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A User's Guide To Discrete Morse Theory
 Proc. of the 2001 Internat. Conf. on Formal Power Series and Algebraic Combinatorics, A special volume of Advances in Applied Mathematics
, 2001
"... this paper we present an adaptation of Morse Theory that may be applied to any simplicial complex (or more general cell complex). There have been other adaptations of Morse Theory that can be applied to combinatorial spaces. For example, a Morse theory of piecewise linear functions appears in [26] a ..."
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this paper we present an adaptation of Morse Theory that may be applied to any simplicial complex (or more general cell complex). There have been other adaptations of Morse Theory that can be applied to combinatorial spaces. For example, a Morse theory of piecewise linear functions appears in [26] and the very powerful \Stratied Morse Theory" was developed by Goresky and MacPherson [19],[20]. These theories, especially the latter, have each been successfully applied to prove some very striking results
Resolutions by mapping cones
 Homology Homotopy Appl
"... Many wellknown free resolutions arise as iterated mapping cones. Prominent examples are the EliahouKervaire resolution of stable monomial ideals (as noted by Evans and Charalambous [10]), and the Taylor resolution. The idea of the iterated mapping cone construction is the following: Let I ⊂ R be a ..."
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Cited by 33 (4 self)
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Many wellknown free resolutions arise as iterated mapping cones. Prominent examples are the EliahouKervaire resolution of stable monomial ideals (as noted by Evans and Charalambous [10]), and the Taylor resolution. The idea of the iterated mapping cone construction is the following: Let I ⊂ R be an ideal generated by
SYZYGIES OF ORIENTED MATROIDS
 DUKE MATHEMATICAL JOURNAL VOL. 111, NO. 2
, 2002
"... We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are StanleyReisner ideals of complexes of independent sets and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular rea ..."
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Cited by 23 (2 self)
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We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids, and oriented matroids. These are StanleyReisner ideals of complexes of independent sets and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of R. Stanley’s formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by D. Bayer, S. Popescu, and B. Sturmfels [3]. We resolve the combinatorial problems posed in [3] by computing Möbius invariants of graphic and cographic arrangements in terms of Hermite polynomials.
COMPUTING OPTIMAL MORSE MATCHINGS
, 2004
"... Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computation ..."
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Morse matchings capture the essential structural information of discrete Morse functions. We show that computing optimal Morse matchings is NPhard and give an integer programming formulation for the problem. Then we present polyhedral results for the corresponding polytope and report on computational results.
Discrete Morse Theory for Manifolds with Boundary
, 2012
"... We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities ” relating the homology of the manifold to the number of interior critical cells. We also ..."
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We introduce a version of discrete Morse theory specific for manifolds with boundary. The idea is to consider Morse functions for which all boundary cells are critical. We obtain “Relative Morse Inequalities ” relating the homology of the manifold to the number of interior critical cells. We also derive a Ball Theorem, in analogy to Forman’s Sphere Theorem. The main corollaries of our work are: (1) For each d ≥ 3andforeachk ≥ 0, there is a PL dsphere on which any discrete Morse function has more than k critical (d − 1)cells. (This solves a problem by Chari.) (2) For fixed d and k, there are exponentially many combinatorial types of simplicial dmanifolds (counted with respect to the number of facets) that admit discrete Morse functions with at most k critical interior (d − 1)cells. (This connects discrete Morse theory to enumerative combinatorics/ discrete quantum gravity.) (3) The barycentric subdivision of any simplicial constructible dball is
ON BOREL FIXED IDEALS GENERATED IN ONE DEGREE
, 2007
"... We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S = k[x1, x2,...,xn]; this includes the case of powers of the homogeneous maximal ideal (x1, x2,...,xn) as a s ..."
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We construct a (shellable) polyhedral cell complex that supports a minimal free resolution of a Borel fixed ideal, which is minimally generated (in the Borel sense) by just one monomial in S = k[x1, x2,...,xn]; this includes the case of powers of the homogeneous maximal ideal (x1, x2,...,xn) as a special case. In our most general result we prove that for any Borel fixed ideal I generated in one degree, there exists a polyhedral cell complex that supports a minimal free resolution of I.
The minimal free resolution of a Borel ideal
"... This paper focuses on the EliahouKervaire minimal free resolution of a Borel ideal. Throughout, S = k[x1,..., xn] is a polynomial ring over a field k. We grade S by deg(xi) = 1 for each i. Let M be an ideal in S. A free resolution of S/M is an exact sequence of homomorphisms of free modules di d1 ..."
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Cited by 9 (0 self)
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This paper focuses on the EliahouKervaire minimal free resolution of a Borel ideal. Throughout, S = k[x1,..., xn] is a polynomial ring over a field k. We grade S by deg(xi) = 1 for each i. Let M be an ideal in S. A free resolution of S/M is an exact sequence of homomorphisms of free modules di d1
Gröbner basis degree bounds on Tor k[Λ] (k, k) and discrete Morse theory for posets
, 2003
"... The purpose of this paper is twofold. ⊲ We give combinatorial bounds on the ranks of the groups Tor R • (k, k) • in the case where R = k[Λ] is an affine semigroup ring, and in the process provide combinatorial proofs for bounds by Eisenbud, Reeves and Totaro on which Tor groups vanish. In additio ..."
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The purpose of this paper is twofold. ⊲ We give combinatorial bounds on the ranks of the groups Tor R • (k, k) • in the case where R = k[Λ] is an affine semigroup ring, and in the process provide combinatorial proofs for bounds by Eisenbud, Reeves and Totaro on which Tor groups vanish. In addition, we show that if the bounds hold for a field k then they hold for K[Λ] and any field K. Moreover, we provide a combinatorial construction for a free resolution of K over K[Λ] which achieves these bounds. ⊲ We extend the lexicographic discrete Morse function construction of Babson and Hersh for the determination of the homotopy type and homology of order complexes of posets to a larger class of facet orderings that includes orders induced by monomial term orders. Since it is known that the order complexes of finite intervals in the poset of monomials in k[Λ] ordered by divisibility in k[Λ] govern the Torgroups, the newly developed tools are applicable and serve as the main ingredients for the proof of the bounds and the construction of the resolution.