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Problems in the Steenrod algebra
 Bull. London Math. Soc
, 1998
"... This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development ..."
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Cited by 29 (1 self)
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This article contains a collection of results and problems about the Steenrod algebra and related algebras acting on polynomials which nonspecialists in topology may find of some interest. Although there are topological allusions throughout the article, the emphasis is on the algebraic development of the Steenrod algebra and its connections to the various topics indicated below. Contents 1 Historical background 4
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 6 (4 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.
A Topological Calculus for Formal Power Series
, 1997
"... . We propose geometric models for performing various computations with formal power series over a commutative ring, including reciprocation, substitution, reversion, and Lagrange inversion. The models are based on a family of complex BottSamelson varieties which may be realized as manifolds of flag ..."
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. We propose geometric models for performing various computations with formal power series over a commutative ring, including reciprocation, substitution, reversion, and Lagrange inversion. The models are based on a family of complex BottSamelson varieties which may be realized as manifolds of flags satisfying appropriate restrictions. We discuss the relationship of the geometric computations with multiple complex cobordism theory, focussing on the dual of the LandweberNovikov algebra and raising delicate issues concerning the construction of explicit cobordisms. We outline extensions of the calculus to Hurwitz series, appealing to the Fa`a di Bruno algebra of algebraic combinatorics. 1. Introduction The methods of formal power series have permeated algebraic topology since the work of Hirzebruch in the 1950s, and have often centered around cobordism theory. The applications became more specific following the work of Novikov and Quillen on the relationship between formal group laws ...
Ring of Polytopes, Quasisymmetric functions and Fibonacci numbers, arXiv:1002.0810v1
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Izvestiya: Mathematics 65:4 687–704 c○2001 RAS(DoM) and LMS Izvestiya RAN: Ser. Mat. 65:4 49–66 DOI 10.1070/IM2001v065n04ABEH000347
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Flag Manifolds and the Landweber–Novikov Algebra Abstract
, 1998
"... We investigate geometrical interpretations of various structure maps associated with the Landweber–Novikov algebra S ∗ and its integral dual S∗. In particular, we study the coproduct and antipode in S∗, together with the left and right actions of S ∗ on S ∗ which underly the construction of the quan ..."
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We investigate geometrical interpretations of various structure maps associated with the Landweber–Novikov algebra S ∗ and its integral dual S∗. In particular, we study the coproduct and antipode in S∗, together with the left and right actions of S ∗ on S ∗ which underly the construction of the quantum (or Drinfeld) double D(S ∗). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincaré duality with respect to double cobordism theory; these lead directly to our main results for the Landweber– Novikov algebra.
Flag Manifolds and the Landweber{Novikov Algebra
, 1998
"... We investigate geometrical interpretations of various structure maps associated with the Landweber{Novikov algebra S and its integral dual S . In particular, we study the coproduct and antipode in S , together with the left and right actions of S on S which underly the construction of the quant ..."
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We investigate geometrical interpretations of various structure maps associated with the Landweber{Novikov algebra S and its integral dual S . In particular, we study the coproduct and antipode in S , together with the left and right actions of S on S which underly the construction of the quantum (or Drinfeld) double D(S). We set our realizations in the context of double complex cobordism, utilizing certain manifolds of bounded flags which generalize complex projective space and may be canonically expressed as toric varieties. We discuss their cell structure by analogy with the classical Schubert decomposition, and detail the implications for Poincare duality with respect to double cobordism theory; these lead directly to our main results for the Landweber{ Novikov algebra.