Ronald Brown

Bio: Ronald Brown is an academic researcher from Bangor University. The author has contributed to research in topics: Homotopy & Homotopy group. The author has an hindex of 37, co-authored 128 publications receiving 5402 citations. Previous affiliations of Ronald Brown include University of Hull & University of Wales.

From Groups to Groupoids: a Brief Survey

Abstract: A groupoid should be thought of as a group with many objects, or with many identities. A precise definition is given below. A groupoid with one object is essentially just a group. So the notion of groupoid is an extension of that of groups. It gives an additional convenience, flexibility and range of applications, so that even for purely group-theoretical work, it can be useful to take a path through the world of groupoids. A succinct definition is that a groupoid G is a small category in which every morphism is an isomorphism. Thus G has a set of morphisms, which we shall call just elements of G, a set Ob(G) of objects or vertices, together with functions s, t : G→ Ob(G), i : Ob(G) → G such that si = ti = 1. The functions s, t are sometimes called the source and target maps respectively. If a,b ∈ G and ta = sb, then a product or composite ab exists such that s(ab) = sa, t(ab) = tb. Further, this product is associative; the elements ix, x ∈ Ob(J), act as identities; and each element a has an inverse a with s(a) = ta, t(a) = sa,aa = isa,aaa = ita. An element a is often written as an arrow a : sa→ ta.

Topology and Groupoids

Abstract: Omar Antoĺın Camarena pointed out a gap in the proofs in [BT&G, Bro06] of a condition for the Phragmen–Brouwer Property not to hold; this note gives the correction in terms of a result on a pushout of groupoids, and some additional background.

G-groupoids, crossed modules and the fundamental groupoid of a topological group

Abstract: THEOREN 1. The categories of 8-groupoids and of crossed modules are equivalent. This result was, we understand, known to Verdier in 1965; it was then used by Duskin [6] ; it was discovered independently by us in 1972. The work of Verdier and Duskin is unpublished, we have found that Theorem 1 is little known, and so we hope that this account will prove useful. We shall also extend Theorem 1 to include in Theorem 2 a comparison of homotopy notions for the two cittegories. As an application of Theorem 1, we consider the fundamental groupoid nX of a topological group X. Clemly nX is a B-groupoid ; its associated crossed module has (as does any crossed module) an obstruction class or k-invariant which in this case lies in @(,X, zl(X, e)). We prove in Theorem 3 that this k-invariant is the fist Postnikov invariant of the classifying space Bsx of the singular complex 8X of X. An example of the use of Theorem 3 is the (possibly well known) result that the first Postnikov invariant of Bob) is zero. Duskin was led to his application of Theorem 1 in the theory of group extensions by an interest in Isbell’s principle (a 9’ in an Lsl is an &’ in a &Y). We were led to Theorem 1 as part of a programme for exploiting double groupoids (thab is, groupoid objects in the category of groupoids) in homotopy theory. Basic results on and applications of double groupoids are given in [3], [2] and [4].

The Convenient Setting of Global Analysis

Abstract: Introduction Calculus of smooth mappings Calculus of holomorphic and real analytic mappings Partitions of unity Smoothly realcompact spaces Extensions and liftings of mappings Infinite dimensional manifolds Calculus on infinite dimensional manifolds Infinite dimensional differential geometry Manifolds of mappings Further applications References Index.

Surgery on compact manifolds

Abstract: Preliminaries: Note on conventions Basic homotopy notions Surgery below the middle dimension Appendix: Applications Simple Poincare complexes The main theorem: Statement of results An important special case The even-dimensional case The odd-dimensional case The bounded odd-dimensional case The bounded even-dimensional case Completion of the proof Patterns of application: Manifold structures on Poincare complexes Applications to submanifolds Submanifolds: Other techniques Separating submanifolds Two-sided submanifolds One-sided submanifolds Calculations and applications: Calculations: Surgery obstruction groups Calculations: The surgery obstructions Applications: Free actions on spheres General remarks An extension of the Atiyah-Singer $G$-signature theorem Free actions of $S^1$ Fake projective spaces (real) Fake lens spaces Applications: Free uniform actions on euclidean space Fake tori Polycyclic groups Applications to 4-manifolds Postscript: Further ideas and suggestions: Recent work Function space methods Topological manifolds Poincare embeddings Homotopy and simple homotopy Further calculations Sullivan's results Reformulations of the algebra Rational surgery References Index.

General theory of lie groupoids and lie algebroids

Abstract: This a comprehensive modern account of the theory of Lie groupoids and Lie algebroids, and their importance in differential geometry, in particular their relations with Poisson geometry and general connection theory. It covers much work done since the mid 1980s including the first treatment in book form of Poisson groupoids, Lie bialgebroids and double vector bundles, as well as a revised account of the relations between locally trivial Lie groupoids, Atiyah sequences, and connections in principal bundles. As such, this book will be of great interest to all those concerned with the use of Poisson geometry as a semi-classical limit of quantum geometry, as well as to all those working in or wishing to learn the modern theory of Lie groupoids and Lie algebroids.

Manin Triples for Lie Bialgebroids

Abstract: In his study of Dirac structures, a notion which includes both Poisson structures and closed 2-forms, T. Courant introduced a bracket on the direct sum of vector fields and 1-forms. This bracket does not satisfy the Jacobi identity except on certain subspaces. In this paper we systematize the properties of this bracket in the definition of a Courant algebroid. This structure on a vector bundle $E\rightarrow M$, consists of an antisymmetric bracket on the sections of $E$ whose ``Jacobi anomaly'' has an explicit expression in terms of a bundle map $E\rightarrow TM$ and a field of symmetric bilinear forms on $E$. When $M$ is a point, the definition reduces to that of a Lie algebra carrying an invariant nondegenerate symmetric bilinear form. For any Lie bialgebroid $(A,A^{*})$ over $M$ (a notion defined by Mackenzie and Xu), there is a natural Courant algebroid structure on $A\oplus A^{*}$ which is the Drinfel'd double of a Lie bialgebra when $M$ is a point. Conversely, if $A$ and $A^*$ are complementary isotropic subbundles of a Courant algebroid $E$, closed under the bracket (such a bundle, with dimension half that of $E$, is called a Dirac structure), there is a natural Lie bialgebroid structure on $(A,A^{*})$ whose double is isomorphic to $E$. The theory of Manin triples is thereby extended from Lie algebras to Lie algebroids. Our work gives a new approach to bihamiltonian structures and a new way of combining two Poisson structures to obtain a third one. We also take some tentative steps toward generalizing Drinfel'd's theory of Poisson homogeneous spaces from groups to groupoids.