Here are some pointers to downloads. Numbers such as [mn] refer to the publication list where many pdf downloads are available.

Updated February 10, 2017.

**BROWN, R. **

ABSTRACT: The aim of this article is to explain a philosophy for applying higher dimensional Seifert-van Kampen Theorems,
and how the use of groupoids and strict higher groupoids resolves some foundational anomalies in algebraic topology at
the border between homology and homotopy.
We explain some applications to filtered spaces,
and special cases of them, while a sequel will show the relevance to *n*-cubes of pointed spaces.

This paper has been accepted for publication in the special issue, forthcoming in 2017, of Indagationes Mathematicae honoring the memory of L. E. J. Brouwer.

**BROWN, R. **

Talk for Category Theory 2015, Aveiro, Portugal, June 15-19, 2015.

handout version .4MB

ABSTRACT:

This philosophy involves homotopically defined functors H from (Topological Data) to (Algebraic Data), and conversely "classifying space" functors B from (Algebraic Data) to (Topological Data). These should satisfy:- H is homotopically defined.
- HB is naturally equivalent to 1.
- The Topological Data has a notion of
*connected*. - For all Algebraic Data A, we have BA is connected.
- H preserves certain colimits of connected Topological Data.
- The algebraic data splits into several equivalent kinds, ranging from "broad" to "narrow", related by Dold-Kan type equivalences. The broad data is used for conjecturing and proving theorems; the narrow data is used for calculations and relating to classical methods.

BROWN, R.

Link to an LMS Education blog: Dec. 28, 2014Alexander Grothendieck: some recollections

**BROWN, R. **

"A homotopical approach to algebraic topology via compositions of cubes"

Talk for "7th De Brun Workshop: Homological Perturbation Theory", NUI Ireland, Galway,
1 December 2014 - 5 December 2014

Abstract: Cubical compositions allow the expression in all dimensions of ``algebraic inverses to subdivision'' and so apply to nonabelian local-to-global problems, such as the Seifert-van Kampen Theorem, now groupoidal in dimension 1. The extension to higher dimensions enables replacing the usual singular chains approach to basic homology. In this theory, one needs two types of algebraic model: ``broad'', using cubes, for geometry, conjectures, and proof; and ``narrow'', more like chain complexes, for relating to classical ideas and for calculation. Also the methods apply not to spaces but to filtered spaces..

**BROWN, R. **

"Intuitions for cubical methods in nonabelian algebraic topology"

Talk for "CONSTRUCTIVE MATHEMATICS AND MODELS OF TYPE THEORY", IHP Paris,
02 June 2014 - 06 June 2014

full version 5MB

handout version 3.7MB

Abstract: The talk will start from the 1-dimensional Seifert-van Kampen Theorem for the fundamental group, then groupoid, and so to a use of strict double groupoids for higher versions. These allow for some precise nonabelian calculations of some homotopy types, obtained by a gluing process. Cubical methods are involved because of the ease of writing multiple compositions, leading to "algebraic inverses to subdivision", relevant to higher dimensional local-to-global problems. Also the proofs involve some ideas of 2-dimensional formulae and rewriting. The use of strict multiple groupoids is essential to obtain precise descriptions as colimits and hence precise calculations. Another idea is to use both a "broad" and a "narrow" model of a particular kind of homotopy types, where the broad model is used for conjectures and proofs, while the narrow model is used for calculations and relation to classical methods. The algebraic proof of the equivalence of the two models then gives a powerful tool.

**BROWN, R. and ANTOLIN CAMARENA,O. **

"Corrigendum to ``Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem'', J. Homotopy and Related Structures 1 (2006) 175-183.

Abstract: Omar Antolin Camarena pointed out a gap in the proofs in "Topology and Groupoids", and in the paper cited in the title, of a condition for the Phragmen--Brouwer Property not to hold; this note gives some background and the correction, with a useful theorem on puchouts of groupoids.

Brouwer correction, 6 pages. arXiv/submitted: 0948151

"Crossed sequences, G-groupoids, and Double Groupoids"

In clearing out my office at the University, I came across a first version of work with Chris Spencer, which was rejected by JPAA, and eventually revised and published as two papers. So I thought I would scan it and let people read it if they want! It includes material on homotopies which did not appear until 1987 as part of work with Philip Higgins.

Brown-Spencer early version pdffile 13.8MB, 57 pp

In further clearing I found this first version of paper:

25. (with P.J. HIGGINS),``On the connection between the second
relative homotopy groups of some related spaces'', *Proc.
London Math. Soc.* (3) 36 (1978) 193-212.

We were informed by Frank Adams, but with no specific mathematical comment, that two international authorities and one other referee had felt the paper was too long and should be cut by a third. So rather than try somewhere else, we cut many figures and shortened some arguments and the Introduction.

My feeling is that the first version was better, so I make this available for the reader to judge, if they want to bother, and as an historical record. Actually by this stage the version in all dimensions had been done, and was published in 1981.

Brown-Higgins paper 1, first version pdffile 15.5MB, 57 pp"My friend John Robinson (sculptor), 1935-2007" Talk given to the Probus Club of Llandudno, May 15, 2013. (I am indebted to John's three volume autobiography "From the Beginning Onwards" (Edition Limitee, 2005) for some of the information and pictures. This version of the presentation as added a short biography. )

Presentation file: pdf 25.6 MB

In memoriam Jean-Louis Loday; 1946-2012

Tribute to Jean-Louis Loday ; pdffile 3 pages

**BROWN, R.**

The intuitions for using multiple categories and groupoids in algebraic topology.

Seminar, Durham, June 5, 2006

pdf handout version .17MBI was reminded of this talk by the paper arXiv:1206.5784 "Parallel Transport on Higher Loop Spaces", by Ivan Horozov, and so thought I would include it here.

While in Durham I visited the John Robinson sculptures with Ivan and his wife.
**BROWN, R.**

Calculating homotopy *n*-types using strict higher homotopy groupoids of some spaces with structure: intuitions, results, limitations.

Chicago University

April 20, 2012

Note: This seminar did not have as much time as was expected, and so did not achieve the goals envisaged in the title!
Also there was lots of lively discussion. Aims achieved were to discuss the Anomalies on the first slide, and to explain intuitions for the
homotopy double groupoid of a pair of spaces.

For more on the aims in the title, see the article R. Brown, ``Computing homotopy types using crossed

**BROWN, R.**

Motion, space, knots, and higher dimensional algebra

William J. Spencer Lecture

Kansas State University

Manhattan

April 17, 2012
pdf handout version .32MB

pdf full version 1.2MB

ABSTRACT: The notion of mathematical space is important for encoding data for motion. To understand the structure of various kinds of space we need forms of representation, and in the end some kind of algebra to perform computations. We illustrate these ideas with the Dirac String Trick and a video explaining it, and then move on to use algebra to study the space around a knot. Mathematics has also come up with new conceptual ways of describing way some complicated structures are put together from simple ones, using analogies from across widely different parts of mathematics. A new method in mathematics is "higher dimensional algebra", in which we can have formulae and their manipulation in dimension 2, or higher, rather than the usual restriction essentially to a line, which is desirable for computation, but not necessarily for description. This way of using abstraction for analogy and comparison is one of the powers of mathematics. (This talk is addressed to a general audience.)

**BROWN, R.**

Applications of a nonabelian tensor product of groups

ABSTRACT: This is an introduction to the area with applications to the obstruction to homotopical excision, and the Schur multiplier, and with some indication of the background in higher structures.

Colloquium at the Mathematics Institute, University of Goettingen, May 5, 2011.

Full beamer pdf file Handout beamer pdf file

**BROWN, R.**

Filtered spaces, crossed complexes and cubical higher homotopy groupoids: a new foundation for algebraic topology

ABSTRACT: We discuss why we might need a new, or different, foundation for algebraic topology, and then describe how one Higher Homotopy Seifert-van Kampen Theorem allows for a deduction of old results in algebraic topology, such as the Relative Hurewicz Theorem, and also new results including nonabelian results on second relative homotopy groups, and on homotopy 2-types. The proof however goes through the route of stict cubical higher homotopy groupoids defined for filtered spaces, avoiding the use of singular homology and of simplicia approximation.

Presentation to the Tbilisi Conference on "Homotopy Theory and Non Commutative Geometry" March 28-April 1, 2011

Full beamer pdf (1.37MB) handout version pdf (.45MB)

**BROWN, R.**

Covering morphisms of groupoids, derived modules and a 1-dimensional Relative Hurewicz Theorem

ABSTRACT: We fill a lacuna in the literature by giving a version in dimension 1 of the Relative Hurewicz Theorem, and relate this to abelianisations of groupoids, covering spaces, covering morphisms of groupoids, and Crowell's notion of derived modules.

**BROWN, R.** **and STREET, R.**

Covering morphisms of crossed complexes and of cubical omega-groupoids with connection are closed under tensor product

ABSTRACT: The aim is the theorems of the title and the corollary that the tensor product of two free crossed resolutions of groups or groupoids is also a free crossed resolution of the product group or groupoid. The route to this corollary is through the equivalence of the category of crossed complexes with that of cubical omega-groupoids with connections where the initial definition of the tensor product lies. It is also in the latter category that we are able to apply techniques of dense subcategories to identify the tensor product of covering morphisms as a covering morphism.

pdf arXiv:1009.5609 in math.AT

Cahier. Top. Geom. Diff. Cat. (to appear).

Possible connections between whiskered categories and groupoids, Leibniz algebras, automorphism structures and local-to-global questions

ABSTRACT: We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered R-categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Leibniz algebras, but the commutator theory here does not satisfy the Leibniz identity. We also discuss potential applications and extensions, for example to resolutions of monoids.

Published in J. Homotopy and Related Structures,5(1) (2010) 305-318 pdf ; erratum

**BROWN, R.**

Some strict higher homotopy groupoids: intuitions, examples, applications, prospects.

ABSTRACT: The aim is to show how the idea of `algebraic inverse to subdivision' led to a family of strict higher homotopy groupoids more intuitive and powerful than the earlier relative homotopy groups, through having structure in a full range of dimensions and also the advantages of symmetry and multiple compositions. These structures help not only to understand traditional structures of such homotopy groups, such as actions but can allow specific calculations of some such groups through calculation of richer structures, modelling the n-types. Even richer structures allow calculations of say Whitehead products and new results such as an n-adic Hurewicz theorem.

Presentation for TTT Manchester, July 5, 2010 (1 hr)

full pdf file handout pdf file

**BROWN, R.**

Some strict higher homotopy groupoids: intuitions, examples, applications, prospects.

ABSTRACT: The aim is to show how the idea of `algebraic inverse to subdivision'
led to a family of strict higher homotopy groupoids more intuitive and powerful
than the earlier relative and *n*-adic homotopy groups, through having
structure in a full range of dimensions. The advantage of strict structures
is that they help not only to understand traditional structures of such homotopy
groups, such as actions and Whitehead products, but allow specific calculations
of some such groups through calculation of richer structures.

Presentation for CT2010, Genoa, June 22, 2010. (1/2 hr)

pdf (handout version)

**BROWN, R.**

Crossed modules and the homotopy 2-type of a free loop space

ABSTRACT:The question was asked by Niranjan Ramachandran: how to describe the fundamental groupoid of LX, the free loop space of a space X? We give an answer by assuming X to be the classifying space of a crossed module over a group, and then describe completely a crossed module over a groupoid determining the homotopy 2-type of LX. The method requires detailed information on the monoidal closed structure on the category of crossed complexes.

MSClass:18D15,55Q05,55Q52

Keywords: free loop space, crossed module, crossed complex, closed category, classifying space, higher homotopies.

pdf file revised 17 May, 2010 http://arxiv.org/abs/1003.5617 (revised version May 8, 2010)

**BROWN, R.**,

What is and what should be `Higher dimensional group theory'?

ABSTRACT: The presentation will show, including some knot demos, some of the problems and intuitions which have led to this question, and how certain cubical algebraic structures with partial operations whose domains are given by geometric conditions have been found quite natural for expressing modes of higher dimensional subdivision and composition which are related to long term concerns in algebraic topology.

Seminar at Liverpool University School of Mathematical Sciences, 4 December, 2009

DISCLAIMER: It is not possible in a one hour seminar to answer fully the questions in the title, nor to give adequate references. For further information, do a web search on "Higher dimensional algebra".

pdf file of beamer presentation 2MB pdfhandoutversion .6 MB

**BROWN, R.**,

Moore rectangles on a space form a strict cubical omega-category

ABSTRACT: A question of Jack Morava is answered by generalising the notion of Moore paths to that of Moore hyperrectangles, so obtaining a strict cubical \omega-category. This also has the structure of connections in the sense of Brown and Higgins, but cancellation of connections does not hold.

pdf file arXiv: 0909.2212v2 (19/09/09)

**BROWN, R.**,

Askloster: This page gives access to information on the seminar July 23-26, 2009, and to pdf files of my two (beamer) presentations:

Some intuitions of higher dimensional algebra, and potential applications pdf (2.5MB) pdfhandoutversion .8MB

Category theory, higher dimensional algebra, groupoid atlases: prospective

descriptive tools in theoretical neuroscience
pdf (2.5MB)
pdfhandoutversion
1.2 MB

**MOSA, G.H. **,

Higher dimensional algebroids and crossed
complexes, (1987).

This is now (May, 2009) available through the above link.

`Diagonals for crossed complexes'

ABSTRACT: This is a scan of handwritten notes dated 26/7/92 , giving some calculations of diagoal maps for the fundamental crossed complexes of some 2-dimensional CW-complexes, namely the Klein bottle and the torus.

pdf 5 pages 1.2 MB

**BAIANU, I.C., GLAZEBROOK, J.F., and BROWN, R.**,

`Algebraic topology foundations of supersymmetry and symmetry breaking in quantum field theory and quantum gravity: a review' SIGMA

ABSTRACT: A novel Algebraic Topology approach to Supersymmetry (SUSY) and Symmetry Breaking in Quantum Field and Quantum Gravity theories is presented with a view to developing a wide range of physical applications. These include: controlled nuclear fusion and other nuclear reaction studies in quantum chromodynamics, nonlinear physics at high energy densities, dynamic Jahn-Teller effects, superfluidity, high temperature superconductors, multiple scattering by molecular systems, molecular or atomic paracrystal structures, nanomaterials, ferromagnetism in glassy materials, spin glasses, quantum phase transitions and supergravity. This approach requires a unified conceptual framework that utilizes extended symmetries and quantum groupoid, algebroid and functorial representations of non--Abelian higher dimensional structures pertinent to quantized spacetime topology and state space geometry of quantum operator algebras. Fourier transforms, generalized Fourier--Stieltjes transforms, and duality relations link, respectively, the quantum groups and quantum groupoids with their dual algebraic structures; quantum double constructions are also discussed in this context in relation to quasitriangular, quasiHopf algebras, bialgebroids, Grassmann-Hopf algebras and Higher Dimensional Algebra. On the one hand, this quantum algebraic approach is known to provide solutions to the quantum Yang-Baxter equation. On the other hand, our novel approach to extended quantum symmetries and their associated representations is shown to be relevant to locally covariant General Relativity theories that are consistent with either nonlocal quantum field theories or local bosonic (spin) models with the extended quantum symmetry of entangled, `string-net condensed' (ground) states.

SIGMA (Symmetry, Integrability, and Geometry: Methods and Applications) 79 pp. pdf (to appear) [167]

**09.02 Ronald Brown **

*Memory Evolutive Systems*

ABSTRACT: This is a review of "Memory Evolutive Systems; Hierarchy, Emergence, Cognition"by A Ehresmann and J.P. Vanbremeersch, in series Studies in Multidisciplinarity, 402 pages, 2007 ISBN-13: 978-0-444-52244-3 ISBN-10: 0-444-52244-1 ELSEVIER

to appear in Axiomathes pdf

**09.01 Ronald Brown **

*`Double modules', double categories and groupoids, and a new homotopical
double groupoid*

We give a rather general construction of double categories and so double groupoids from a structure we call a `double module'.

We also give a homotopical construction of a double groupoid from a triad consisting of a space, two subspaces, and a set of base points, under a condition which also implies that this double groupoid contains two second relative homotopy groups.

AMSCLass2000: 18B40,18D05,55E30

KEYWORDS: double categories, double groupoids, crossed modules

Algebraic colimit calculations in homotopy theory using fibred and cofibred categories

ABSTRACT: Higher Homotopy van Kampen Theorems allow the computation as colimits of certain homotopical invariants of glued spaces. One corollary is to describe homotopical excision in critical dimensions in terms of induced modules and crossed modules over groupoids. This paper shows how fibred and cofibred categories give an overall context for discussing and computing such constructions, allowing one result to cover many cases. A useful general result is that the inclusion of a fibre of a fibred category preserves connected colimits. The main homotopical applications are to pairs of spaces with several base points, but we also describe briefly the situation for triads.

arXiv:0809.4192v2 math.AT, math.CT pdf of final version [164]

Exact sequences of fibrations of crossed complexes, homotopy classification of maps, and nonabelian extensions of groups

ABSTRACT: The classifying space of a crossed complex generalises the
construction of Eilenberg-Mac Lane spaces. We show how the theory of fibrations
of crossed complexes allows the analysis of homotopy classes of maps from
a free crossed complex to such a classifying space. This gives results on
the homotopy classification of maps from a CW-complex to the classifying
space of a crossed module and also, more generally, of a crossed complex
whose homotopy groups vanish in dimensions between 1 and *n*. The results
are analogous to those for the obstruction to an abstract kernel in group
extension theory.

Journal of Homotopy and Related Structures 3 (2008) 331-343.pdf file math/0802.4537 [164]

**06.04 R. Brown, I. Morris, J. Shrimpton and C.D. Wensley**

Graphs of morphisms of graphs

ABSTRACT: This is an account for the combinatorially minded reader of various categories of directed and undirected graphs, and their analogies with the category of sets. As an application, the endomorphisms of a graph are in this context not only composable, giving a monoid structure, but also have a notion of adjacency, so that the set of endomorphisms is both a monoid and a graph. We extend Shrimpton's (unpublished) investigations on the morphism digraphs of reflexive digraphs to the undirected case by using an equivalence between a category of reflexive, undirected graphs and the category of reflexive, directed graphs with reversal. In so doing, we emphasise a picture of the elements of an undirected graph, as involving two types of edges with a single vertex, namely `bands' and `loops'. Such edges are distinguished by the behaviour of morphisms with respect to these elements.

Electronic Journal of Combinatorics, A1 of Volume 15(1), 2008. 1-28.

pdf file [163]

Possible connections between whiskered categories and groupoids, many object Lie algebras, automorphism structures and local-to-global questions

ABSTRACT: We define the notion of whiskered categories and groupoids, showing that whiskered groupoids have a commutator theory. So also do whiskered R-categories, thus answering questions of what might be `commutative versions' of these theories. We relate these ideas to the theory of Liebniz algebras, but the commutator theory here does not satisfy the Liebniz identity. We also discuss potential applications and extensions, for example to resolutions of monoids.

http://arxiv.org/abs/0708.1677
version 3 rewritten: 26 July, 2010
pdf

A new higher homotopy groupoid: the fundamental globular $\omega$-groupoid of a filtered space

MSC Classification:18D10, 18G30, 18G50, 20L05, 55N10, 55N25.

KEY WORDS: filtered space, higher homotopy van Kampen theorem, cubical singular
complex, free globular groupoid

ABSTRACT: We show that the graded set of filter homotopy classes rel vertices
of maps from the $n$-globe to a filtered space may be given the structure
of globular $\omega$--groupoid. The proofs use an analogous fundamental cubical
$\omega$--groupoid due to the author and Philip Higgins. This method also
relates the construction to the fundamental crossed complex of a filtered
space, and this relation allows the proof that the crossed complex associated
to the free globular $\omega$-groupoid on one element of dimension $n$ is
the fundamental crossed complex of the $n$-globe.

pdf

Homology, homotopy and applications 10 (2008), No. 1, pp.327-343.

[162]

Normalisation for the fundamental crossed complex of a simplicial set

ABSTRACT. The algebra of crossed complexes is shown to be sufficiently rich
to model the 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.

J. Homotopy and Related Structures, Special Issue devoted to the memory of
Saunders Mac Lane, 2 (2007) 49-79. math.AT/0611728

[152]

Analogy, concepts and methodology, in Mathematics

Summary: A general answer as to why mathematics has lots of applications could be in the form: mathematics has over the centuries developed a language, or even a set of evolving and interacting languages, for expression, description, deduction, verification and calculation. In this context, we discuss the terms in the title, and their potential interaction with teaching and research.

**Three themes in the work of Charles Ehresmann: Local-to-global;
Groupoids; Higher dimensions.**

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.

Expansion of an invited talk given to the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo 8.05.2005-15.05.2005 (Poland). (See the discussion site on the web.)

Published in: Proceedings of the 7th Conference on the Geometry and Topology of Manifolds: The Mathematical Legacy of Charles Ehresmann, Bedlewo (Poland) 8.05.2005-15.05.2005, Banach Centre Publications 76, Institute of Mathematics Polish Academy of Sciences, Warsaw, (2007) 51-63. (math.DG/0602499).

[147]

**Global actions, groupoid atlases, and related topics**

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 developed 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.

Published in: J. Homotopy and Related Structures, 1 (2006) 101-167.

[149]

**Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve
Theorem**

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.

J. Homotopy and Related Structures 1 (2006) 175-183

[151]

*Complex nonlinear biodynamics in categories*

Abstract: A categorical, higher dimensional algebra and generalized topos framework for \L ukasiewicz--Moisil Algebraic--Logic models of nonlinear dynamics in complex functional genomes and cell interactomes is proposed. \L ukasiewicz--Moisil Algebraic--Logic models of neural, genetic and neoplastic cell networks, as well as signaling pathways in cells are formulated in terms of nonlinear dynamic systems with n--state components that allow for the generalization of previous logical models of both genetic activities and neural networks. An algebraic formulation of variable `next-state functions' is extended to a \L ukasiewicz--Moisil Topos with an <>n--valued \L ukasiewicz--Moisil Algebraic Logic subobject classifier description that represents non-random and nonlinear network activities as well as their transformations in developmental processes and carcinogenesis. The unification of the theories of organismic sets, molecular sets and Robert Rosen's (M,R)--systems is also considered here in terms of natural transformations of organismal structures which generate higher dimensional algebras based on consistent axioms, thus avoiding well known logical paradoxes occuring with sets. Quantum bionetworks, such as quantum neural nets and quantum genetic networks, are also discussed and their underlying, noncommutative quantum logics are considered in the context of an emerging Quantum Relational Biology.

Download: 05_13.pdf (revised version 17/10/05).

Published in: Axiomathes 16 (2006) 65-122.

[145]

*Category Theory: an abstract setting for analogy and comparison*

Abstract: `Comparison' and `Analogy' are fundamental aspects of knowledge acquisition. We argue that one of the reasons for the usefulness and importance of Category Theory is that it gives an abstract mathematical setting for analogy and comparison, allowing an analysis of the process of abstracting and relating new concepts. This setting is one of the most important routes for the application of Mathematics to scientific problems. We explore the consequences of this through some examples and thought experiments.

Download: 05_10.pdf

Published in: What is Category Theory? Advanced Studies in Mathematics and Logic, Polimetrica Publisher, Italy, (2006) 257-274.

[146]

*String rewriting for double coset systems*

ABSTRACT: In this paper we show how string rewriting methods can be applied to give a new method of computing double cosets. Previous methods for double cosets were enumerative and thus restricted to finite examples. Our rewriting methods do not suffer this restriction and we present some examples of infinite double coset systems which can now easily be solved using our approach. Even when both enumerative and rewriting techniques are present, our rewriting methods will be competitive because they i) do not require the preliminary calculation of cosets; and ii) as with single coset problems, there are many examples for which rewriting is more effective than enumeration.

Automata provide the means for identifying expressions for normal forms in infinite situations and we show how they may be constructed in this setting. Further, related results on logged string rewriting for monoid presentations are exploited to show how witnesses for the computations can be provided and how information about the subgroups and the relations between them can be extracted. Finally, we discuss how the double coset problem is a special case of the problem of computing induced actions of categories which demonstrates that our rewriting methods are applicable to a much wider class of problems than just the double coset problem.

Download: * 05_07.pdf

Published in: J. Symbolic Comp. 41 (2006) 573-590..

[139]

*Nonabelian algebraic topology*

ABSTRACT: This is a version of a talk presented to the Workshop on n-categories: foundations and applications, IMA Minneapolis, June 7-18, 2004, which gave a sketch of the contents and background to a book with the title `Nonabelian algebraic topology' being written under support of a Leverhulme Emeritus Fellowship (2002-2004) by the speaker and Rafael Sivera (Valencia). The aim is to give in one place a full account of work by R. Brown and P.J. Higgins since the 1970s which defines and applies crossed complexes and cubical higher homotopy groupoids in algebraic topology and group cohomology.

*A homotopy double groupoid of a Hausdorff space II: a van Kampen theorem*

Theory and Applications of Categories, Vol. 14, 2005, No. 9, pp 200-220.

ABSTRACT: This paper is the second in a series exploring the properties of a functor which assigns a homotopy double groupoid with connections to a Hausdorff space. We show that this functor satisfies a version of the van Kampen theorem, and so is a suitable tool for nonabelian, 2-dimensional, local-to-global problems. The methods are analogous to those developed by Brown and Higgins for similar theorems for other higher homotopy groupoids. An integral part of the proof is a detailed discussion of commutative cubes in a double category with connections, and a proof of the key result that any composition of commutative cubes is commutative. These last results have recently been generalised to all dimensions by Philip Higgins.

Keywords: double groupoid, double category, thin structure, connections,
commutative cube, van Kampen theorem.

2000 MSC: 18D05, 20L05, 55Q05, 55Q35

Link to volume 14-2005 of TAC for downloads.

[140]

*Category theory and higher dimensional algebra: potential descriptive
tools in neuroscience *

ABSTRACT: We explain the notion of colimit in category theory as a potential tool for describing structures and their communication,and the notion of higher dimensional algebra as potential yoga for dealing with processes and processes of processes.

This is a development of a lecture given by the first author at the International Conference on Theoretical Neuroscience, Delhi, February, 2003.

Proceedings of the International Conference on Theoretical Neurobiology, Delhi, February 2003, edited by Nandini Singh, National Brain Research Centre, Conference Proceedings 1 (2003) 80-92.

Download pdf file: Here

[136]

**02.26 : BROWN, R. **

*Crossed complexes and homotopy groupoids as non commutative tools for
higher dimensional local-to-global problems *

Abstract: (This is an extended account of a lecture given at the meeting on `Categorical Structures for Descent and Galois Theory ...', Fields Institute, September 23-28, 2002.) We outline the main features of the definitions and applications of crossed complexes and cubical omega-groupoids with connections. These give forms of higher homotopy groupoids, and new views of basic algebraic topology and the cohomology of groups, with the ability to obtain non commutative results and compute homotopy types.

Published in: Fields Institute Communications 43 (2004) 101-130.

Revised version July 2008.

[132]

**02.25 : BROWN, R. & HIGGINS, P.J. **

*The fundamental groupoid of the quotient of a Hausdorff space by a
discontinuous action of a discrete group is the orbit groupoid *

Abstract: The main result is that the fundamental groupoid of the orbit space of a discontinuous action of a discrete group on a Hausdorff space is the orbit groupoid of the fundamental groupoid of the space. This result, which is related to work of Armstrong, is due to Brown and Higgins in 1985 and was published in sections 9 and 10 of Chapter 9 of the first author's book on Topology [Brown:1988]. Since the book is out of print, and the result seems not well known, we now advertise it here.We also describe work of Higgins and of Taylor which makes this result usable for calculations. As an example, we compute the fundamental group of the symmetric square of a space.

This is a somewhat edited, and in one point (on normal closures) corrected, version of those sections of [Brown:1988]. It is also hoped that this publication will allow wider views of this result, for example in topos theory and descent theory. Because of its provenance, this should be read as a graduate text rather than an article. This explains also the inclusion of exercises. It is expected that this material will be part of a new edition of the book.

Published in: This is now part of Topology and Groupoids, Booksurge 2006.

Download: pdf file: 02_25.pdf

02.24
**R. Brown, P.J.Higgins**

*Cubical abelian groups with connections are equivalent to chain complexes
*

**Abstract:** The theorem of the title is shown to be a consequence of
the equivalence between crossed complexes and cubical omega-groupoids with
connections proved by us in [BH3]. We assume the definitions given in [BH3].
Thus this paper is a companion to others, for example [T1], which show that
a deficit of the traditional theory of cubical sets and cubical groups has
been the lack of attention paid to the ``connections'', defined in [BH3].
Indeed the traditional degeneracies of cubical theory identify certain opposite
faces of a cube, unlike the degeneracies of simplicial theory which identify
adjacent faces. The connections allow for a fuller analogy with the methods
available for simplicial theory by giving forms of `degeneracies' which identify
adjacent faces of cubes. They are used in [BH3] and [ABS] to give a definition
of a `commutative cube'.

Part of the interest of these results is that the family of categories equivalent to that of crossed complexes can be regarded as a foundation for a non-abelian approach to algebraic topology and the cohomology of groups. These results show that a form of abelianisation of these categories leads to well-known structures.

Homology, Homotopy and Applications, 5(1) (2003) 49-52.

[131]

**02.22 : BROWN, R. & GLAZEBROOK, J.F**.

*Connections, local subgroupoids, and a holonomy Lie groupoid of a line
bundle gerbe *

Abstract: Our main aim is to associate a holonomy Lie groupoid to the connective structure of an abelian gerbe. The construction has analogies with a procedure for the holonomy Lie groupoid of a foliation, in using a locally Lie groupoid and a globalisation procedure. We show that path connections and 2-holonomy on line bundles may be formulated using the notion of a connection pair on a double category, due to Brown-Spencer, but now formulated in terms of double groupoids using the thin fundamental groupoids introduced by Caetano-Mackaay-Picken. To obtain a locally Lie groupoid to which globalisation applies, we use methods of local subgroupoids as developed by Brown-Icen-Mucuk.

Univ. Iagel. Acta Math. XLI (2003) 283-296 math.DG/0210322

[134]

**02.18** **R. Brown, G.Janelidze**

*Galois theory and a new homotopy double groupoid of a map of spaces*

ABSTRACT: The authors have used generalised Galois Theory to construct a
homotopy double groupoid of a surjective fibration of Kan simplicial sets.
Here we apply this to construct a new homotopy double groupoid of a map of
spaces, which includes constructions by others of a 2-groupoid,
cat^{1}-group or crossed module. An advantage of our construction
is that the double groupoid can give an algebraic model of a foliated
bundle.

Applied Categorical Structures 12 (2004) 63-80.

[129]

**02.09** **R. Brown, E.J. Moore, T. Porter, C.D. Wensley**

*Crossed complexes, and free crossed resolutions for amalgamated sums and
HNN-extensions of groups*

ABSTRACT: The category of crossed complexes gives an algebraic model of
CW-complexes and cellular maps. Free crossed resolutions of groups contain
information on a presentation of the group as well as higher homological
information. We relate this to the problem of calculating non-abelian extensions.
We show how the strong properties of this category allow for the computation
of free crossed resolutions for amalgamated sums and HNN-extensions of groups,
and so obtain computations of higher homotopical syzygies in these cases.

Georgian Mathematical
Journal: Special Issue for Hvedri Inassaradize's 70th birthday 9 (2002)
623--644.

[125]

R. Brown, I. Icen, O. Mucuk

*Holonomy and monodromy groupoids*

ABSTRACT: We outline the construction of the holonomy groupoid of a locally Lie groupoid and the monodromy groupoid of a Lie groupoid. These specialise to the well known holonomy and monodromy groupoids of a foliation, when the groupoid is just an equivalence relation.

Published in Lie Algebroids, Banach Center Publications Institute of Mathematics, Polish Academy of Sciences, Warsaw, 54 (2001) 9-20.

[119]

**00.17 : BROWN, R., BULLEJOS, M. & PORTER,
T.**

*Free crossed resolutions for graph products of groups*

**Abstract:** The category of crossed complexes gives an algebraic model
of the category of $CW$-complexes and cellular maps. We explain basic results
on crossed complexes which allow the computation of free crossed resolutions
of graph products of groups, given free crossed resolutions of the individual
groups.

Proceedings Workshop Korea 2000, J. Mennicke, Moo Ha Woo (eds.) Recent Advances in Group Theory, Heldermann Verlag Research and Exposition in Mathematics 27 (2002) 8--23.

[125]

**02.04****: R. Brown & C.D.Wensley**

*Computation and Homotopical Applications of Induced Crossed Modules *

**Abstract: **We explain how the computation of induced crossed modules
allows the computation of certain homotopy 2-types and, in particular, second
homotopy groups. We discuss various issues involved in computing induced
crossed modules and give some examples and applications.

J. Symbolic Computation 35 (2003) 59-72.

[124]

**00.14 : BROWN, R. & ICEN, I.**

*Towards a 2-dimensional notion of holonomy*

**Abstract: **Previous work (Pradines, 1966, Aof & Brown, 1992) has
given a setting for a holonomy Lie groupoid of a locally Lie groupoid. Here
we develop analogous 2-dimensional notions starting from a locally Lie crossed
module of groupoids. This involves replacing the Ehresmann notion of a local
smooth coadmissible section of a groupoid by a local smooth coadmissible
homotopy (or free derivation) for the crossed module case. The development
also has to use corresponding notions for certain types of double groupoids.
This leads to a holonomy Lie groupoid rather than double groupoid, but one
which involves the 2-dimensional information.

Advances in Mathematics,178 (2003) 141-175.

[123]

**BROWN,R.**

*Constructing free crossed resolutions for some graphs of groups*

A zipped .ps file of my slides for the talk at CT2000, Como, July 2000:

**00.15 : BROWN, R., ICEN, I. & MUCUK, O.**

*Local subgroupoids II: Examples and properties *

**Abstract: **The notion of local subgroupoid as a generalisation of a
local equivalence relation was defined in a previous paper by the first two
authors. Here we use the notion of star path connectivity for a Lie groupoid
to give an important new class of examples, generalising the local equivalence
relation of a foliation, and develop in this new context basic properties
of coherence, due earlier to Rosenthal in the special case. These results
are required for further applications to holonomy and monodromy.

Topology and its Applications 127 (2003) 393-408.

[122]

**00.13 : BROWN, R. & ICEN, I.**

*Homotopies and automorphisms of crossed modules of groupoids*

Abstract: We give a detailed description of the structure of the actor 2-crossed module related to the automorphisms of a crossed module of groupoids. This generalises work of Brown and Gilbert for the case of crossed modules of groups, and part of this is needed for work on 2-dimensional holonomy to be developed elsewhere.

Applied Categorical Structures 11 (2003) 185-206.

[115]

**00.11 : AL-AGL, A.A., BROWN, R. & STEINER,
R.**

*Multiple categories: the equivalence of a globular and a cubical approach
*

Abstract: We show the equivalence of two kinds of strict multiple category, namely: the well known globular $\omega$-categories, the cubical $\omega$-categories with connections.

Published in Advances in Mathematics, 170 (2002) 71-118.

[116]

**99.14 : BROWN, R. & HEYWORTH, A**

*Using rewriting systems to compute left Kan extensions and induced actions
of categories. *(revised version of 98.14)

J Symbolic Computation 29 (2000) 5-31.

**Abstract:** The basic method of rewriting for words in a free monoid
given a monoid presentation is extended to rewriting for paths in a free
category given a `Kan extension presentation'. This is related to work of
Carmody-Walters on the Todd-Coxeter procedure for Kan extensions, but allows
for the output data to be infinite, described by a language. The result also
allows rewrite methods to be applied in a greater range of situations and
examples, in terms of induced actions of monoids, categories, groups or
groupoids.

[107]

**98.24 : Ronald Brown & A. Razak Salleh**

*Free crossed resolutions of groups and presentations of modules of identities
among relations*

**Abstract:** We give formulae for a module presentation of the module
of identities among relations for a presentation of a group, in terms of
information on 0- and 1-combings of the Cayley graph. This is seen as a special
case of extending a partial free crossed resolution of a group given a partial
contracting homotopy of its universal cover.

LMS J Computation and Math
2 (1999) 28-61.

Subj-class: Group Theory; Algebraic Topology

MSC-class: 20F05,20J05,20L10,18G50,57M07

**97.20 : R. Brown**

*Groupoids and crossed objects in algebraic topology*

Notes for lectures at the Summer School in Algebraic Topology, Grenoble,
June 15 - July 5, 1997, (74 pages).

**Abstract:** The notes concentrate on the background, intuition, proof
and applications of the 2-dimensional Van Kampen Theorem (for the fundamental
crossed module of a pair), with sketches of extensions to higher dimensions.One
of the points stressed is how the extension from groups to groupoids leads
to an extension from the abelian homotopy groups to non abelian higher
dimensional generalisations of the fundamental group, as was sought by the
topologists of the early part of this century.

This links with J.H.C. Whitehead's efforts to extend combinatorial group
theory to higher dimensions in terms of combinatorial homotopy theory, and
which analogously motivated his simple homotopy theory. In particular, we
advertise the definition of a homotopy double groupoid of a pair of spaces,
which was defined by Brown and Higgins in 1974, forty-two years after Cech's
definition of higher homotopy groups.

[102]

**93.09 : R. Brown & O. Mucuk**

*The monodromy groupoid of a Lie groupoid*,

Cah. Top. G\'eom. Diff. Cat, 36 (1995) 345-369.

**Abstract:** We show that under general circumstances, the disjoint union
of the universal covers of the stars of a Lie groupoid admits the structure
of a Lie groupoid, such that the projection has a monodromy property on the
extension of local smooth morphisms. This completes a detailed account of
results announced by J Pradines.

[88]

**93.10 : R. Brown & O. Mucuk**

*Foliations, locally Lie groupoids, and holonomy*

Cah. Top. G\'eom. Diff. Cat, 37 (1996) 61-71.

**Abstract:** We show that a paracompact foliated manifold determines
a locally Lie groupoid (or piece of a differentiable groupoid, in the sense
of Pradines). This allows for the construction of holonomy and monodromy
groupoids of a foliation to be seen as particular cases of constructions
for locally Lie groupoids.

[89]

**93.18 : R. Brown**

*Representation and computation for crossed modules*,

Proceedings Cat\'egories, Algebres, Esquisses, Neo-esquisses

Caen, 1994, 6pp.

**Abstract:** This paper discusses the notion of *structural
computation*, and illustrates it with the problem of translating a notion
of tensor product between several equivalent categories, in this case crossed
modules, cat-1-groups, double groupoids with connection, 2-groupoids. In
this case, the easy definition of tensor product is for double groupoids
with connection.

http://www.informatics.bangor.ac.uk/public/math/research/ftp/algtop/rb/caen.ps.gz

**95.18 : R. Brown**

*Homotopy theory, and change of base for groupoids and multiple groupoids*

Applied Categorical Structures, 4 (1996) 175-193.

**Abstract:** This survey article is an expanded version of a talk given
at the European Category Theory Meeting, Tours, July 1995. It shows how the
notion of *change of base*, used in some applications to homotopy theory
of the fundamental groupoid, has surprising higher dimensional analogues,
through the use of certain higher homotopy groupoids with values in forms
of multiple groupoids.

[93]

**93.19 : R. Brown**

*Higher order symmetry of graphs*

Bulletin Irish Math. Soc., 32 (1994) 46-59.

**Abstract:** This paper advertises the joining of two themes: groups
and symmetry; and categorical methods and analogues of set theory. The basic
idea is that the `symmetry object' of a directed graph should be both a group
and a directed graph. From this is obtained the notion of `inner automorphism'
of a directed graph. The work of J. Shrimpton completely describes these.

[85]

**95.08 : Brown, M. Golasinski, T.Porter, &
A.P.Tonks,**

*Spaces of maps into classifying spaces for equivariant crossed
complexes.*

Indag. Math., 8 (1997) 157-172.

**Abstract:** We give an equivariant version of the homotopy theory of
crossed complexes. The applications generalize work on equivariant Eilenberg-Mac
Lane spaces, including the non abelian case of dimension 1, and on local
systems. It also generalizes the theory of equivariant 2-types, due to Moerdijk
and Svensson. Further, we give results not just on the homotopy classification
of maps but also on the homotopy types of certain equivariant function spaces.

Paper 2 is given below.

[94]

**98.16 : R Brown, M Golasinski, T Porter & A Tonks,
**

*Spaces of maps into classifying spaces for equivariant crossed complexes,
II: The general topological group case*, (1998) 29pp, K-theory 23
(2001)129-155.

**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 from Tonks' thesis, 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.

[114]

**95.03 : R. Brown & C.D. Wensley**

*On finite induced crossed modules and the homotopy 2-type of mapping
cones*,

**Abstract:** Results on the finiteness of induced crossed modules are
proved both algebraically and topologically. Using the Van Kampen type theorem
for the fundamental crossed module, applications are given to the 2-types
of mapping cones of classifying spaces of groups. Calculations of the cohomology
classes of some finite crossed modules are given, using crossed complex methods.

Theory and Applications of Categories, Vol. 1, No. 3, 54-71..

[92]

**95.04 : R. Brown & C.D.Wensley**

*Computing crossed modules induced by an inclusion of a normal subgroup,
with applications to homotopy 2-types*,

**Abstract:** We obtain some explicit calculations of crossed Q-modules
induced from a crossed module over a normal subgroup P of Q. By virtue of
theorems of Brown and Higgins, this enables the computation of the homotopy
2-types and second homotopy modules of certain homotopy pushouts of maps
of classifying spaces of discrete groups.

[95]

**97.15 : R. Brown & G. Janelidze**

*Van Kampen theorems for categories of covering morphisms in lextensive
categories*,

J. Pure Applied Algebra, 119 (1997) 255-263.

**Abstract:** We show that lextensive categories are a natural setting
for statements and proofs of the *tautologous* Van Kampen theorem, in
terms of coverings of a space.

http://www.informatics.bangor.ac.uk/public/math/research/ftp/algtop/rb/gj-vkt.ps.gz

**97.16 : R. Brown & G. Janelidze**

*Galois theory of second order covering maps of simplicial sets*,

J. Pure Applied Algebra (135)1 (1999) 83-91.

**Abstract:** We give a version for simplicial sets of a second order
notion of covering map, which bears the same relation to the usual coverings
as do groupoids to sets. The Generalised Galois theory of the second author
yields a classification of such coverings by the action of a certain kind
of double groupoid.

[129]

**95.34 : R.Brown & W.Dreckmann**

*Domains of data and domains of terms in AXIOM*, 23pp, January 1995.

**Abstract:** This paper discusses the advantages of the AXIOM symbolic
computation system, and illustrates them with some AXIOM2.0 code for directed
graphs and free categories and groupoids on directed graphs. In order to
implement the latter, we have to make a distinction between domains of data
and domains of terms, where, for example, the first gives the data for a
finite directed graph, whereas the latter converts this data into an object
of Axiom category DirectedGraphCategory, where the terms range over the objects
and arrows of the directed graph.

**95.17 : R. Brown & T.Porter**

*On the Schreier theory of non abelian extensions : generalisations
and calculations*

Proceedings Royal Irish Acad., 96A (1966) 213-227.

**Abstract:** We use presentations and identities among relations to give
a generalisation of the Schreier theory of nonabelian extensions of groups.
This replaces the usual multiplication table for the extension group by more
efficient, and often geometric, data. The methods utilise crossed modules
and crossed resolutions. This work is related to work of Turing in 1938.

[97]

**91.08 : R. Brown**

*Computing homotopy types using crossed N-cubes of groups*

**Abstract:** This paper is a slightly edited version in LaTeX of the
paper of the same title which appeared in the *Adams Memorial Symposium
on Algebraic Topology*, Vol 1, edited N. Ray and G Walker, Cambridge
University Press, 1992, 187-210.

It gives a survey of some computational uses in homotopy theory of various
structures related to multiple groupoids. In particular, it advertises various
of `higher homotopy groupoids'.

math.AT/0109091

[74]

**98.15 : R. Brown & I. Icen **

*Lie local subgroupoids and their monodromy*

**Abstract:** The notion of local equivalence relation on a topological
space is generalised to that of local subgroupoid. Properties of coherence
are considered. The main result is notions of holonomy and monodromy groupoid
for certain Lie local subgroupoids.

[113]

**00.03 : Ronald Brown & Ilhan Icen**

*Lie local subgroupoids and their holonomy and monodromy Lie
groupoids*

(revised version of 98.15)

**Abstract:** The notion of local equivalence relation on a topological
space is generalised to that of local subgroupoid. The main result is the
construction of the holonomy and monodromy groupoids of certain Lie local
subgroupoids, and the formulation of a monodromy principle on the extendability
of local Lie morphisms.

*Topology and its Applications* 115 (2000) 125-138.

[113]

**93.09 : R. Brown & K.C.H. Mackenzie**

*Determination of a double Lie groupoid by its core diagram*

J. Pure Appl. Algebra, 80 (3): 237--272, 1992.

**Abstract:** In a double groupoid *S* we show that there is a canonical
groupoid structure on the set of those squares of *S* for which the
two source edges are identities; we call this the *core groupoid *of
*S*. The target maps from the core groupoid to the groupoids of horizontal
and vertical edges of *S* are now base--preserving morphisms whose kernels
commute, and we call the diagram consisting of the core groupoid and these
two morphisms the {\em core diagram} of *S*. If *S* is a double
Lie groupoid, and each groupoid structure on *S* satisfies a natural
double form of local triviality, we show that the core diagram determines
*S* and, conversely, that a locally trivial double Lie groupoid may
be constructed from an abstractly given core diagram satisfying some natural
additional conditions.

In the algebraic case, the corresponding result includes the known equivalences
between crossed modules, special double groupoids with special connection
(Brown and Spencer), and cat^{1}-groups (Loday). These cases correspond
to core diagrams for which both target morphisms are (compatibly) split
surjections.

[76]

**90.28 : R. Brown & M. A.-E. S.-F. Aof**

*The holonomy groupoid of a locally topological groupoid*,

Top. Appl., 47 (1992) 97-113.

This is a slightly edited version in LaTeX of the above paper, with updated references. It should be read in conjunction with the papers with Mucuk on monodromy groupoids and on foliations.

**Abstract:** The well known holonomy groupoid of a foliation is here
generalised to the holonomy groupoid of a locally topological groupoid. This
gives an account of an important theorem of J. Pradines (1966) on the
globalisation of locally topological groupoids.

[74]

**93.07 : BROWN, R. & MUCUK, O.**

*Covering groups of non-connected topological groups revisited*

**Abstract:** In general a universal covering of a non connected topological
group need not admit a topological group structure such that the covering
map is a morphism of topological groups. This result is due to R.L. Taylor
(1953). We generalise this result and relate it to the theory of obstructions
to group extensions. The methods use: the equivalence between covering maps
of X and covering groupoids of the fundamental groupoid of X; the equivalence
between group groupoids and crossed modules; and descriptions of cohomology
in terms of crossed complexes.

**Published in:** *Math. Proc Camb. Phil. Soc. *115 (1994) 97-110

[82]

**93.09 : R. Brown,** (Latex, edited version
of 89.07)

*Some problems in non-abelian homotopical and homological algebra*

**Abstract:** This is an edited Latex version of the paper *Some problems
in non-Abelian homotopical and homological algebra*, {\em Homotopy theory
and related topics, Proceedings Kinosaki, 1988}, ed. M. Mimura, Springer
Lecture Notes in Math., 1418 (1990) 105-129.

Part of the motivation of the paper was to give a kind of survey of the area
in terms of what had not been done, thus allowing a lighter touch on what
had been done (the exterior rather than the interior).

Many of the problems came from grant proposals unsupported as either `irrelevant
to mainstream algebraic topology', or because `calculating integral homotopy
types is not a central problem of homotopy theory'. It is hoped now to be
able to see how or if the mainstream is catching up.

Comments on progress on some of the problems since 1989 are given, with an
additional Bibliography.

[62]

To Ronald Brown's Publications page

School of Computer Science home page

Mathematics home page

U. W. Bangor home page