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57
Higherdimensional algebra and topological quantum field theory
 Jour. Math. Phys
, 1995
"... For a copy with the handdrawn figures please email ..."
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Cited by 140 (14 self)
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For a copy with the handdrawn figures please email
On the Betti numbers of real varieties
 Proc. Amer. Math. Soc
, 1964
"... The object of this note will be to give an upper bound for the sum of the Betti numbers of a real affine algebraic variety. (Added in proof. Similar results have been obtained by R. Thom [lO].) Let F be a variety in the real Cartesian space Rm, defined by polynomial equations /i(xi, ■ ■ ■ , xm) ..."
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Cited by 132 (0 self)
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The object of this note will be to give an upper bound for the sum of the Betti numbers of a real affine algebraic variety. (Added in proof. Similar results have been obtained by R. Thom [lO].) Let F be a variety in the real Cartesian space Rm, defined by polynomial equations /i(xi, ■ ■ ■ , xm) = 0, ■ • • , fP(xx, ■ ■ ■ , xm) = 0. The qth Betti number of V will mean the rank of the Cech cohomology group Hq(V), using coefficients in some fixed field F. Theorem 2. If each polynomial f, has degree S k, then the sum of the Betti numbers of V is ^k(2k — l)m1. Analogous statements for complex and/or projective varieties will be given at the end. I wish to thank W. May for suggesting this problem to me. Remark A. This is certainly not a best possible estimate. (Compare Remark B.) In the examples which I know, the sum of the Betti
On the connection between the second relative homotopy groups of some related spaces
 PROC. LONDON MATH. SOC
, 1978
"... The title of this paper is chosen to imitate that of the paper by van Kampen [10] which gave some basic computational rules for the fundamental group TTX { Y, £) of a based space (an earlier more special result is due to Seifert [14]). In [1] results more general than van Kampen's were obtained in t ..."
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Cited by 52 (35 self)
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The title of this paper is chosen to imitate that of the paper by van Kampen [10] which gave some basic computational rules for the fundamental group TTX { Y, £) of a based space (an earlier more special result is due to Seifert [14]). In [1] results more general than van Kampen's were obtained in terms of fundamental groupoids. The advantage of the use of groupoids is that one obtains an easy description of the fundamental groupoid of a union of spaces even when the spaces and their intersections are not pathconnected; in such cases, the computation of the fundamental group is greatly simplified by using groupoids. To obtain analogous results in dimension 2 we make essential use of a kind of double groupoid first described in [4]. A major aim is to introduce the homotopy double groupoid p(X, Y,Z) defined for any triple (X, Y,Z) of spaces such that every loop in Z is contractible in Y. The methods of [1] are generalized to give results on p(X, Y,Z). We obtain, as algebraic consequences, results on the second relative homotopy group 7T2{X, Y, t) in the form of computational rules for the crossed module We are grateful to referees for helpful comments.
Higher dimensional algebra V: 2groups
 Theory Appl. Categ
"... A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to tw ..."
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Cited by 25 (2 self)
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A 2group is a ‘categorified ’ version of a group, in which the underlying set G has been replaced by a category and the multiplication map m: G×G → G has been replaced by a functor. Various versions of this notion have already been explored; our goal here is to provide a detailed introduction to two, which we call ‘weak ’ and ‘coherent ’ 2groups. A weak 2group is a weak monoidal category in which every morphism has an inverse and every object x has a ‘weak inverse’: an object y such that x ⊗ y ∼ = 1 ∼ = y ⊗ x. A coherent 2group is a weak 2group in which every object x is equipped with a specified weak inverse ¯x and isomorphisms ix: 1 → x ⊗ ¯x, ex: ¯x ⊗ x → 1 forming an adjunction. We describe 2categories of weak and coherent 2groups and an ‘improvement ’ 2functor that turns weak 2groups into coherent ones, and prove that this 2functor is a 2equivalence of 2categories. We internalize the concept of coherent 2group, which gives a quick way to define Lie 2groups. We give a tour of examples, including the ‘fundamental 2group ’ of a space and various Lie 2groups. We also explain how coherent 2groups can be classified in terms of 3rd cohomology classes in group cohomology. Finally, using this classification, we construct for any connected and simplyconnected compact simple Lie group G a family of 2groups G � ( � ∈ Z) having G as its group of objects and U(1) as the group of automorphisms of its identity object. These 2groups are built using Chern–Simons theory, and are closely related to the Lie 2algebras g � ( � ∈ R) described in a companion paper. 1 1
Crossed Complexes And Homotopy Groupoids As Non Commutative Tools For Higher Dimensional LocalToGlobal Problems
"... We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections. ..."
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Cited by 18 (7 self)
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We outline the main features of the definitions and applications of crossed complexes and cubical #groupoids with connections.
Quillen Closed Model Structures for Sheaves
, 1995
"... In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisim ..."
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Cited by 14 (0 self)
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In this chapter I give a general procedure of transferring closed model structures along adjoint functor pairs. As applications I derive from a global closed model structure on the category of simplicial sheaves closed model structures on the category of sheaves of 2groupoids, the category of bisimplicial sheaves and the category of simplicial sheaves of groupoids. Subsequently, the homotopy theories of these categories are related to the homotopy theory of simplicial sheaves. 1 Introduction There are two ways of trying to generalize the well known closed model structure on the category of simplicial sets to the category of simplicial objects in a Grothendieck topos. One way is to concentrate on the local aspect, and to use the Kanfibrations as a starting point. In [14] Heller showed that for simplicial presheaves there is a local (there called right) closed model structure. In [2] K. Brown showed that for a topological space X the category of "locally fibrant" sheaves of spectra on ...
On open 3manifolds proper homotopy equivalent to geometrically simply connected polyhedra
 Topology Appl
, 1998
"... ..."
Spaces of maps into classifying spaces for equivariant crossed complexes
 Indag. Math. (N.S
, 1997
"... Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using ..."
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Cited by 10 (7 self)
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Abstract. The results of a previous paper on the equivariant homotopy theory of crossed complexes are generalised from the case of a discrete group to general topological groups. The principal new ingredient necessary for this is an analysis of homotopy coherence theory for crossed complexes, using detailed results on the appropriate Eilenberg–Zilber theory, and of its relation to simplicial homotopy coherence. Again, our results give information not just on the homotopy classification of certain equivariant maps, but also on the weak equivariant homotopy type of the corresponding equivariant function spaces. Mathematics Subject Classifications (2001): 55P91, 55U10, 18G55. Key words: equivariant homotopy theory, classifying space, function space, crossed complex.
Computations and homotopical applications of induced crossed modules
 J. Symb. Comp
"... We explain how the computation of induced crossed modules allows the computation of certain homotopy 2types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications. ..."
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Cited by 10 (8 self)
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We explain how the computation of induced crossed modules allows the computation of certain homotopy 2types and, in particular, second homotopy groups. We discuss various issues involved in computing induced crossed modules and give some examples and applications.
About Local Refinement of Tetrahedral Grids based on Bisection
 In 5th International Meshing Roundtable
, 1996
"... . We present a novel approach to the development of a threedimensional simplicial refinement strategies based on ideas that have been shown to be effective for twodimensional refinement. This procedure can be applied with other refinement algorithms with minimal adjustments and can be extended rec ..."
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Cited by 10 (2 self)
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. We present a novel approach to the development of a threedimensional simplicial refinement strategies based on ideas that have been shown to be effective for twodimensional refinement. This procedure can be applied with other refinement algorithms with minimal adjustments and can be extended recursively to even higherdimensional spaces. keywords. mesh refinement, 3D bisection, tetrahedra, adaptivity 1 Introduction As is well known, there are two main steps in local adaptive refinement: the refinement of a subset of elements, and the achievement of the conformity of the mesh. The elements that offer the simplest choice in any dimension are the simplices: triangles in two dimensions, tetrahedra in three dimensions and their analog in even higher dimensions. There are also some algorithms based on bisection or longestside midpoint insertion [1, 2, 3, 4, 5, 6]. 1.1 Notation and definitions Definition 1 (nsimplex) Suppose X0 ; X1 ; : : : ; Xn are n + 1 points such that the set of ve...