Ed Swartz - Papers

Matroids and quotients of spheres, Math. Zeit., 241 (2002), 247-269.     pdf file

g-elements of matroid complexes, Journal of Comb. Theory Ser. B, 88 (2003), 369-375. pdf file

Lower bounds for h-vectors of k-CM, independence and broken circuit complexes, SIAM J. Disc. Math., 18 (2004/05), 647-661. pdf file

Topological representations of matroids, J. Amer. Math. Soc., 16 (2003), 427-442.  pdf file

(with Tamas Hausel)  Intersection forms of toric hyperkaehler varieties, Proc. Amer. Math. Soc., 134 (2006), 2403-2409   math.AG/0306369

(with Kathryn Nyman)  Inequalities for the h- and flag h-vecotrs of geometric lattices, Disc. and Comp. Geom., 32 (2004), 533-548.  pdf file

g-elements, finite buildings and higher Cohen-Macaulay connectivity, J. Combin. Theory Ser. A, 113 (2006), 1305-1320. math.CO/0512086

Face enumeration: from spheres to manifolds. J. Europ. Math. Soc., 11 (2009), 449-485. math/0709.3998

(with I. Novik)  Face ring multiplicity via CM-connectivity sequences,  Canadian J. of Mathematics, 61 (2009), 888-903, arXiv:math.AC/0606.5246

(with P. Hersh) Coloring complexes and arrangements, J. Algebraic Comb., 27 (2008), 205-214. arXiv/math/0706.3657

(with J.Chestnut and J. Sapir)  Enumerative properties of triangulations of spherical bundles over S^1, European J. Comb., 29 (2008), 662-671. arXiv:math.CO/0611.5039

(with I. Novik)  Socles of Buchsbaum modules, complexes and posets  Adv. in Math., 222 (2009), 2059-2084. arXiv: mathCO/0711.0783

(with I. Novik)  Applications of Dehn-Sommerville relations,  Disc. and Comp. Geom., 42 (2009), 261-276. pdf file

Topological finiteness for edge-vertex enumeration, Adv. in Math., 219 (2008), 1722-1728. pdf file

(with I. Novik)) Gorenstein rings through face rings of manifolds, Composit. Math., 145 (2009), 993-1000. arXiv:0806.1017

(with F. Lutz and T. Sulanke) f-vectors of 3-manifolds, Elec. J. of Comb. 16 (2)(2009), R13. pdf file

(with E. Miller and I. Novik) Face rings of simplicial complexes with singularities, Math. Ann., 351 (2011), pg. 857-875. arXiv:1001.2812   (replaces arXiv 0908.1433)

(with C. Klivans) Projection volumes of hyperplane arrangements.  Discrete and Computational Geometry, 46 (2011), 417-426. arXiv:1001.5095

(with I. Novik) Face numbers of pseudomanifolds with isolated singularities, Math. Scan. 110 (2012), 198-222,  arXiv:1004.5100

(with M. Hughes) Quotients of spheres by linear actions of tori, arXiv:1205.6387
To every linear quotient of a sphere by a real torus there is an associated rationally represented matroid whose Tutte polynomial determines the integral homology of both the full quotient space and the subspace of rationally singular points.   We also determine when the quotient space is a manifold and, more specifically, when it is a sphere.

The average dual surface of a cohomology class and  minimal simplicial decompositions of infinitely many lens spaces,  arXiv:1310.1991
  Discrete normal surfaces are normal surfaces whose intersection with each tetrahedron of a triangulation has at most one component.  They are natural Poincar\'e duals to $1$-cocycles with $\ZZ/2\ZZ$-coefficients.  We show that for a simplicial poset and  a fixed cohomology class the average Euler characteristic of the associated discrete normal surfaces only depends on the $f$-vector of the triangulation.  As an application we determine the minimum simplicial poset representations, also known as crystallizations, of lens spaces $L(2k,q),$ where $2k=qr+1.$  Higher dimensional analogs of discrete normal surfaces are directly connected to the Charney-Davis conjecture for flag spheres.

Thirty-five years and counting,  arXiv: 1411.0987
A survey of attmepts to extend the g-theorem to other classes of spheres.  Includes previously unpublished material on g-conjecture and bistellar moves, small g_2, the topology of stacked manifolds and counterexamples to over optimistic extensions of the g-theorem.

(with B. Basak) Three dimensional normal pseudomanifolds with relatively few edges, Adv. Math. 365 (2020),  arxiv:1803.08942

We classify all three-dimensional normal pseudomanifolds X with two or fewer singularities and relatively minimal g_2.   X has relatively minimal g_2 if g_2(X) = g_2(link v) for some vertex v.  Such an X is always homeomorphic to a compression body with its boundary components coned off.  As a consequense we also prove that if g_2(X) is less than or equal to three, then X is homeomorphic to S^3 or the suspension of RP^2.

(with I. Novik) g-vectors of manifolds with boundary, Algebr. Comb., 3 (2020), 887-911. arxiv:1909.06729

We extend several g-theorems for manifolds without boundary to manifolds with boundary.  Included is a discussion of a connection between lower bound theorems and PL-handle decompositions and surgery .