[Show abstract][Hide abstract] 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 CW-complexes and geometric
lattices.
Journal of Algebra 06/2014; 439. DOI:10.1016/j.jalgebra.2015.05.007 · 0.60 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: These lectures are an informal elementary introduction to buildings. They are
written for, and by, a non-expert. 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 2-4 firm-up 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 Eilenberg-MacLane spaces and is thus
determined by Khovanov homology. By using the Dold-Thom 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
Jarnik-Besicovitch in the metrical theory of Diophantine approximation are
established for quaternions by applying results on the measure of general `lim
sup' sets.
Mathematical Proceedings of the Cambridge Philosophical Society 09/2011; 157(3). DOI:10.1017/S0305004114000462 · 0.64 Impact Factor
[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 join-semilattices (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 spin-off is a general presentation for the
Renner monoid of an algebraic monoid, which we illustrate in the special case
of the "classical" algebraic monoids.
Proceedings of the London Mathematical Society 04/2011; DOI:10.1112/plms/pds068 · 1.11 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: By gluing together the sides of eight copies of an all-right angled
hyperbolic 6-dimensional polytope, two orientable hyperbolic 6-manifolds with
Euler characteristic -1 are constructed. They are the first known examples of
orientable hyperbolic 6-manifolds having the smallest possible volume.
[Show abstract][Hide abstract] ABSTRACT: We define a homology theory for a certain class of posets equipped with a pre-sheaf of modules. We show that when restricted to Boolean lattices this homology is isomorphic to the homology of the “cube” complex defined by Khovanov.
Journal of Algebra 07/2009; 322(2-322):429-448. DOI:10.1016/j.jalgebra.2009.04.005 · 0.60 Impact Factor
[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.
Advances in Mathematics 12/2008; 223(5-223):1782-1814. DOI:10.1016/j.aim.2009.10.008 · 1.29 Impact Factor
[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 Leray-Serre 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.
Transactions of the American Mathematical Society 08/2008; DOI:10.1090/S0002-9947-2012-05459-6 · 1.12 Impact Factor
[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 spin-off 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.
[Show abstract][Hide abstract] ABSTRACT: By gluing together copies of an all-right angled Coxeter polytope a number of open hyperbolic 6-manifolds with Euler characteristic -1 are constructed. They are the first known examples of hyperbolic 6-manifolds having the smallest possible volume.
Electronic Research Announcements of the American Mathematical Society 11/2004; 11. DOI:10.1090/S1079-6762-05-00145-9 · 0.46 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: The rich theory of Coxeter groups is used to provide an algebraic construction of finite volume hyperbolic n-manifolds. 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 6-dimensions, are given, each of very small volume, and in one case of smallest possible volume.
[Show abstract][Hide abstract] ABSTRACT: The problem of classifying, up to isometry, the orientable spherical and hyperbolic 3-manifolds 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].
Topology and its Applications 05/2001; 138(1-3-138):253-263. DOI:10.1016/j.topol.2003.08.025 · 0.55 Impact Factor
[Show abstract][Hide abstract] ABSTRACT: It is shown that any finitely generated, non-elementary Fuchsian group has among its homomorphic images all but finitely many of the alternating groups An. This settles in the affirmative a long-standing conjecture of Graham Higman.
Journal of Algebra 01/2000; 223(2-223):457-476. DOI:10.1006/jabr.1999.8014 · 0.60 Impact Factor
[Show abstract][Hide abstract] 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 torsion-free and thus the quotient a hyperbolic n-manifold.
[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.