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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.
Spaces of polytopes and cobordism of quasitoric manifolds
, 2006
"... Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as th ..."
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Cited by 15 (10 self)
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Our aim is to bring the theory of analogous polytopes to bear on the study of quasitoric manifolds, in the context of stably complex manifolds with compatible torus action. By way of application, we give an explicit construction of a quasitoric representative for every complex cobordism class as the quotient of a free torus action on a real quadratic complete intersection. We suggest a systematic description for omnioriented quasitoric manifolds in terms of combinatorial data, and explain the relationship with nonsingular projective toric varieties (otherwise known as toric manifolds). By expressing the first and third authors ’ approach to the representability of cobordism classes in these terms, we simplify and correct two of their original proofs concerning quotient polytopes; the first relates to framed embeddings in the positive cone, and the second involves modifying the operation of connected sum to take account of orientations. Analogous polytopes provide an informative setting for several of the details.
The polyhedral product functor: a method of computation for momentangle complexes, arrangements and related spaces
, 2008
"... This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the ..."
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Cited by 13 (2 self)
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This article gives a natural decomposition of the suspension of generalized momentangle complexes or partial product spaces which arise as polyhedral product functors described below. In the special case of the complements of certain subspace arrangements, the geometrical decomposition implies the homological decomposition in GoreskyMacPherson [20], Hochster[22], Baskakov [3], Panov [36], and BuchstaberPanov [7]. Since the splitting is geometric, an analogous homological decomposition for a generalized momentangle complex applies for any homology theory. This decomposition gives an additive decomposition for the StanleyReisner ring of a finite simplicial complex and generalizations of certain homotopy theoretic results of Porter [39] and Ganea [19]. The spirit of the work here follows that of DenhamSuciu in [16].
Categorical aspects of toric topology
 Contemporary Mathematics series
"... Abstract. We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces XK, such as momentangle complexes ZK, quasitoric manifolds M, and DavisJanuszkiewicz spaces DJ(K). We first exhibit XK a ..."
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Cited by 3 (0 self)
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Abstract. We argue for the addition of category theory to the toolkit of toric topology, by surveying recent examples and applications. Our case is made in terms of toric spaces XK, such as momentangle complexes ZK, quasitoric manifolds M, and DavisJanuszkiewicz spaces DJ(K). We first exhibit XK as the homotopy colimit of a diagram of spaces over the small category cat(K), whose objects are the faces of a finite simplicial complex K and morphisms their inclusions. Then we study the corresponding cat(K)diagrams in various algebraic Quillen model categories, and interpret their homotopy colimits as algebraic models for XK. Such models encode many standard algebraic invariants, and their existence is assured by the Quillen structure. We provide several illustrative calculations, often over the rationals, including proofs that quasitoric manifolds (and various generalisations) are rationally formal; that the rational Pontrjagin ring of the loop space ΩDJ(K) is isomorphic to the quadratic dual of the StanleyReisner algebra Q[K] for flag complexes K; and that DJ(K) is coformal precisely when K is flag. We conclude by describing algebraic models for the loop space ΩDJ(K) for any complex K, which mimic our previous description as a homotopy colimit of topological monoids. 1.