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18
Torus actions and their applications in topology and combinatorics
 University Lecture Series 24, American Mathematical Society
, 2002
"... Staneley–Reisner rings, torus actions, toric varieties, quasitoric manifolds, momentangle complexes, subspace arrangements Abstract. The aim of this book is to present torus actions as a connecting bridge between combinatorial and convex geometry on one side, and commutative and homological algebra ..."
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Cited by 45 (1 self)
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Staneley–Reisner rings, torus actions, toric varieties, quasitoric manifolds, momentangle complexes, subspace arrangements Abstract. The aim of this book is to present torus actions as a connecting bridge between combinatorial and convex geometry on one side, and commutative and homological algebra, algebraic geometry and topology on the other. The established link helps to understand the geometry and topology of a toric space by studying the combinatorics of its orbit quotient. Conversely, subtlest properties of a combinatorial object can be recovered by realizing it as the orbit structure for a proper manifold or complex acted on by the torus. The latter can be a symplectic manifold with Hamiltonian torus action, a toric variety or manifold, a subspace arrangement complement etc., while the combinatorial objects involved include simplicial and cubical complexes, polytopes and arrangements. Such an approach also provides a natural topological interpretation of many constructions from commutative and homological algebra used in the combinatorics in terms of torus actions.
ON DAVISJANUSZKIEWICZ HOMOTOPY TYPES II; COMPLETION AND GLOBALISATION
, 2009
"... For any finite simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CWcomplexes whose integral cohomology rings are isomorphic to the StanleyReisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be hom ..."
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Cited by 20 (6 self)
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For any finite simplicial complex K, Davis and Januszkiewicz have defined a family of homotopy equivalent CWcomplexes whose integral cohomology rings are isomorphic to the StanleyReisner algebra of K. Subsequently, Buchstaber and Panov gave an alternative construction, which they showed to be homotopy equivalent to the original examples. It is therefore natural to investigate the extent to which the homotopy type of a space X is determined by such a cohomology ring. Having analysed this problem rationally in Part I, we here consider it prime by prime, and utilise Lannes’ T functor and BousfieldKan type obstruction theory to study the pcompletion of X. We find the situation to be more subtle than for rationalisation, and confirm the uniqueness of the completion whenever K is a join of skeleta of simplices. We apply our results to the global problem by appealing to Sullivan’s arithmetic square, and deduce integral uniqueness whenever the StanleyReisner algebra is a complete intersection.
Topology of Matching, Chessboard, and General Bounded Degree Graph Complexes
 Algebra Universalis, Special Issue in Memory of GianCarlo Rota
, 2003
"... We survey results and techniques in the topological study of simplicial complexes of (di, multi, hyper)graphs whose node degrees are bounded from above. These complexes have arisen is a variety of contexts in the literature. The most wellknown examples are the matching complex and the chessbo ..."
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Cited by 13 (2 self)
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We survey results and techniques in the topological study of simplicial complexes of (di, multi, hyper)graphs whose node degrees are bounded from above. These complexes have arisen is a variety of contexts in the literature. The most wellknown examples are the matching complex and the chessboard complex. The topics covered here include computation of Betti numbers, representations of the symmetric group on rational homology, torsion in integral homology, homotopy properties, and connections with other fields.
Poset fiber theorems
 TRANS. AMER. MATH. SOC
, 2004
"... Suppose that f: P → Q is a poset map whose fibers f −1 (Q≤q) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences o ..."
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Cited by 12 (2 self)
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Suppose that f: P → Q is a poset map whose fibers f −1 (Q≤q) are sufficiently well connected. Our main result is a formula expressing the homotopy type of P in terms of Q and the fibers. Several fiber theorems from the literature (due to Babson, Baclawski and Quillen) are obtained as consequences or special cases. Homology, CohenMacaulay, and equivariant versions are given, and some applications are discussed.
Colimits, StanleyReisner Algebras, and Loop Spaces
, 2003
"... We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: rightangled Artin and Coxeter groups (and their complex analogues, which ..."
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Cited by 4 (3 self)
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We study diagrams associated with a finite simplicial complex K, in various algebraic and topological categories. We relate their colimits to familiar structures in algebra, combinatorics, geometry and topology. These include: rightangled Artin and Coxeter groups (and their complex analogues, which we call circulation groups); StanleyReisner algebras and coalgebras; Davis and Januszkiewicz’s spaces DJ(K) associated with toric manifolds and their generalisations; and coordinate subspace arrangements. When K is a flag complex, we extend wellknown results on Artin and Coxeter groups by confirming that the relevant circulation group is homotopy equivalent to the space of loops ΩDJ(K). We define homotopy colimits for diagrams of topological monoids and topological groups, and show they commute with the formation of classifying spaces in a suitably generalised sense. We deduce that the homotopy colimit of the appropriate diagram of topological groups is a model for ΩDJ(K) for an arbitrary complex K, and that the natural projection onto the original colimit is a homotopy equivalence when K is flag. In this case, the two models are compatible.
KTheory of nonlinear projective toric varieties
, 2005
"... ... homotopysheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic Ktheory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to ..."
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... homotopysheaves of topological spaces) on projective toric varieties and prove a splitting result for its algebraic Ktheory, generalising earlier results for projective spaces. The splitting is expressed in terms of the number of interior lattice points of dilations of a polytope associated to the variety. The proof uses combinatorial and geometrical results on polytopal complexes. The same methods also give an elementary explicit calculation of the cohomology groups of a projective toric variety over any commutative ring.
Discrete Morse complexes
 In ACCOTA: Workshop on Combinatorial and Computational Aspects of Optimization, Topology and Algebra
, 2002
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Equivariant Topology of Configuration Spaces
, 2012
"... We study the Fadell–Husseini index of the configuration space F (R d, n) with respect to different subgroups of the symmetric group Sn. For p prime and k ≥ 1, we completely determine IndexZ/p(F (R d, p); Fp) and partially describe Index (Z/p) k(F (R d, p k); Fp). In this process we obtain results of ..."
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We study the Fadell–Husseini index of the configuration space F (R d, n) with respect to different subgroups of the symmetric group Sn. For p prime and k ≥ 1, we completely determine IndexZ/p(F (R d, p); Fp) and partially describe Index (Z/p) k(F (R d, p k); Fp). In this process we obtain results of independent interest, including: (1) an extended equivariant Goresky–MacPherson formula, (2) a complete description of the top homology of the partition lattice Πp as an Fp[Zp]module, and (3) a generalized Dold theorem for elementary abelian groups. The results on the Fadell–Husseini index yield a new proof of the Nandakumar & Ramana Rao conjecture for a prime. For n = p k
DESCRIBING TORIC VARIETIES AND THEIR EQUIVARIANT COHOMOLOGY
, 2009
"... Topologically, compact toric varieties can be constructed as identification spaces: they are quotients of the product of a compact torus and the order complex of the fan. We give a detailed proof of this fact, extend it to the noncompact case and draw several, mostly cohomological conclusions. In ..."
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Topologically, compact toric varieties can be constructed as identification spaces: they are quotients of the product of a compact torus and the order complex of the fan. We give a detailed proof of this fact, extend it to the noncompact case and draw several, mostly cohomological conclusions. In particular, we show that the equivariant integral cohomology of a toric variety can be described in terms of piecewise polynomials on the fan if the ordinary integral cohomology is concentrated in even degrees. This generalises a result of Bahri–Franz–Ray. We also investigate torsion phenomena in integral cohomology.
THE AUSLANDERREITEN TRANSLATE ON MONOMIAL QUOTIENT RINGS
, 802
"... Abstract. For t in N n, E.Miller has defined a category of tdetermined modules over the polynomial ring S in n variables. We consider the AuslanderReiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is tdetermined if every generator x a has a ≤ t. We compute the mu ..."
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Abstract. For t in N n, E.Miller has defined a category of tdetermined modules over the polynomial ring S in n variables. We consider the AuslanderReiten translate, Nt, on the (derived) category of such modules. A monomial ideal I is tdetermined if every generator x a has a ≤ t. We compute the multigraded cohomology and betti spaces of N k t (S/I) for every iterate k, and also the Smodule structure of these cohomology modules. This comprehensively generalizes results of Hochster and Gräbe on local cohomology of StanleyReisner rings. Contents