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17
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 41 (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 19 (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.
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 11 (3 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.
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 8 (1 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.
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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Cited by 3 (3 self)
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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
"... We investigate properties of the set of discrete Morse functions on a simplicial complex as defined by Forman [4]. It is not difficult to see that the pairings of discrete Morse functions of ∆ again form a simplicial complex, the discrete Morse complex of ∆. It turns out that several known results f ..."
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Cited by 2 (0 self)
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We investigate properties of the set of discrete Morse functions on a simplicial complex as defined by Forman [4]. It is not difficult to see that the pairings of discrete Morse functions of ∆ again form a simplicial complex, the discrete Morse complex of ∆. It turns out that several known results from combinatorial topology and enumerative combinatorics, which previously seemed to be unrelated, can be reinterpreted in the setting of these discrete Morse complexes.
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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Cited by 1 (1 self)
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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
RESEARCH STATEMENT
"... I enjoy working on mathematical problems that syncretize ideas from topology, combinatorics and algebra. Usually these arise in subjects such as algebraic topology, topological or algebraic combinatorics and geometric group theory. The theory of arrangements (of hyperplanes, subspaces, points etc.) ..."
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I enjoy working on mathematical problems that syncretize ideas from topology, combinatorics and algebra. Usually these arise in subjects such as algebraic topology, topological or algebraic combinatorics and geometric group theory. The theory of arrangements (of hyperplanes, subspaces, points etc.) and configuration spaces are some such areas in modern mathematics that lie at the intersection of these subjects. My thesis project is about generalizing some results about hyperplane arrangements to smooth manifolds. My thesis work suggests several natural directions for further research, some regarding continuation of the current work and some about the applications to other areas. Here is a summary of my thesis, work in progress and future projects. 1. Arrangements of Submanifolds For basic notions in the theory of hyperplane arrangements and also for foundational results we refer the reader to the classical book of Orlik and Terao [14]. My thesis starts by a generalization of hyperplane arrangements. The results in my thesis are inspired by the work of Deligne on topology of Artin groups [6], Salvetti’s work on the same lines [16], Mike Davis’ work on manifold reflection groups [3, Chapter 10] and finally, Zaslavsky’s work on topological
Arrangements of symmetric products of spaces
, 2003
"... We study the combinatorics and topology of general arrangements of subspaces of the form D + SP n−d (X) in symmetric products SP n (X) where D ∈ SP d (X). Symmetric products SP m (X): = X m /Sm, also known as the spaces of effective “divisors ” of order m, together with their companion spaces of div ..."
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We study the combinatorics and topology of general arrangements of subspaces of the form D + SP n−d (X) in symmetric products SP n (X) where D ∈ SP d (X). Symmetric products SP m (X): = X m /Sm, also known as the spaces of effective “divisors ” of order m, together with their companion spaces of divisors/particles, have been studied from many points of view in numerous papers, see [7] and [21] for the references. In this paper we approach them from the point of view of geometric combinatorics. Using the topological technique of diagrams of spaces along the lines of [34] and [37], we calculate the homology of the union and the complement of these arrangements. As an application we include a computation of the homology of the homotopy end space of the open manifold SP n (Mg,k), where Mg,k is a Riemann surface of genus g punctured at k points, a problem which was originally motivated by the study of commutative (m + k, m)groups [32]. 1 Arrangements of symmetric products