T. Porter
Mathematics Division, School of Informatics,
University of Wales Bangor, Gwynedd LL57 1UT, Wales, U.K.
e-mail: t.porter@bangor.ac.uk

Atlases of Groupoids and Global Actions.

Atlases of Groupoids and Global Actions.

Introduction


(Report on joint work with A.Bak, R.Brown and G.Minian.)


This page introduces and advertises some of the categorical aspects of the new area of groupoid atlases. Groupoid atlases are natural generalisations of A. Bak's global actions, which originated in his study of algebraic analogues of topological spaces. His aim was to develop a setting where the algebraic aspects of algebraic K-theory could be developed without the heavy topological machinery that that subject usually involves. A global action is a structure obtained by patching together `local' group actions. Any group action gives rise to a groupoid in a standard way and applying this to each patch gives the associated groupoid atlas

The initial development of global actions was by Bak, and his students, but has been continued in a series of workshops in a collaboration between Bangor and Bielefeld (Bak, Brown, Minian, Porter).1 The paper will concentrate on the results of this joint work, in particular revealing the links between certain classes of global actions/ groupoid atlases and other structures (complexes of groups, orbihedra). We will discuss the fundamental groupoid of a groupoid atlas and examine its links with coverings. Elementary examples will be given of groupoid atlases and global actions, illustrating their homotopy invariants.

1  Groupoid atlases.

Summary: Definition and examples. Single domain groupoid atlases = multiple groupoids.
The simplest example of a groupoid atlas is one with a single `domain'. Such an object is a family of independant groupoids all having the same set of objects, so is the same as a multiple groupoid. In general a groupoid atlas is a set with a covering by patches, each part of which carries the structure of a single domain global atlas. Certain patches may be specified to be compatible with each other, but there is no requirement for such compatibility for arbitrary overlapping patches. A morphism of global atlases is a mapping between the structures in the obvious way. (These come in two flavours, weak and strong, but the difference between them will be played down in the talk.)

The examples will include the line L with underlying set \mathbb Z and with local patches the pairs {n, n+1} with groupoid Note that no composition takes place between separate local groupoids. We will use this line to give a notion of path and of homotopy and thus to give one approach to the fundamental groupoid of a groupoid atlas.

Another example of a groupoid atlas which is not a global action is obtained by passing to connected components/orbits. Given any groupoid G, we can pass to the underlying equivalence relation corresponding to the partition of the set of objects by connected components. Applying this to each patch of a groupoid atlas G yields Equiv(G). Even if G is a global action, Equiv(G) will usually only be a groupoid atlas.

2  Global actions.

Summary: Examples: Gln(R), A(G, H) for different groups G and families of subgroups, H.

Restricting attention to groupoid atlases in which each local groupoid is the action groupoid corresponding to a group action gives back the original concept of global action as introduced by Bak. These will be discussed using examples only.

Given an associative ring R, the group Gln(R) of non-singular n ×n matrices carries a global action structure given by the action of its subgroups of elementary matrices. This structure, here denoted Gln(R), has p0Gln(R) @ K1(n,R), p1Gln(R) @ K2(n,R) and in general piGln(R) @ Ki+1(n,R). Here Kk(n,R) is the kth-unstable algebraic K-group of R. (This was the original motivating example of Bak, developed for attacking some of the outstanding conjectures in algebraic K-theory.) This example is one of a large family of examples that are fun and interesting in their own right.

The way that a global action extends local information to become global information can be observed from the simplest cases of the A(G, H). The underlying set of this single domain global action is the group G, the actions are by multiplication by elements of the various subgroups in the family H.

If H has just a single group H in it, then the global action is just the collection of orbits, i.e. right cosets. There is no interaction between them.

If H = {H1, H2}, then any H1-orbit intersects with any H2-orbit, so now orbits do interact. How they interact can be very influential on the homotopy properties of the situation.

EXAMPLE 1.

As an example consider the symmetric group S3 generated by a, b with a3 = b2 = (ab)2 = 1 with a denoting the 3-cycle (1 2 3) and b the transposition (1  2). Take H1 = < a > = { 1, (1 2 3), (1 3 2) } yielding two orbits for its left action on S3, H1 and H1 b. Similarly take H2 = < b >, giving local orbits H2, H2 a, H2a2. Any H1-orbit intersects with any H2-orbit, but of course they do not overlap among themselves. This gives an intersection diagram:


Here the three top points represent the < b > cosets, whilst the bottom two points are the < a > cosets. An edge indicates a non-empty intersection. This graph makes it clear that, even in such a simple case, it is possible to find loops and circuits within the global action, following an element through a local orbit and within an intersection crossing to the next orbit, eventually getting back to the starting position. The circuit relates the structure of the single domain global action with the combinatorial information encoded in the presentation.

3  Complexes, groups actions and complexes of groups.

Examples. Link with work of Haefliger on orbihedra, etc.

To each global action or groupoid atlas, there are associated two simplicial complexes, related to the nerve, N(A), and the Vietoris complex, V(A), of the relation given by the covering by components of the underlying set. These are homotopically equivalent so one chooses between them only on the grounds of which presents the data more easily for the application that is being considered. The above intersection diagram is one such nerve complex, the corresponding Vietoris complex is

Here is a summary of some related current projects:

Bak's global actions, Volodin models and the Kapranov-Saito conjecture

(joint with A. Bak (Bielefeld), R. Brown (Bangor), and G. Minian (Buenos Aires)) Bak generalised Volodin's models for the classifying spaces of algebraic K-theory. He developed a well structured, but as yet somewhat undeveloped, abstract homotopy theory in which the homotopy groups of the general linear global action Gln(R) are the unstable algebraic K-groups of R. Using ideas from combinatorial group theory, and homological algebra, we have an outline proof of a conjecture of Kapranov and Saito on combinatorial ways of building the classifying spaces, linked to the higher homotopical syzygies for the Steinberg groups and various of the Stasheff polytopes. (This has been partially supported by British Council/ARC and INTAS grants.)

Homotopical syzygies and presentations of groups

(with R. Brown and C.D. Wensley). This is an extension of the methods developed for the attack on the Kapranov-Saito study above, investigating the way in which higher syzygies can be induced from subpresentations and subgroups.

n-fold Cech derived functors and applications of crossed models for homotopy types in non-cohomological algebra

(joint with N. Inassaridze (Tbilisi), R. Brown and others). Cech derived functors, as introduced by Pirashvili, allow the possibility of using analogues of Cech homology methods in algebraic situations to aid in the calculation of classical derived functors, however they are useful only in low dimensions. N. Inassaridze and myself, with the assistence of a student of Inassaridze, have generalised these to n-fold Cech derived functors, using the technology of simplicial groups and crossed n-cubes that I developed some years ago. We feel we have just scratched the surface of this theory, as yet, and further study is needed. One paper has been written, see publications and preprint listings.
This page was last modified on 26-10-2005. T.P.