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14
Cellular resolutions of powers of monomial ideals, preprint arXiv:1212.2146 [math.AC
"... Abstract. There are many connections between the invariants of the different powers of an ideal. We investigate how to construct minimal resolutions for all powers at once using methods from algebraic and polyhedral topology with a focus on ideals arising from combinatorics. In one construction, we ..."
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Abstract. There are many connections between the invariants of the different powers of an ideal. We investigate how to construct minimal resolutions for all powers at once using methods from algebraic and polyhedral topology with a focus on ideals arising from combinatorics. In one construction, we obtain cellular resolutions for all powers of edge ideals of bipartite graphs on n vertices, supported by (n − 2)–dimensional complexes. Our main result is an explicit minimal cellular resolution for all powers of edge ideals of paths. These cell complexes are constructed by first subdividing polyhedral complexes and then modifying them using discrete Morse theory. 1.
Combinatorial stratifications and minimality of 2arrangements
, 2014
"... We prove that the complement of any affine 2arrangement in Rd is minimal, that is, it is homotopy equivalent to a cell complex with as many icells as its ith rational Betti number. For the proof, we provide a Lefschetztype hyperplane theorem for complements of 2arrangements, and introduce Alexa ..."
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We prove that the complement of any affine 2arrangement in Rd is minimal, that is, it is homotopy equivalent to a cell complex with as many icells as its ith rational Betti number. For the proof, we provide a Lefschetztype hyperplane theorem for complements of 2arrangements, and introduce Alexander duality for combinatorial Morse functions. Our results greatly generalize previous work by Falk, Dimca–Papadima, Hattori, Randell, and Salvetti–Settepanella and others, and they demonstrate that in contrast to previous investigations, a purely combinatorial approach suffices to show minimality and the Lefschetz Hyperplane Theorem for complements of complex hyperplane arrangements.
Minimality of toric arrangements
"... We prove that the complement of a toric arrangement has the homotopy type of a minimal CW complex. As a corollary we obtain that the integer cohomology of these spaces is torsion free. We use Discrete Morse Theory, providing a sequence of cellular collapses that leads to a minimal complex. ..."
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We prove that the complement of a toric arrangement has the homotopy type of a minimal CW complex. As a corollary we obtain that the integer cohomology of these spaces is torsion free. We use Discrete Morse Theory, providing a sequence of cellular collapses that leads to a minimal complex.
Tight complexes in 3space admit perfect discrete Morse functions
, 2012
"... In 1967, Chillingworth proved that all convex simplicial 3balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth’s theorem b ..."
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In 1967, Chillingworth proved that all convex simplicial 3balls are collapsible. Using the classical notion of tightness, we generalize this to arbitrary manifolds: We show that all tight simplicial 3manifolds admit some perfect discrete Morse function. We also strengthen Chillingworth’s theorem by proving that all convex simplicial 3balls are nonevasive. In contrast, we show that many nonevasive 3balls are not convex.