##
Leverhulme Emeritus Fellowship: Professor Ronnie Brown

August 1, 2002, for two years.

### Background

In the early part of the 20th century, topologists were aware that in the
connected case, the first homology group of a space was the fundamental group
made abelian, and that homology groups existed in all positive dimensions.
Further, the fundamental group gave more information in geometric and analytic
contexts than did the first homology group. They were therefore interested
in seeking higher dimensional versions of the *non abelian* fundamental
group.

E. Cech submitted to the 1932 ICM at Zurich a paper on higher homotopy groups, using maps of spheres. However these groups were quickly proved to be abelian in dimensions > 1, and Cech was asked to withdraw his paper, so that only a short paragraph appeared in the Proceedings. Thus the dream of these topologists seemed to fail, and since then this dream has widely been felt to have been a mirage.

However work of J. H. C. Whitehead in the 1940s introduced the notion of crossed module, using the boundary of the second relative homotopy group of a pair and the action of the fundamental group. He and Mac Lane showed that crossed modules classified (connected) homotopy 2-types. Crossed modules are indeed more complicated than groups, and there is a case for regarding them as `2-dimensional groups'. They are being increasingly used in combinatorial group theory, homological algebra, algebraic topology, and differential geometry.

In the 1967, Brown introduced the fundamental groupoid of a space on a set of base points, and the writing of his 1968 book on topology suggested to him that all of 1-dimensional homotopy theory was better expressed in terms of goupoids rather than groups. This raised the question of the putative use of groupoids in higher homotopy theory, based on squares and cubes rather than paths.

An initial question was the potential extent of an algebraic theory of double groupoids. Relations of certain double groupoids to crossed modules were worked out with C.B. Spencer in the early 1970s. A definition of a homotopy double groupoid of a pair of pointed spaces was made with P.J. Higgins in 1974, and shown to yield a 2-dimensional Van Kampen type theorem which could compute some homotopy 2-types. This work was published in 1978. Extensions of this work to all dimensions were worked out and announced in 1977, 1978, and a full account of the basic theory was published in 1981. Successive years gave further developments and applications, particularly tensor products and homotopies for these higher dimensional objects, and the notion and applications of classifying space. A crucial part of this work was the technically difficult proof of the major properties of the:

### cubical homotopy groupoid of a filtered space

It is this structure which enabled the proof of a Generalised Van Kampen Theorem, based on the slogan of `algebraic inverses to subdivision', and to apply tensor products to topological situations. For example, two key technical and surprising results are that if X_* is a filtered space, then:

- (i) the natural map p: R(X_*) → ρ(X_*) is, under a weak condition, a Kan fibration of cubical sets, and
- (ii) the compositions on R(X_*) are inherited by ρ(X_*).

The proofs use methods of collapsing for subcomplexes of cubes.

New applications of this technology are regularly being discovered. For more background see our web article referred to below, see for example the programme proposed by Peter May on http://www.math.uchicago.edu/~may/ForWeb.pdf , or else do a web search on higher dimensional algebra.

### Aims of the project

The aim is the first monograph to deal with this area of `higher dimensional algebra (HDA)'. It will be at a level appropriate for postgraduate students as well as advanced researchers. The content will mainly be work of the proposer with P.J. Higgins and others since 1970, and which is scattered over various journals and not in a common format. It will start with the intuitive background and the 2-dimensional case, which allows for pictures. Its timeliness reflects the recent explosion of interest in HDA across mathematics, physics and computer science (for more information, do a web search on `Higher Dimensional Algebra'). The monograph will give an exposition of some basic algebraic topology using algebraic tools which closely model the geometry, and give new results and calculations.

The monograph will enable students and researchers to understand these new methods in algebra and topology and the way they relate to basic intuitions of algebraic structures whose operations are defined under geometric conditions. This enables the algebra to more closely reflect the geometry, and has the serendipitous effect of leading to more calculability. The novelty of the ideas for most mathematicians makes it important to treat in full detail this special case of higher dimensional algebra, since it has had and will continue to have a role as a model for future work ([31] has inspired recent work on concurrency). The book will be of the order of 350 pages.

Some of the key papers, such as [31,32], are written in a clear but spare style and need more explanation and pointers to the structure of the proofs, for example of the sophisticated methods of proof using cubical collapsing.

The book will be written with Dr R V Sivera (Valencia) who has been working for many years on a version of this project, but it has been uncompleted because of Brown's responsibilities up to October 1999 in teaching, and administration; his continued responsibility for research and other projects (for example a major EC Public Understanding project in WMY2000); and serious ill health from an acoustic neuroma over the years 2000 and 2001, now well recovered through excellent treatment.

The major work is in organising and retyping where necessary the current work, improving proofs and the structure of the exposition, giving a common style and viewpoint, and providing a full bibliography to this large and expanding area. Historical notes will also be provided.

The first part of the work will cover aspects of the papers [20,21,25,35,43,92,95] referred to below, and some of [50], while the later parts will cover [31,32,48,57,61,71,72,104,120].

The term `higher dimensional algebra' was coined by Brown in 1987 and has caught on, as a web search shows. While the prime aim of the Fellowship will be the monograph, the support would allow the possibility of developing other research in this area which requires secretarial, research assistance, and travel and which has hung fire for reasons stated above.

### Comments May 2003

The title of the book is likely to be `Nonabelian Algebraic Topology', as
this gives a clear idea of aims and philosophy. For recent references, see

http://www.bangor.ac.uk/~mas010/fields-art3.pdf

http://www.bangor.ac.uk/~mas010/noncommut-at.pdf

### Major Publications of Brown related to the area of research

(Discussion and a wider list of publications on the area can be found in the web article `Higher dimensional group theory'.

Elements of modern topology, McGraw Hill, Maidenhead, 1968.

Topology: a geometric account of general topology, homotopy types, and the
fundamental groupoid, Ellis Horwood, Chichester (1988) 460 pp.

(This seems to be still the only text on basic topology to use groupoids fully in the area of 1-dimensional homotopy theory, and the topics of the Van Kampen theorem, covering spaces, orbit spaces, and subgroup theorems in group theory.)

20. (with C.B. SPENCER), `` 𝑮-groupoids, crossed modules and the
fundamental groupoid of a topological group'', — Proc. Kon. Ned. Akad.
v. Wet.} 7 (1976) 296-302.

21. (with C.B. SPENCER), ``Double groupoids and crossed modules'', —Cah. Top. Géom. Diff. 17 (1976) 343-362.

22. (with P.J. HIGGINS), ``Sur les complexes croisés, ω-groupoïdes et 𝑻-complexes'',
— C.R. Acad. Sci. Paris Sèr. A. 285 (1977) 997-999.

23. (with P.J. HIGGINS), ``Sur les complexes croisés d'homotopie associés
a quelques espaces filtrés'', — C.R. Acad. Sci. Paris Sér. A.286
(1978) 91-93.

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.

31. (with P.J. HIGGINS), ``On the algebra of cubes'', —J. Pure Appl. Algebra
21 (1981) 233-260.

32. (with P.J. HIGGINS), ``Colimit theorems for relative homotopy groups'',
—J. Pure Appl. Algebra 22 (1981) 11-41.

33. (with P.J. HIGGINS),``The equivalence
of Ω-groupoids and cubical Τ-complexes'', —Cah. Top. Géom. Diff. 22 (1981) 349-370.

34. (with P.J. HIGGINS), ``The equivalence
of ∞-groupoids and crossed complexes'', —Cah. Top. Géom. Diff. 22 (1981) 371-386.

35. (with J. HUEBSCHMANN), ``Identities among relations'', —in Low dimensional
topology, London Math. Soc. Lecture Note Series 48 (ed. R. Brown and T.L.
Thickstun, Cambridge University Press, 1982), pp. 153-202.

36. ``Higher dimensional group theory'', —in Low dimensional topology, London
Math Soc. Lecture Note Series 48 (ed. R. Brown and T.L. Thickstun, Cambridge
University Press, 1982), pp. 215-238.

42. (with J.-L. LODAY), ``Excision homotopique en basse dimension'', —C.R.
Acad. Sci. Paris Sér. I 298 (1984) 353-356.

43. ``Coproducts of crossed P-modules: applications to second homotopy groups
and to the homology of groups'', —Topology 23 (1984) 337-345.

48. (with P.J. HIGGINS), ``Tensor products and homotopies for ω-groupoids
and crossed complexes'', —J. Pure Appl. Alg. 47 (1987) 1-33.

50. ``From groups to groupoids: a brief survey'', —Bull. London Math. Soc.
19 (1987) 113-134.

57. (with M. GOLASINSKI), ``A model structure for the homotopy theory of
crossed complexes'', —Cah. Top. Géom. Diff. Cat. 30 (1989) 61-82.

59. (with N.D. GILBERT), ``Algebraic models of 3-types and automorphism
structures for crossed modules'', —Proc. London Math. Soc. (3) 59 (1989)
51-73.

61. (with P.J. HIGGINS), ``Crossed complexes and chain complexes with
operators'', —Math. Proc. Camb. Phil. Soc. 107 (1990) 33-57.

71. (with P.J.HIGGINS), ``The classifying space of a crossed complex'', —Math.
Proc. Camb. Phil. Soc. 110 (1991) 95-120.

72. (with H.J.BAUES), ``On the relative homotopy groups of the product filtration
and a formula of Hopf'', —J. Pure Appl. Algebra 89 (1993) 49-61.

74. ``Computing homotopy types using crossed n-cubes of groups'', —Adams Memorial
Symposium on Algebraic Topology, Vol 1, edited N. Ray and G Walker, Cambridge
University Press, 1992, 187-210.

78. ``Out of line'', Royal Institution Proceedings, 64 (1992) 207-243.

92. (with C.D.WENSLEY), ``On finite induced crossed modules and the homotopy
2-type of mapping cones'', —Theory and Applications of Categories 1(3) (1995)
54-71.

95. (with C.D.WENSLEY), ``Computing crossed modules induced by an inclusion
of a normal subgroup, with applications to homotopy 2-types'', —Theory and
Applications of Categories 2 (1996) 3-16.

97. (with T. PORTER), ``On the Schreier theory of non-abelian extensions:
generalisations and computations''. —Proceedings Royal Irish Academy 96A (1996)
213-227.

104. (with A. RAZAK SALLEH), `Free crossed crossed resolutions of groups
and presentations of modules of identities among relations', —LMS J. Comp.
and Math. 2 (1999) 28-61.

105. (with G.H. MOSA), `Double categories, 2-categories, thin structures
and connections', —Theory and Applications of Categories 5 (1999) 163-175.

107. (with ANNE HEYWORTH), `Using rewriting systems to compute left Kan
extensions and induced actions of categories', —J. Symbolic Computation 29
(2000) 5-31.

114. (with M. GOLASINSKI, T.PORTER and A.P.TONKS), ``On function spaces of
equivariant maps and the equivariant homotopy theory of crossed complexes
II: the general topological group case'', —K-Theory 23 (2001)129-155.

115. (with İ. İÇEN ), `Automorphisms of crossed modules of groupoids',
—Applied Categorical Structures (to appear).

116. (with F.A. AL-AGL and R. STEINER), ``Multiple categories: the equivalence
between a globular and cubical approach'', —Advances in Mathematics 170 (2002)
71-118. http://arXiv.org/abs/math.CT/0007009.

118. (with K.HARDIE, H.KAMPS, T. PORTER), ` The homotopy double groupoid
of a Hausdorff space', —Theory and Applications of Categories, 10 (2002) 71-93.

122. (withİ. İÇEN, and O. MUCUK), `Local subgroupoids II: Examples
and properties', —Topology and its Applications 127 (2003) 393-408.

123. (with I.ICEN) `Towards a 2-dimensional notion of holonomy' , —Advances
in Math. (to appear). math.DG/0009082

124. (with C.D.WENSLEY), `Induced crossed modules and computational group
theory', —J. Symbolic Computation 35 (2003) 59-72.

125. (with M. BULLEJOS and T.PORTER),`Crossed complexes, free crossed resolutions
and graph products of 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.

126. (with E. MOORE, T.PORTER, C.D.WENSLEY), `Crossed complexes, and free
crossed resolutions for amalgamated sums and HNN-extensions of groups', —Georgian
Math. J. 9 (2002) 623-644.

129. (with G. JANELIDZE), `A new homotopy double groupoid of a map of spaces',
—Applied Categorical Structures (to appear).

130. (with T. PORTER), `The intuitions of higher dimensional algebra for
the study of structured space', —Revue de Synthése, (to appear 1-2,
2003)`.

131. (with HIGGINS, P.J.), `Cubical abelian groups with connections are
equivalent to chain complexes', —Homology, Homotopy and Applications, 5(1)
(2003) 49-52.

The following may be found on http://www.informatics.bangor.ac.uk/public/mathematics/research/preprints/02/02prep.html

02.22 BROWN, R. & GLAZEBROOK, J.F. Connections, local subgroupoids, and a holonomy Lie groupoid of a line bundle gerbe

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

02.26 BROWN, R. Crossed complexes and homotopy groupoids as non commutative tools for higher dimensional local-to-global problems

COMMENTS: September 11, 2012

The main results of the support by the Leverhulme Trust were the publication of the books:

"Topology and Groupoids" (2006), a revised version of RB's previous topology book;

"Nonabelian algebraic topology:filtered spaces, crossed complexes, cubical homotopy groupoids" (2011)>, R. Brown, P.J. Higgins, R. Sivera, EMS Tracts in Mathematics, Vol 15.

Also in the period from 2002 a number of articles listed in my Publication List were published, and also a number of preprints and presentations.

Last updated 11 September, 2012