Showing papers by "Ronald Brown published in 2006"
Book•
24 Feb 2006
TL;DR: Camarena as discussed by the authors 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.
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.
269 citations
Posted Content•
TL;DR: A Bak developed a combinatorial approach to higher $K$-theory, in which control is kept of the elementary operations involved, through paths and ''paths of paths' in what he called a global action as mentioned in this paper.
Abstract: A Bak developed a combinatorial approach to higher $K$-theory, in which control is kept of the elementary operations involved, through paths and `paths of paths' in what he called a global action The homotopy theory of these was developed by G Minian R Brown and T Porter developed applications to identities among relations for groups, and also the extension to groupoid atlases This paper is intended as an introduction tothis circle of ideas, and so to give a basis for exploration and development of this area
21 citations
Journal Article•
TL;DR: In this article, the homotopy theory of global actions is introduced, which one obtains naturally from the notion of path of elementary operations, and the concept of groupoid atlas plays a clarifying role.
Abstract: Global actions were introduced by A. Bak to give a combinatorial approach to higher K-theory, in which control is kept of the elementary operations through paths and paths of paths. This paper is intended as an introduction to this circle of ideas, including the homotopy theory of global actions, which one obtains naturally from the notion of path of elementary operations. Emphasis is placed on developing examples taken from combinatorial group theory, as well as K-theory. The concept of groupoid atlas plays a clarifying role.
13 citations
Posted Content•
TL;DR: In this article, van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids are discussed in terms of the themes of the title.
Abstract: This paper illustrates the themes of the title in terms of: van Kampen type theorems for the fundamental groupoid; holonomy and monodromy groupoids; and higher homotopy groupoids. Interaction with work of the writer is explored.
6 citations
Posted Content•
TL;DR: Crossed complexes have been shown to have an algebraic rich to model the geometric inductive definition of simplices as discussed by the authors, and they have been used to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex.
Abstract: Crossed complexes are shown to have an algebra suciently rich to model the geometric inductive definition of simplices, and so to give a purely algebraic proof of the Homotopy Addition Lemma (HAL) for the boundary of a simplex This leads to the fundamental crossed complex of a simplicial set The main result is a normalisation theorem for this fundamental crossed complex, analogous to the usual theorem for simplicial abelian groups, but more complicated to set up and prove, because of the complications of the HAL and of the notion of homotopies for crossed complexes We start with some historical background, and give a survey of the required basic facts on crossed complexes
6 citations
TL;DR: This paper shows how string rewriting methods can be applied to give a new method of computing double cosets and discusses how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that the rewriting methods are applicable to a much wider class of problems than just the double Coset problem.
3 citations
Posted Content•
TL;DR: In this paper, a proof of the Jordan Curve Theorem was presented, which relates it to the Phragmen-Brouwer Property and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points.
Abstract: We publicise a proof of the Jordan Curve Theorem which relates it to the Phragmen-Brouwer Property, and whose proof uses the van Kampen theorem for the fundamental groupoid on a set of base points.
3 citations
01 Jan 2006
TL;DR: Brown and Porter as mentioned in this paper developed a combinatorial approach to higher K-theory, in which control is kept of the elementary operations involved, through paths and "paths of paths" in what he called a global action.
Abstract: A. Bak developed a combinatorial approach to higher K-theory, in which control is kept of the elementary operations involved, through paths and ‘paths of paths’ in what he called a global action. The homotopy theory of these was developed by G. Minian. R. Brown and T. Porter developed applications to identities among relations for groups, and also the extension to groupoid atlases. This paper is intended as an introduction to this circle of ideas, and so to give a basis for exploration and development of this area.
1 citations