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47
Lectures on Polytopes
, 1994
"... uptodate electronically. Thus, this is an electronic preprint, the newest, latest and hottest version of which you should always be able to get via our WWWserver, at ..."
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Cited by 359 (5 self)
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uptodate electronically. Thus, this is an electronic preprint, the newest, latest and hottest version of which you should always be able to get via our WWWserver, at
Real root conjecture fails for five and higherdimensional spheres, Discrete Comput
 Geom
"... Abstract: A construction of convex flag triangulations of five and higher dimensional spheres, whose hpolynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than realrootedness is conjectured and some evidence is p ..."
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Cited by 24 (1 self)
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Abstract: A construction of convex flag triangulations of five and higher dimensional spheres, whose hpolynomials fail to have only real roots, is given. We show that there is no such example in dimensions lower than five. A condition weaker than realrootedness is conjectured and some evidence is provided. Let the fpolynomial fX of a simplicial complex X be defined by the formula fX(t): = ∑ t #σ. There is a classical problem: what can be said in general about the fpolynomials of (a certain class of) simplicial complexes? In particular, it is well known what polynomials appear as fpolynomials of • general simplicial complexes, or • triangulations of spheres that are the boundary complexes of convex polytopes (the reader may consult [St1] for ample discussion). The question concerning all triangulations of spheres remains still open. However the answer is conjecturally the same. What we are interested in is the special case of the latter. Namely, what can be said in general about
Some Aspects Of The Combinatorial Theory Of Convex Polytopes
, 1993
"... . We start with a theorem of Perles on the kskeleton, Skel k (P ) (faces of dimension k) of d polytopes P with d+b vertices for large d. The theorem says that for fixed b and d, if d is sufficiently large, then Skel k (P ) is the kskeleton of a pyramid over a (d \Gamma 1)dimensional polytope. ..."
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Cited by 18 (3 self)
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. We start with a theorem of Perles on the kskeleton, Skel k (P ) (faces of dimension k) of d polytopes P with d+b vertices for large d. The theorem says that for fixed b and d, if d is sufficiently large, then Skel k (P ) is the kskeleton of a pyramid over a (d \Gamma 1)dimensional polytope. Therefore the number of combinatorially distinct kskeleta of dpolytopes with d + b vertices is bounded by a function of k and b alone. Next we replace b (the number of vertices minus the dimension) by related but deeper invariants of P , the gnumbers. For a dpolytope P there are [d=2] invariants g1 (P ); g2 (P ); :::; g [d=2] (P ) which are of great importance in the combinatorial theory of polytopes. We study polytopes for which g k is small and carried away to related and slightly related problems. Key words: Convex polytopes, skeleton, simplicial sphere, simplicial manifold, fvector, g theorem, ranked atomic lattices, stress, rigidity, sunflower, lower bound theorem, elementary poly...
Tight submanifolds, smooth and polyhedral
 Tight and Taut Submanifolds
, 1997
"... We begin by defining and studying tightness and the twopiece property for smooth and polyhedral surfaces in threedimensional space. These results are then generalized to surfaces with boundary and with singularities, and to surfaces in higher dimensions. Later sections deal with generalizations t ..."
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Cited by 15 (4 self)
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We begin by defining and studying tightness and the twopiece property for smooth and polyhedral surfaces in threedimensional space. These results are then generalized to surfaces with boundary and with singularities, and to surfaces in higher dimensions. Later sections deal with generalizations to the case of smooth and polyhedral submanifolds of higher dimension and codimension, in particular highly connected submanifolds. Twentysix open
Intersection homology of toric varieties and a conjecture of
 Kalai,” Comment. Math. Helv
, 1999
"... Suppose that a ddimensional convex polytope P ⊂ R d is rational, i.e. its vertices are all rational points. Then P gives rise to a polynomial g(P) = 1 + g1(P)q + g2(P)q 2 + · · · with nonnegative coefficients as follows. Let XP be the associated toric variety (see §6 – our variety XP is d + 1d ..."
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Cited by 15 (3 self)
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Suppose that a ddimensional convex polytope P ⊂ R d is rational, i.e. its vertices are all rational points. Then P gives rise to a polynomial g(P) = 1 + g1(P)q + g2(P)q 2 + · · · with nonnegative coefficients as follows. Let XP be the associated toric variety (see §6 – our variety XP is d + 1dimensional and affine). The coefficient gi is the rank of the 2ith intersection homology group of XP. The polynomial g(P) turns out to depend only on the face lattice of P, (see §1). It can be thought of as a measure of the complexity of P; for example, g(P) = 1 if and only if P is a simplex. Suppose that F ⊂ P is a face of dimension k. We construct an associated polyhedron P/F as follows (see the figure below): choose an (n − k − 1)plane L whose intersection with P is a single point p of the interior of F. Let L ′ be a small parallel displacement of L that intersects the interior of P. Then P/F is the intersection of P with L ′ ; it is only welldefined up to a projective transformation, but its combinatorial type is welldefined. Faces of P/F are in onetoone correspondence with faces of P which contain F.
FACE ENUMERATION  FROM SPHERES TO MANIFOLDS
, 2007
"... We prove a number of new restrictions on the enumerative properties of homology manifolds and semiEulerian complexes and posets. These include a determination of the affine span of the fine hvector of balanced semiEulerian complexes and the toric hvector of semiEulerian posets. The lower bounds ..."
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Cited by 13 (2 self)
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We prove a number of new restrictions on the enumerative properties of homology manifolds and semiEulerian complexes and posets. These include a determination of the affine span of the fine hvector of balanced semiEulerian complexes and the toric hvector of semiEulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup’s 3dimensional constructions [47], allow us to give a complete characterization of the fvectors of arbitrary simplicial triangulations of S¹ × S³, CP², K3 surfaces, and (S² × S²)#(S² × S²). We also establish a principle which leads to a conjecture for homology manifolds which is almost logically equivalent to the gconjecture for homology spheres. Lastly, we show that with sufficiently many vertices, every triangulable homology manifold without boundary of dimension three or greater can be triangulated in a 2neighborly fashion.
Socles of Buchsbaum modules, complexes and posets
 Advances in Math. 222
, 2009
"... The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and po ..."
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Cited by 11 (4 self)
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The socle of a graded Buchsbaum module is studied and is related to its local cohomology modules. This algebraic result is then applied to face enumeration of Buchsbaum simplicial complexes and posets. In particular, new necessary conditions on face numbers and Betti numbers of such complexes and posets are established. These conditions are used to settle in the affirmative Kühnel’s conjecture for the maximum value of the Euler characteristic of a 2kdimensional simplicial manifold on n vertices as well as Kalai’s conjecture providing a lower bound on the number of edges of a simplicial manifold in terms of its dimension, number of vertices, and the first Betti number. 1
Face Numbers of 4Polytopes and 3Spheres
 Proceedings of the international congress of mathematicians, ICM 2002
, 2002
"... Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to s ..."
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Cited by 10 (2 self)
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Steinitz (1906) gave a remarkably simple and explicit description of the set of all fvectors f(P ) = (f0 , f1 , f2) of all 3dimensional convex polytopes. His result also identifies the simple and the simplicial 3dimensional polytopes as the only extreme cases. Moreover, it can be extended to strongly regular CW 2spheres (topological objects), and further to Eulerian lattices of length 4 (combinatorial objects).
Skeletal Rigidity of Simplicial Complexes, I
 Symplicial Complexes I, II, European Journal of Combinatorics
"... This is the first part of a twopart paper, with the second part to appear in a later volume of this journal. The concept of infinitesimal rigidity concerns a graph (or a 1dimensional simplicial complex, which we regard as a barandjoint framework) realized in ddimensional Euclidean space. We gen ..."
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Cited by 10 (0 self)
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This is the first part of a twopart paper, with the second part to appear in a later volume of this journal. The concept of infinitesimal rigidity concerns a graph (or a 1dimensional simplicial complex, which we regard as a barandjoint framework) realized in ddimensional Euclidean space. We generalize this notion to rrigidity of higherdimensional simplicial complexes, again realized in ddimensional space. Roughly speaking, rrigidity means lack of nontrivial rmotion, and an rmotion amounts to assigning a velocity vector to each r \Gamma 2dimensional simplex in such a way that all r \Gamma 1dimensional volumes of r \Gamma 1simplices are instantaneously preserved. We give three different, but equivalent, elementary formulations of rmotions and the related idea of rstresses in this part of the paper, and two additional ones in part II. We also give a homological interpretation of these concepts, in a special case. This homological interpretation can be extended to the gene...
Remarks on the combinatorial intersection cohomology of fans, preprint
"... Abstract. This partly expository paper reviews the theory of combinatorial intersection cohomology of fans developed by BarthelBrasseletFieselerKaup, BresslerLunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal prope ..."
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Cited by 9 (0 self)
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Abstract. This partly expository paper reviews the theory of combinatorial intersection cohomology of fans developed by BarthelBrasseletFieselerKaup, BresslerLunts, and Karu. This theory gives a substitute for the intersection cohomology of toric varieties which has all the expected formal properties but makes sense even for nonrational fans, which do not define a toric variety. As a result, a number of interesting results on the toric g and h polynomials have been extended from rational polytopes to general polytopes. We present explicit complexes computing the combinatorial IH in degrees one and two; the degree two complex gives the rigidity complex previously used by Kalai to study g2. We present several new results which follow from these methods, as well as previously unpublished proofs of Kalai that gk(P) = 0 implies gk(P ∗ ) = 0 and gk+1(P) = 0. For a ddimensional convex polytope P, Stanley [St2] defined a polynomial invariant h(P, t) = ∑d k=0 hk(P)tk of the face lattice of P, usually referred to as the “generalized ” or “toric ” hpolynomial of P. It was “generalized ” in that it extended a previous definition