .

[2]  David W. Barnette,et al.  The projection of the f-vectors of 4-polytopes onto the (E, S)-plane , 1974, Discret. Math..

[3]  D. M. Y. Sommerville,et al.  The Relations Connecting the Angle-Sums and Volume of a Polytope in Space of n Dimensions , 1927 .

[4]  Robert W. Irving.  On a bound of Graham and Spencer for a graph-coloring constant , 1973 .

[5]  Richard P. Stanley,et al.  An Introduction to Cohen-Macaulay Partially Ordered Sets , 1982 .

[6]  Curtis Greene,et al.  The Möbius Function of a Partially Ordered Set , 1982 .

[7]  R. Stanley.  The Upper Bound Conjecture and Cohen‐Macaulay Rings , 1975 .

[8]  Louis J. Billera,et al.  Counting Faces and Chains in Polytopes and Posets , 1983 .

[9]  Carl W. Lee,et al.  A Proof of the Sufficiency of McMullen's Conditions for f-Vectors of Simplicial Convex Polytopes , 1981, J. Comb. Theory, Ser. A.

[10]  A. Brøndsted.  An Introduction to Convex Polytopes , 1982 .

[11]  G. C. Shephard,et al.  Stellar subdivisions of boundary complexes of convex polytopes , 1974 .

[12]  R. Stanley.  Combinatorics and commutative algebra , 1983 .

[13]  V. Klee.  A Combinatorial Analogue of Poincaré's Duality Theorem , 1964, Canadian Journal of Mathematics.

[14]  Richard P. Stanley,et al.  Balanced Cohen-Macaulay complexes , 1979 .

[15]  R. Stanley.  The number of faces of a simplicial convex polytope , 1980 .

[16]  Carl W. Lee.  Counting the faces of simplicial convex polytopes , 1981 .

[17]  A. G. Kouchnirenko.  Polyèdres de Newton et nombres de Milnor , 1976 .

[18]  Kenneth Baclawski,et al.  Cohen-Macaulay ordered sets , 1980 .

[19]  Richard P. Stanley,et al.  Some Aspects of Groups Acting on Finite Posets , 1982, J. Comb. Theory, Ser. A.

[20]  V. Danilov,et al.  THE GEOMETRY OF TORIC VARIETIES , 1978 .

[21]  G. A. Reisner,et al.  Cohen-Macaulay quotients of polynomial rings , 1976 .

[22]  David Barnette,et al.  Projections of f-Vectors of Four-Polytopes , 1973, J. Comb. Theory, Ser. A.

[23]  Bernt Lindström,et al.  On the realization of convex polytopes, Euler's formula and Möbius functions , 1971 .

[24]  R. Stanley.  Hilbert functions of graded algebras , 1978 .

[25]  L. Billera,et al.  Sufficiency of McMullen's conditions for $f$-vectors of simplicial polytopes , 1980 .

[26]  Margaret M. Bayer.  Facial enumeration in polytopes, spheres and other complexes , 1983 .

[27]  P. McMullen.  The numbers of faces of simplicial polytopes , 1971 .