Publications (18)10.65 Total impact
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ABSTRACT: We describe a "cellular" approach to the computation of the cohomology of a poset with coefficients in a presheaf. A cellular cochain complex is constructed, described explicitly and shown to compute the cohomology under certain circumstances. The descriptions are refined further for certain classes of posets including the cell posets of regular CWcomplexes and geometric lattices.  [Show abstract] [Hide abstract]
ABSTRACT: These lectures are an informal elementary introduction to buildings. They are written for, and by, a nonexpert. The aim is to get to the definition of a building and feel that it is an entirely natural thing. To maintain the lecture style examples have replaced proofs. The notes at the end indicate where these proofs can be found. Most of what we say has its origins in the work of Jacques Tits, and our account borrows heavily from the books of Abramenko and Brown and of Ronan. Lecture 1 illustrates all the essential features of a building in the context of an example, but without mentioning any building terminology. In principle anyone could read this. Lectures 24 firmup and generalize these specifics: Coxeter groups appear in Lecture 2, chambers systems in Lecture 3 and the definition of a building in Lecture 4. Lecture 5 addresses where buildings come from by describing the first important example: the spherical building of an algebraic group.  [Show abstract] [Hide abstract]
ABSTRACT: We show that the spectrum constructed by Everitt and Turner as a possible Khovanov homotopy type is a product of EilenbergMacLane spaces and is thus determined by Khovanov homology. By using the DoldThom functor it can therefore be obtained from the Khovanov homotopy type constructed by Lipshitz and Sarkar.  [Show abstract] [Hide abstract]
ABSTRACT: We show that the unnormalised Khovanov homology of an oriented link can be identified with the derived functors of the inverse limit. This leads to a homotopy theoretic interpretation of Khovanov homology.  [Show abstract] [Hide abstract]
ABSTRACT: Analogues of the classical theorems of Khintchine, Jarnik and JarnikBesicovitch in the metrical theory of Diophantine approximation are established for quaternions by applying results on the measure of general `lim sup' sets.  [Show abstract] [Hide abstract]
ABSTRACT: This is the second in a series of papers that develops the theory of reflection monoids, motivated by the theory of reflection groups. Reflection monoids were first introduced in arXiv:0812.2789. In this paper we study their presentations as abstract monoids. Along the way we also find general presentations for certain joinsemilattices (as monoids under join) which we interpret for two special classes of examples: the face lattices of convex polytopes and the geometric lattices, particularly the intersection lattices of hyperplane arrangements. Another spinoff is a general presentation for the Renner monoid of an algebraic monoid, which we illustrate in the special case of the "classical" algebraic monoids.  [Show abstract] [Hide abstract]
ABSTRACT: By gluing together the sides of eight copies of an allright angled hyperbolic 6dimensional polytope, two orientable hyperbolic 6manifolds with Euler characteristic 1 are constructed. They are the first known examples of orientable hyperbolic 6manifolds having the smallest possible volume.  [Show abstract] [Hide abstract]
ABSTRACT: We define a homology theory for a certain class of posets equipped with a presheaf of modules. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the “cube” complex defined by Khovanov.  [Show abstract] [Hide abstract]
ABSTRACT: This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders.  [Show abstract] [Hide abstract]
ABSTRACT: The decorated hypercube found in the construction of Khovanov homology for links is an example of a Boolean lattice equipped with a presheaf of modules. One can place this in a wider setting as an example of a coloured poset, that is to say a poset with a unique maximal element equipped with a presheaf of modules. In this paper we initiate the study of a bundle theory for coloured posets, producing for a certain class of base posets a LeraySerre type spectral sequence. We then show how this theory finds application in Khovanov homology by producing a new spectral sequence converging to the Khovanov homology of a given link.  [Show abstract] [Hide abstract]
ABSTRACT: By studying the action of the Weyl group of a simple Lie algebra on its root lattice, we construct torsion free subgroups of small and explicitly determined index in a large infinite class of Coxeter groups. One spinoff is the construction of hyperbolic manifolds of very small volume in up to 8 dimensions. Comment: 18 pages  [Show abstract] [Hide abstract]
ABSTRACT: This is the first of a series of papers in which we initiate and develop the theory of reflection monoids, motivated by the theory of reflection groups. The main results identify a number of important inverse semigroups as reflection monoids, introduce new examples, and determine their orders. 
Article: The smallest hyperbolic 6manifolds
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ABSTRACT: By gluing together copies of an allright angled Coxeter polytope a number of open hyperbolic 6manifolds with Euler characteristic 1 are constructed. They are the first known examples of hyperbolic 6manifolds having the smallest possible volume.  [Show abstract] [Hide abstract]
ABSTRACT: The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic nmanifolds. Combinatorial properties of finite images of these groups can be used to compute the volumes of the resulting manifolds. Three examples, in 4,5 and 6dimensions, are given, each of very small volume, and in one case of smallest possible volume. 
Article: 3manifolds from Platonic solids
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ABSTRACT: The problem of classifying, up to isometry, the orientable spherical and hyperbolic 3manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. This allows us to complete the classification begun by Best [Canad. J. Math. 23 (1971) 451], Lorimer [Pacific J. Math. 156 (1992) 329], Richardson and Rubinstein [Hyperbolic manifolds from a regular polyhedron, Preprint].  [Show abstract] [Hide abstract]
ABSTRACT: It is shown that any finitely generated, nonelementary Fuchsian group has among its homomorphic images all but finitely many of the alternating groups An. This settles in the affirmative a longstanding conjecture of Graham Higman. 
Article: Constructing Hyperbolic Manifolds
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ABSTRACT: In this paper we show how to obtain representations of Coxeter groups acting on H^n to certain classical groups. We determine when the kernel of such a representation is torsionfree and thus the quotient a hyperbolic nmanifold.  [Show abstract] [Hide abstract]
ABSTRACT: We continue our development of the theory of reflection monoi ds by first deriving a presentation for a general reflection monoid from a result of Easdown, East and Fitzgerald for factorizable inverse monoids. We then derive "Popova" style presentations for reflection mon oids built from Boolean hyperplane arrangements and reflection arrangements.
Publication Stats
101  Citations  
10.65  Total Impact Points  
Top Journals
Institutions

20012014

The University of York
 Department of Mathematics
York, England, United Kingdom


2000

Bielefeld University
Bielefeld, North RhineWestphalia, Germany
