Problems in homotopy theory
Ronnie Brown

Ronnie Brown

Emeritus Professor of Mathematics, Bangor University.

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Contributed page to this site: Nonabelian algebraic topology

My main entry into this area was through writing a book in the 1960’s published as “Elements of Modern Topology”, Mc Graw? Hill, (1968). In trying to give an account of the traditional van Kampen theorem, it became irritating that the usual version did not yield the fundamental group of the circle, which is after all the basic example in algebraic topology. On discovering work of P.J. Higgins on applications of groupoids to group theory, I found a version of the van Kampen theorem for the fundamental groupoid π 1(X,A) on a set A of base points; for the circle one takes A to consist of two points!

A discussion on groupoids with George W. Mackey in 1967 led me to develop a chapter on covering spaces using groupoid theory. In this exposition, a covering map of spaces is modelled by a covering morphism of groupoids, following the use of this idea by Higgins: the theory seemed to me more convenient than the usual exposition in terms of actions.

The overall view arising from the book (now available as “Topology and Groupoids”) seemed to be that all of 1-dimensional homotopy theory was better expressed using groupoids rather than groups. This is also quite intuitive: one does not express a railway timetable in terms of return journeys and changes from one set of return journeys to another!

This raised the question of the potential use of groupoids in higher homotopy theory. Further, it seemed that the proof of the van Kampen Theorem could generalise to higher dimensions if one had the correct homotopical gadget. This led in 1965 to the search for a homotopy double groupoid of a space, and so to repeated but abortive attempts to define such a gadget.

It gradually became clear that the step from dimension 1 to dimension 2 was a considerable one. Part of the background is the realisation that whereas group objects in the category of groups are abelian groups, group objects in the category of groupoids are equivalent to J.H.C. Whitehead’s crossed modules. This suggests the possibility that multiple groupoids could help to fulfil the dreams of the topologists of the early 20th century, to find higher dimensional generalisations of the fundamental group, whose non commutativity was well known to have useful applications in geometry and analysis.

Indeed on this sort of grounds Alexandroff and Hopf persuaded E. Cech to withdraw a paper on higher homotopy groups for the 1932 ICM at Zurich, so that only a small paragraph appeared in the Proceedings.

A fortunate Research Council grant led to a Research Assistantship for Chris Spencer at Bangor in 1971-2, and so to a lot of new results on double groupoids and crossed modules.

In 1974, Philip Higgins visited Bangor to work in this area, and near the end of this visit we did a strategic analysis of the situation as follows:

  • Whitehead’s tricky theorem on free crossed modules, proved by him using transversality and knot theory, was evidence for the possibility of universal properties in 2-dimensional homotopy theory.

  • If a putative 2-dimensional van Kampen theorem was to be any good it should have Whitehead’s theorem as a Corollary.

  • Whitehead’s theorem was about relative homotopy groups.

  • We should therefore try to define a define a homotopy double groupoid for a pair (X,A) of pointed spaces.

  • This immediately led to the idea of forming homotopy classes rel vertices of maps of a square into X which took the edges to A and the vertices to the base point.

  • This worked out and between us we soon had a complete theorem and proof written out.

This paper was finally published in 1978 in the teeth of opposition.

The natural extension of this theory to all dimensions was to use filtered spaces and maps f of a cube into a filtered space such that f mapped the r-skeleton of the cube to the r‘th level of the filtration for all r. The theory was quite tricky to develop, and was published in two papers in 1981, in the JPAA.

One intuition behind this work is to avoid the use of the notion of “free abelian group on the oriented, or ordered, simplices” which is the foundation of the usual singular homology, and which allows the notion of cycle and boundary, and which goes back to Poincaré. Instead a “chain” in dimensions 2 is an element of a relative homotopy group π n(X n,X n1,x) for a filtered space X *, and in dimension 1 is an element of the fundamental groupoid π 1(X 1,X 0). These “chains” have the structure of a crossed complex ΠX * and calculability arises from a Higher Homotopy Seifert-van Kampen Theorem (HHSv KT) which implies freeness of this crossed complex in the case X * is the skeletal filtration of a CW-complex.

In that same year I visited Strasbourg and explained these ideas to Jean-Louis Loday in his seminar. He became convinced that there was a van Kampen Theorem for his n-cat-groups, which we later called catn-groups, of a cube of spaces.

A key overall point is that these higher van Kampen theorems work for structured spaces. Part of the philosophy of this is that to specify a space one needs some kind of data, which will usually have some kind of structure. It is not unreasonable to suppose that the invariants we construct should be formed so as to take account of this structure.

An important aspect of this work is that homotopy groups give but a pale shadow of the actual homotopy type, and any results which help to penetrate the essentially nonabelian homotopy type, and allow new calculations of this, should be seen as of value.