Ed Swartz

g -Elements, finite buildings and higher Cohen-Macaulay connectivity

Chari proved that if Δ is a (d - 1)-dimensional simplicial complex with a convex ear decomposition, then h0 ≤ ... ≤ h⌊d/2⌋ [M.K. Chari, Two decompositions in topological combinatorics with applications to matroid complexes, Trans. Amer. Math. Soc. 349 (1997) 3925-3943]. Nyman and Swartz raised the problem of whether or not the corresponding g-vector is an M-vector [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533-548]. This is proved to be true by showing that the set of pairs (ω, Θ), where Θ is a l.s.o.p. for k[Δ], the face ring of Δ, and ω is a g-element for k[Δ]/Θ, is nonempty whenever the characteristic of k is zero.Finite buildings have a convex ear decomposition. These decompositions point to inequalities on the flag h-vector of such spaces similar in spirit to those examined in [K. Nyman, E. Swartz, Inequalities for h- and flag h-vectors of geometric lattices, Discrete Comput. Geom. 32 (2004) 533-548] for order complexes of geometric lattices. This also leads to connections between higher Cohen-Macaulay connectivity and conditions which insure that h0 < ... < hi for a predetermined i.

Face enumeration - from spheres to manifolds

We prove a number of new restrictions on the enumerative properties of homology manifolds and semi-Eulerian complexes and posets. These include a determination of the affine span of the fine h-vector of balanced semi-Eulerian complexes and the toric h-vector of semi-Eulerian posets. The lower bounds on simplicial homology manifolds, when combined with higher dimensional analogues of Walkup�s 3-dimensional constructions [47], allow us to give a complete characterization of the f-vectors 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 g-conjecture 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 2-neighborly fashion.

Lower Bounds for h -Vectors of k -CM, Independence, and Broken Circuit Complexes

We present a number of lower bounds for the h-vectors of k-Cohen--Macaulay (k-CM), broken circuit, and independence complexes. These lead to bounds on the coefficients of the characteristic and reliability polynomials of matroids. The main techniques are the use of series and parallel constructions on matroids and the short simplicial h-vector for pure complexes.

g -Elements of matroid complexes

A g-element for a graded R-module is a one-form with properties similar to a Lefschetz class in the cohomology ring of a compact complex projective manifold, except that the induced multiplication maps are injections instead of bijections. We show that if k(Δ) is the face ring of the independence complex of a matroid and the characteristic of k is zero, then there is a nonempty Zariski open subset of pairs (Θ, Ω) such that Θ is a linear set of parameters for k(Δ) and Ω is a g-element for k(Δ)/ 〈Θ〉. This leads to an inequality on the first half of the h-vector of the complex similar to the g-theorem for simplicial polytopes.

Gorenstein rings through face rings of manifolds

The face ring of a homology manifold (without boundary) modulo a generic system of parameters is studied. Its socle is computed and it is verified that a particular quotient of this ring is Gorenstein. This fact is used to prove that the algebraic g -conjecture for spheres implies all enumerative consequences of its far-reaching generalization (due to Kalai) to manifolds. A special case of Kalai’s conjecture is established for homology manifolds that have a codimension-two face whose link contains many vertices.

Face Ring Multiplicity via CM-Connectivity Sequences

The multiplicity conjecture of Herzog, Huneke, and Srinivasan is verified for the face rings of the following classes of simplicial complexes: matroid complexes, complexes of dimension one and two, and Gorenstein complexes of dimension at most four. The lower bound part of this conjec- ture is also established for the face rings of all doubly Cohen-Macaulay complexes whose 1-skeletons connectivitydoes not exceed the codimension plusone as wellas forall(d −1)-dimensional d-Cohen- Macaulay complexes. The main ingredient of the proofs is a new interpretation of the minimal shifts in the resolution of the face ring k(�) via the Cohen-Macaulay connectivity of the skeletons of �.

Projection Volumes of Hyperplane Arrangements

We prove that for any finite real hyperplane arrangement the average projection volumes of the maximal cones are given by the coefficients of the characteristic polynomial of the arrangement. This settles the conjecture of Drton and Klivans that this held for all finite real reflection arrangements. The methods used are geometric and combinatorial. As a consequence, we determine that the angle sums of a zonotope are given by the characteristic polynomial of the order dual of the intersection lattice of the arrangement.

Inequalities for the h-Vectors and Flag h-Vectors of Geometric Lattices

Abstract We prove that the order complex of a geometric lattice has a convex ear decomposition. As a consequence, if Δ(L) is the order complex of a rank (r + 1) geometric lattice L, then for all i ≤ r/2 the h-vector of Δ(L) satisfies hi-1 ≤ hi and hi ≤ hr-i. We also obtain several inequalities for the flag h-vector of Δ(L) by analyzing the weak Bruhat order of the symmetric group. As an application, we obtain a zonotopal cd-analogue of the Dowling–Wilson characterization of geometric lattices which minimize Whitney numbers of the second kind. In addition, we are able to give a combinatorial flag h-vector proof of hi-1 ≤ hi when i ≤ (2/7)(r + (5/2)).

Intersection forms of toric hyperkahler varieties

This note proves combinatorially that the intersection pairing on the middle-dimensional compactly supported cohomology of a toric hyperkahler variety is always definite, providing a large number of non-trivial L 2 harmonic forms for toric hyperkahler metrics on these varieties. This is motivated by a result of Hitchin about the definiteness of the pairing of L 2 harmonic forms on complete hyperkahler manifolds of linear growth.