last updated January 8, 2001

INTRODUCTION: This is a slightly edited and updated version of the Final Report approved by INTAS, ommitting some adminstrative details not of general relevance.

- TITLE: Algebraic K-theory, groups and categories
- REF: INTAS93-436 ext
- PROJECT COORDINATOR: Professor R. Brown
- PERIOD COVERED: April, 1997 to February, 2000

The origin of this project was the amalgamation in 1995 of two separate proposals for INTAS support in the areas of Algebraic K-theory from A. Bak at Bielefeld, and of Categorical Methods in Algebraic Homotopy and related topics from R. Brown at Bangor, in the general context of Grothendieck's programme in Galois Theory, homotopical algebra, and multiple categories. The INTAS Scientific Committee ruled that these proposals should be amalgamated. The accepted proposal was extended in 1997 and this is the report on the extension.

The agreed title of the joint proposal `Algebraic K-theory, groups and
categories' indicates well the variety of interconnections and analogies
which were envisaged. `Algebraic K-theory' is an area which has been notable
from the start for its interactions and the problems it has produced. `Groups'
occur as algebraic groups, classical groups, homology groups, homotopy groups,
Galois groups, abstract groups, K-groups, and in many other ways. Further
the Bangor scientific programme has long investigated and developed higher
dimensional analogues of groups, including crossed modules,
cat^{1}-groups, crossed complexes, and various forms of multiple
groupoids.

In all these areas categorical methods are vital, both for guiding the theory as well as achieving specific calculations. The interaction across methods and techniques has been fully vindicated.

In each of the following sections, the references and links are to the Annexe pages with the publications for each group.

The support of INTAS has been valuable for all the groups, both in terms of extended contact and in terms of support for mathematical activity in the NIS. In particular, some of the NIS supported members would have found it difficult to keep in mathematical research without this support.

The amalgamation has been a notable success. In particular, members from the Bak proposal and the Brown proposal are collaborating in ways not originally envisaged.

As examples of this we mention:

(i) the appointment of T. Porter (Bangor) to the Editorial Board of the journal
*K-theory*, of which Bak is Managing Editor, with the aim of extending
publication into homotopical algebra and its applications,

(ii) a successful British Council/ARC supported collaboration between Bangor
and Bielefeld, including a number of visits both ways and several workshops
on `Global actions and algebraic homotopy', which invited members from teams
in the original proposals of both Bangor and Bielefeld, and had external
participants,

(iii) a successful INTAS proposal `Algebraic homotopy, Galois theory and
Descent' (Bangor, with Coimbra (Portugal) and the Georgian Academy of Sciences),

(iv) a successful DFG/RFBR proposal `Structure of classical-like groups over
rings, nonabelian K-theory, and algebraic homotopy theory' (Bielefeld with
St. Petersburg State University).

Bak's notion of `Global action' has been exploited by members of St.Petersburg State University, and has been developed in a broader context in collaboration with Bangor. `Higher dimensional algebra' as developed at Bangor has been exploited by workers at Tbilisi, who have themselves developed a new range of categorical techniques related to Galois theory and are incorporating into their work nonabelian aspects of the theory of global actions..

The joint Bangor/Bielefeld workshops have been notable for the range of discussion and the free exchange of ideas.

The group at Bangor working on mathematics related to the INTAS project consists of R. Brown (as Project Coordinator), T. Porter, C.D. Wensley, and research students I. Içen, M. Alp, A. Mutlu, Anne Heyworth, M.A.Kadir, Emma Moore. Prof. L. A. Lambe (Rutgers and Stockholm) has also advised on symbolic computation, under other support, and he was appointed Honorary Professor at Bangor in 1996. In the year 2000, M.V. Lawson (Bangor) has started a collaboration with A. Patchkoria (Tbilisi) on inverse monoids and simplicial methods, and so some of Lawson's work and the theses of his students have been added to the publication report. His work is becoming more related to the overall programme because of the relations between inverse semigroups and ordered groupoids, and the influence of homotopical methods in inverse semigroup theory.

As a result of this INTAS project, an extensive collaboration has been developed between Bangor and Bielefeld supported additionally from 1998 by a British Council/ARC Grant. This has led to a number of visits in both directions for research discussions and workshops, and the development of joint research on `Global actions and groupoid atlases', in which a long paper is in preparation [5]. This paper gives a broader basis and also a detailed expository account of this new area, which has applications to unstable higher K-theory and to combinatorial group theory, particularly identities among relations.

Work at Bangor over many years has investigated the extension of the notion of abstract group from group to groupoid and to multiple groupoid, where the latter is viewed as a form of `higher dimensional group'. This has led to new results, new calculations, new constructions and new viewpoints in algebraic topology, cohomology theory, group theory and differential topology. The programme is related to Grothendiek's programme of non abelian methods in homological algebra, and to recent increasing use of n-categories, for example in theoretical physics and in computer science. A recent aspect is the use of current tools of symbolic computation for examples and experimentation.

Considerable progress has been made in utilising the non linear methods of
crossed modules and crossed complexes, notably in:

(i) recent work on applying crossed complexes to compute algebraically the
module of identities among relations for group presentations
[37,38] and for the fundamental groupoid of a
graph of groups [21,59];

(ii) work of Brown and Wensley giving finiteness theorems and a range of
determinations and computations of induced crossed modules (a construction
of Brown and Higgins, 1978), with applications to homotopy 2-types
[21];

(iii) work of Brown, Golasi\'nski (Toru\'n), Porter, Tonks applying crossed
complexes and homotopy coherence (Cordier and Porter
[27]) to equivariant homotopy theory, obtaining
results on function spaces of equivariant maps out of reach of previous methods
[9,10];

(iv) the application of crossed modules and double groupoids to second order
holonomy, in Içen's thesis [14];

(v) a paper of Brown with Janelidze (Tbilisi), applying the latter's generalised
Galois theory to a general Van Kampen theorem in lextensive categories
[16], and further work on second order covering
maps of simplicial sets [17];

(vi) work in Ehler's thesis on simplicial groupoids (papers by Ehlers and
Porter [28,29]);

(vii) work in Arvasi's thesis (Arvasi and Porter
[3,4]) on higher Peiffer identities in commutative
algebras, using Carasco and Cegarra's notion of *hypercrossed complex*
which generalised 1978 work of Bangor student Ashley on a non Abelian Dold-Kan
theorem;

(viii) Porter's paper on TQFTs [56], which uses
simplicial groups to generalise to all dimensions low dimensional work of
Yetter;

(ix) work of Brown and Porter applying crossed resolutions to recast in modern
form work by Turing (1938) on non-abelian extensions and identities among
relations, and apply it to computations(cf [21]);

(x) work of A. Mutlu in his thesis on higher order Peiffer operations, resulting
in the publications [50-55].

We should also mention that the notion of *non abelian tensor product
of groups* found by Brown and Loday in 1987 continues to have a wide
range of applications and extensions, and is applied by N.Inassaridze of
the Tbilisi group. A bibliography of 74 papers on this tensor product may
be found on

` http://www.bangor.ac.uk/~mas010/nonabtens.html`, including papers
by members of the Tbilisi group.

Closely related to the above is work of Porter on abstract homotopy theory,
homotopy coherence, proper homotopy theory, and shape theory. A paper with
J.-M. Cordier (Amiens) [27] (in the * Trans.
Amer. Math. Soc.*) on homotopy coherence has appeared, and has had
substantial applications as mentioned above (e.g. (iii)). Another application
is in the thesis of M.A. Kadir, which gives fundamental coherence results
for Cech and Vietoris complexes, and hypercoverings. Porter has written a
substantial survey article on proper homotopy theory for the *Handbook
of Algebraic Topology*, ed. I. M. James, and a book with H. K. Kamps
(Hagen) on `Abstract homotopy theory', (World Scientific) has been published.
The work with Kamps, Kieboom, and Hardie continues with the development of
double groupoid methods in homotopy theory.

Wensley has worked extensively with Brown on the theory and calculation of induced crossed modules, obtaining new determinations, unobtainable by other methods, of homotopy 2-types of mapping cones(published in 1995-6). With their joint student, M. Alp, he has produced a substantial GAP package for computation of crossed modules which has been accepted by the GAP Council as a share package [58]. Current work is with research student Emma Moore on GAP code for normal forms in the fundamental groupoid of a graph of groups, and the construction of free crossed resolutions for this fundamental groupoid in terms of the free crossed resolutions of the individual groups [21,59].

Other work at Bangor on symbolic computation has involved initially the package AXIOM, and collaborations with W. Dreckmann and Prof. L. Lambe. This led to substantial work of research student Anne Heyworth on generalising rewriting theory to left Kan extensions [10] and to a variety of applications of this and of Gröbner bases [30-36], including the guidance system for a mechanical excavator [26]. Most of this work is done in the Computer Algebra System GAP. One aim of this work is the computation of free crossed resolutions of a group from a presentation: a new algorithm for this is presented in the paper with Razak Salleh [22], thus solving a problem going back to Reidemeister (1933) and this algorithm has been implemented [37,38]. A GAP package `IDREL' is in preparation by Heyworth and Wensley and planned to be submitted in 2001.

An extension of the important notion of local equivalence relation has been
obtained with the notion of *local subgroupoid*
[12,13,14]. The main results use delicate previous
work on holonomy and monodromy groupoids. It is expected that this work will
give a unification of ideas of holonomy from foliations and from bundle theory,
which have previously been unrelated, despite the same name and some related
intuitions. The main point is that earlier work on holonomy based on ideas
of J. Pradines allows for an algebraic expression of `iteration of local
procedures'. This work is part of the overall programme and relied on funds
from the Bangor allocation to support Dr Içen's visit. Work on higher
order holonomy and monodromy is underway with Içen.

The paper with Al-Agl and Steiner [1] solves
a 10 year old problem on the equivalence between two notions of (strict)
multiple category, and so allows for useful descriptions of tensor products
and internal homs for globular w-categories. This
work has recently been applied in concurrency theory in computer science.
For a general survey of work on `Higher dimensional group theory', see the
web article

` http://www.bangor.ac.uk/~mas010/hdaweb2.html`.

Professors Mikhalev and Artamonov visited Bangor for one week in February, 2000, and this visit revealed a number of possibilities for future work, particularly in the fields of Gröbner bases, and of identities among relations.

This INTAS programme also led to an INTAS grant *Descent Theory and its
Higher Dimensional Analogues* involving Bangor, Coimbra and Tbilisi,
coordinated by T.Porter from Bangor, which develops ideas of descent, Galois
Theory, and homotopical algebra .

**Interactions**

The main interaction of Bangor is with Tbilisi and Bielefeld.

Joint papers of Brown and Janelidze on coverings and Van Kampen theorems and on extensions of Galois theory have been published (the latest in 1999). This continues the original submission on the extension of Grothendieck's programme in Galois theory.

The extension of the project allowed for the development and planning of
increased and important interactions between these overall areas of research,
with regard to the application of categorical and computational methods in
K-theory, and in particular the more general application of methods of the
theme of *global actions* developed by Bak at Bielefeld. This INTAS
Project has led to a successful Bangor/ Bielefeld collaboration supported
by the British Council/ARC, with visits to Bielefeld in December, 1997, June
1998, and planned in April, 1999 and later, and from Bielefeld to Bangor
in December 1997, and January 1999. A series of joint papers has been planned,
leading from global actions to a new concept of *groupoid atlas* which
seems more suited to homotopy questions.

The Bielefeld group has cooperated closely with members of the groups at St. Petersburg State University, Steklov Institute, Moscow State University, and Bangor University and is beginning cooperation with the Mathematical Institute of the Georgian Academy of Sciences. Members from all of the universities and institutes in the INTAS project have made research visits to Bielefeld and 5 members (Izhboldin, Merkurjev, Nenashev, Panin, and Vavilov) have been or are currently Humboldt Fellows in Bielefeld. The cooperation with St. Petersburg State University has produced 3 joint articles [5], [6], and [7] and others are being written. There is one joint article with Bangor [8] and others are being written as well. Furthermore, work at Moscow State University overlaps with joint work between Bielefeld University and St. Petersburg State University. A. Nenashev from the Steklov group has been for over a year in Bielefeld and has cooperated with A. Bak and members of the Bangor group who made several short research visits to Bielefeld during the past year. Members of the Bielefeld and Bangor group are cooperating in a joint British-German research project centered around global actions.

The principal activities in the Bielefeld group have been centered around
global actions [3], [4], [9], [10], [17],
[18], dimension theory [1], [7], [22],
the structure of classical-like groups [5],
[7], [11], [12], [14], [15], [22], Hermitian K-theory
[11], [12], [29], and K_{1} and
K_{2} of exact categories [23] - [28].

The joint papers [5], [6] and [7] with the St. Petersburg State University group have been commented on already in the report on that group. The results in [7] overlap with those in [23] (of the Moscow State University references). Nenashev's articles are discussed in the report on the Steklov Institute group.

Global actions are the algebraic counterpart of topological spaces. Putting a global action structure on an algebraic object such as a group allows one to construct paths in the objects and to develop in a classical way a homotopy theory of the objects. The papers [3], [4],[8], and [9] develop the foundations of the subject and [10] provides a completely algebraic construction of algebraic K-theory using global actions. The papers [17] and [18] give a model categorical account in the tradition of Quillen and Baues of the homotopy theory of global actions and simplicial complexes.

The papers [7], [15] and [22] develop a notion of dimension in categories and apply it to determining the structure of group valued functors on categories with dimension. The papers [7] and [22] have their focus on the general linear group and the paper [15] on the general quadratic group.

The paper [29] provides foundations for the K-theory of not necessarily even Hermitian forms. The articles [11] and [12] establish basic results for this theory.

Several members of the Moscow State University group have made research visits to Bielefeld. Below is a summary of their research output.

A. V. Mikhalev and coworkers obtained a full solution [1], [2], [4], [6] of the Riesz-Radon problem (1908) for integral representations of a Radon measure on an arbitrary Hausdorff space. Results on Frobenius type theorems for semilinear mappings of matrices over skew fields were established in [5]. Very recently, results giving a description of universal central extensions of matrix Lie algebras were published in [39] and a book [40] `Differential and Difference Dimension Polynomials' was published by Kluwer.

V. A. Artamonov has carried out extensive work on quantum polynomials in [7] - [16]. The paper [17] provides a useful survey of recent results on quantum polynominals and their applications to K-theory and quantum groups and [10] a detailed survey of quantum polynomials and their role in noncommutative algebra including their K-theory and relations with Hopf algebras and noncommutative geometry. The paper [16] shows that if a general quantum polynomial ring is Morita equivalent to a quantum polynomial ring then the rings are isomorphic and solves the Zarisky problem for quantum polynomials. The paper [15] written jointly with R. Wisbauer determines all automorphisms of general quantum polynomial rings and finds invariants and trace maps. The article [12] written jointly with P. M. Cohn shows that the division ring of a coordinate ring of a quantum plane has the property that a centralizer of any nonconstant is commutative and finds generators of the automorphism group of this division ring.

The article [14] gives a necessary and sufficient condition for the triviality of the center of division rings of coordinate rings of quantum spaces.

The paper [13] surveys new results on division rings of coordinate rings of quantum affine spaces.

Very recently, a classification of automorphisms of division rings of quantum rational functions was given in [42] and a survey of recent results on identities in various classes of algebras was provided in [43].

Y. P. Solovjev and coworkers have carried out work on elliptic functions and Feinman integrals in their publications [17] - [21]. The article [17] presents a generalization of perturbation theory with convergent series for Feinman integrals. [17] - [18] taken together provide a new construction of Hermitian K-theory based on a root system. The articles [2] and [21] supply a new method of approximative calculation of Euclidean functional integrals with arbitrary accuracy.

I. Z. Golubchik has conducted research on the Schreier- van der Waerden problem [22] and on the structure of linear groups over P. I. and related rings [23] - [25]. This has relations to work carried out by A. Bak and A. Stepanov and reported on in the Bielefeld University and St. Petersburg State University groups. The article [22] provides a complete solution of the Schreier-van der Waerden problem on determining all isomorphisms of projective and linear groups over arbitrary associative rings. The paper [23] gives a description of normal subgroups of linear groups over P. I. and weakly Noetherian rings and the paper [25] establishes analogs of this result for groups of Lie type.

A. A. Mikhalev and coworkers have done extensive work on noncommutative algebras, free algebras, Lie algebras, and Leibnitz algebras. The paper [26] describes algorithms for symbolic computation in Lie superalgebras. The paper [27] surveys results on orbits of elements of free groups and algebras under the actions of automorphism groups. The article [29] proves that test elements of a free Lie algebra are elements not contained in proper retracts. The article [34] characterizes test elements in free algebra satisfying the Artin-Schreier property. The paper [31] shows that the variety of Leibnitz algebras has the property of differential separability for subalgebras, that the Jacobian conjecture is true for free Leibnitz algebras and that free Leibnitz algebras are finitely solvable. The papers [32], [33], [36] and [37] obtain algorithms for standard bases of ideals in various algebras.

G. Janelidze

Categorical Galois theory (called CGT below for short) was developed by G. Janelidze in 1984-90, and one of the major objectives of this project was to investigate its various connections with higher-dimensional homotopical algebra developed by R. Brown, T. Porter and other members of the Bangor team. Since CGT adequately extends Galois theory of commutative rings (see [6] - [9], [30], [32], [33], [41]) and the theory of central extensions of groups and more general algebraic structures ([30], [36], [42]), those connections should help to realize our extended version of the Grothendieck program, which is supposed to provide a unified foundation not only to algebraic geometry and algebraic topology, but also to the commutator/homology theory of ``group-like" algebraic structures. The two most important results in this area of Bangor-Tbilisi collaboration are described in [3] and [4]. The first of them is a new extension of the Van Kampen theorem, based on Grothendieck's Descent theory: it turns out to be a consequence of the so-called lextensivity property of the category of topological spaces, which simplifies the usual complicated form of the descent data. The second one applies CGT to the adjunction between simplicial sets and groupoids; the resulting second order covering maps of simplicial sets are classified by the internal actions of a new double groupoid, which turned out to a ``many-object version" of certain known constructions, notably of Quillen and Loday. Recently, the geometrical description of that double groupoid was also obtained ([5]). Independently of that, many new results in CGT, its non-homotopical examples, and in related areas of category theory and categorical algebra have been obtained in collaboration with colleagues from Australia, Canada, France, Italy, Hungary, USA, and Portugal. Among those are:

1. CGT was extended in three directions ([31], [32], [35]) with new interesting examples. Let us just mention that as shown in [35], CGT contains the so-called Tannaka duality as a special case, and that the results of [31] establish a deep link between CGT and the Kurosh-Amitsur theory of radicals.

2. The fundamental theorem of Galois theory of commutative rings was originally proved in full generality by A. R. Magid about thirty years ago. Later he found a mistake, and now - together with him - it was corrected, and the correct formulation and proof based on CGT was obtained ([6]).

3. Separability in lextensive and general categories was investigated. It was shown that various basic results on (commutative) separable algebras and decidable objects in a topos whose known proofs involved specific techniques (like projective modules or internal logic of the topos) have simple purely categorical proof. The relationship with the categorical notion of a covering morphism (used in CGT) was established. See [7], [33], [43].

4. Grothendieck's Descent theory. A second expository paper with simplified presentations of various known results (often previously unpublished), and some new results, was written ([28]). All existing problems of finite topological descent were solved, and for the general topological descent theory some counter-examples to open problems are constructed, and it is shown that the finite case provides simple motivations for all existing results ([40]).

5. Factorizations in Galois theory. The categorical version of the classical purely inseparable-separable factorization of finite algebraic field extensions was obtained. The well known monotone-light factorization in topology turned out to be another very special case of this: it was obtained by applying CGT to the adjunction between all compact Hausdorff spaces and the 0-dimensional ones ([8]). More complicated cases (commutative rings, locally connected spaces with local homeomorphisms, and others), where the factorization still exists, but no "good" description of the purely inseparable morphisms can be given, were also investigated ([30]).

6. Central extensions, internal groupoids, and commutators. In several steps during several trips of G. Janelidze to Australia, it was finally proved that the three approaches to central extensions of ``group-like" algebraic structures - homological-algebraic (Froehlich's school), universal-algebraic (commutator theory), and of CGT - perfectly agree ([36], [42]). The Galois theory of central extensions involves internal groupoids in algebraic categories. And there are several important levels of generality, where it is desirable to have certain simplified descriptions of internal groupoids (for example it is well known that in the category of groups they are precisely crossed modules). A reasonably wide description was obtained in [29]. In connection with this a categorical reformulation of commutator theory with many new results was given in [34].

7. Semi-abelian categories were discovered in [39]. S. Mac Lane, who proposed the first version of definition of abelian category in 1950, in fact proposed to find a nonabelian version of it, where the isomorphism theorems and some other basic facts and constructions of group/ring/module theory could be formulated. In the next twenty years various attempt have been made, especially for the purposes of homological algebra and theory of radicals. However all the proposed definitions used ``strange" conditions on normal mono-/epimorphisms, which excluded any reasonable use of general-categorical methods. On the other hand as shown in [1] using descent theory, there is a categorical notion of semi-direct product in Bourn protomodular categories - and that was one of ingredients that helped to prove that the above mentioned old attempts yield a notion that can be equivalently expressed with modern categorically natural axioms. The resulting "semi-abelian" categories provide a good environment to simplify and unify many algebraic constructions. The results of [2] can be considered as an example of this.

8. Absolute homological algebra in general additive categories with kernels was obtained in [37]. It uses the old work of Yoneda and Grothendieck's descent Theory, and includes the known constructions for abelian topological groups and modules. The reason why such a long-standing problem was solved only in 1998 should probably be attributed to the various improvements in descent theory and to previous work on the semi-abelian case (see above).

9. The Kurosh-Amitsur radical theory was already mentioned in connection with CGT and with semi-abelian categories. It was also understood that it has a non-trivial purely combinatorial aspect, related to the known fact that the pairs consisting of a radical class and the corresponding semisimple class sometimes do not occur from a Galois connection (non-associative rings), and an appropriate combinatorial structure to replace the Galois connection was obtained in [38]. Moreover, it was shown there that the categorical setting of the theory of radicals is based on the more fundamental and rather simple combinatorial setting.

A. Patchkoria

The notion of a Schreier internal category in the category of monoids was introduced and it was proved that the category of Schreier internal categories in the category of monoids is equivalent to the category of crossed semimodules. This extends a well-known equivalence of categories between the category of internal categories in the category of groups and the category of crossed modules [55].

Homology and cohomology monoids of presimplicial semimodules (in particular, presimplicial abelian monoids) were introduced and some algebraic and topological applications of them are given. (Constructions of homology and cohomology monoids of topological spaces with coefficients in abelian monoids, a generalisation of the construction of derived functors via simplicial resolutions to semimodule-valued functors, etc.) Relations between our homology monoids and the classical homotopy groups of simplicial abelian monoids are studied. These and other results on homological algebra of monoids and semimodules are included in [51] - [54] and [56]. This work is perceived as being potentially important in homological algebra and its applications where finer invariants than abelian groups are needed (for example where a Grothendieck group loses too much information). The origin of this work was in fact for applications to Cousin's problem in analysis.

In joint work of M. Lawson (Bangor team), homological descriptions of the homotopy groups arising from 0-dimensional idempotents of simplicial inverse semigroups are obtained in some special cases [43].

T. Datuashvili

During the INTAS project period among the other questions the problem of the internal Kan extension, suggested by G. Janelidze, was investigated. The crossed module approach to this question enabled us to obtain under certain conditions the necessary and sufficient conditions for the existence of internal Kan extensions [11]. Later the same results were obtained under more general conditions [13]. Working on the topological approach to the above mentioned problem the well-known equivalence of the category of 3-types (in he sense of [J. H. C. Whitehead, Combinatorial homotopy I, Bull. A.M.S. 55 (1949) 214-245]) with the localized category of crossed modules [H. J. Baues, Combinatorial homotopy and 4-dimensional complexes (Max Plank Institut 1990), preprint], [J.-L. Loday, Spaces with finitely many non-trivial homotopy groups, J. Pure Appl. Algebra 24 (1982) 179-202] is not useful. One needs to deal with (internal) equivalence of internal categories (equivalently, crossed modules), not with a weak equivalence of corresponding crossed modules. This procedure leads to the search of more general relation between crossed modules and connected cell complexes. The main result obtained here (see [12]) is the existence of adjoint pair of functors between homotopy category of internal categories (=crossed modules) and the category of 3-types. The constructions given in [S. Mac Lane, Cohomology theory in abstract groups III, Ann. Math., 50 (1949) 736-761], [S. Mac Lane and J. H. C. Whitehead, On the 3-type of a complex, Proc. Nat. Acad. Sci. USA 36 (1950) 41-48], [J. H. C. Whitehead, Combinatorial homotopy II, Bull. A.M.S. 55 (1949), 453-496] were used and their functoriality was shown. It was important to pay attention to the ``middle" category between crossed modules and homotopy systems, in the process of realization of algebraic 3-types. This is a subcategory XModF of crossed modules XMod, in which the objects are crossed modules with free group of operators. We show that correspondence between algebraic 3-types, crossed sequences of a special type, 3-dimensional homotopy systems and connected cell complexes define functorial relations, equivalence or adjoint between corresponding homotopy categories. These results can be applied to the problem of existence of internal Kan extension by reducing it to the problem of the unique extension of a continuous map between connected cell complexes. T. Datuashvili is also working in collaboration with T. Pirashvili [14] on homology of crossed modules, and with J.-L. Loday on Leibniz algebras.

N. Inassaridze

The important problem of the derived functors of the non abelian tensor product
of Brown and Loday has been solved using methods of non abelian derived functors.
This requires extending the original definition to the case of non compatible
actions [1], and this also leads to new problems
such as those of finiteness of this new product
[2]. Further the existence of these derived
functors leads to new notions of non abelian homology H_{n}(G,A)
where G and A are two groups acting on each other
[1]. They coincide with the usual Eilenberg-Mac
Lane homology groups when A is a G-module. A number of new results on this
nonabelian homology are obtained in the papers
[3,4,5,6,7]. These include: A Mayer-Vietoris
sequence; a description of H_{n}(G,A) as the left derived functors
of the functor H_{1}(G,A); explicit formulas for the second and third
nonabelian homology groups using ^Cech resolutions; sufficient conditions
for the nonabelian homology groups in dimensions
³ 2 to be finitely generated, finite, p-groups,
torsion groups or groups of exponent q. For instance, suppose that the action
of A on G is trivial, that G is finite, and that A is a finite group (or
p-group or finitely generated group). Then H_{n}(G,A), n
³ 2, is a finite group (or p-group or finitely
generated group).

Some properties of the nonabelian tensor product modulo q of two crossed modules, introduced by Conduché and Rodriguez-Fernandez, are established (commutativity, compatibility with the direct limit of crossed modules) [8]. The extension of the tensor product to a tensor product modulo q leads to introduce and develop certain aspects of a q-modular version (q-homology) of the classical Eilenberg-Mac Lane homology theory of groups, where q is a nonnegative integer [8]. Its functorial properties (exactness, universal coefficient formulas) and calculations (for free groups, finite cyclic groups) are given [8]. The relationship between q-homology groups and derived functors of tensor product modulo q is studied [8].

Homology groups modulo q of a precrossed P-module in any dimensions are defined in terms of nonabelian derived functors [9]. The Hopf formula is proved for the second homology group modulo q of precrossed P-modules which shows that for q = 0 our definition is a natural extension of Conduché and Ellis' definition of the second homology group of precrossed P-modules [9]. Other properties of homology groups modulo q of precrossed P-modules are investigated, in particular for any short exact sequence of precrossed P-modules a five term exact homology sequence modulo q is obtained [9].

Some properties of the nonabelian tensor product of two Lie algebras M and N acting on each other are established [10]. Using techniques of nonabelian homological algebra a nonabelian homology of Lie algebra M with coefficients in any Lie algebra N (here M and N act on each other) are constructed as the nonabelian left derived functors of the nonabelian tensor product of Lie algebras, which generalize the classical theory of the homology of Lie algebras [10]. Functorial properties of nonabelian homology of Lie algebras are established.

Defining and using higher (n-fold) ^Cech resolutions of groups and abelianization of crossed n-cubes a new approach to the classical Hopf formula for higher homology of groups is given [11].

The notion of q-modular cohomology of a group G with coefficients in a G-module A is introduced [12], where q is a nonnegative integer. Its description in terms of extensions, some its properties and calculations are given. For a finite group G Tate cohomology modulo q are defined [11].

B. Mesablishvili

The connection between abstract Galois theories of Ligon and of Chase and Sweedler was investigated. In particular, it was shown that they give the same Galois theory in the category of modules over an elementary topos [44]. A.Grothendieck's descent theorem was extended to a wider class of morphisms of schemes [45].

D. Zangurashvili

D. Zangurashvili was working on construction of various factorization systems in general categories, and prepared the papers [57] - [60]; one of them [59] was prepared during her visit in Coimbra University (Portugal) in collaboration with M. Sobral.

Z. Omiadze

Z. Omiadze continued his work on II-categories, which he introduced before ([46], [47]), and on their higher-dimensional versions ([48], [50]). He also began to investigate a new type of 2-dimensional categorical enrichment ([49]).

All members of this group have made research visits to Bielefeld. N. Vavilov visited on a Humboldt Fellowship for a year.

The members of this group have cooperated much with one another and with members of the Bielefeld group. Nine joint papers [2], [7], [8], [10], [24] - [26], [28] of this kind were written. Several articles will appear in K-Theory.

The paper [3] provides a detailed and extensive study of weight elements in Chevalley groups. This theme is carried further in [11] and [16].

The papers [5], [6] are part of a continuing cooperation between N. Vavilov and L. Di Martino on (p, q)-generation of subgroups and groups of Lie type.

The paper [7] with A. Bak poses a very interesting and deep extension of Milnor's conjecture relating K-theory to quadratic forms and provides evidence for the conjecture.

The paper [8] is part of an ongoing cooperation between A. Bak and N. Vavilov on the structure of hyperbolic unitary groups. The current paper is the first in a planned series and is concerned with definitions and basic results, including the normality of the elementary subgroup. A successor is currently being written.

The paper [10] of N. Vavilov and A. Stepanov provides new insight of a geometric nature into the normality of elementary subgroups. These ideas are carried further in [14]. Both papers are related to joint work described above of A. Bak and N. Vavilov.

The papers [17] - [19], [21] and [23] of E. V. Dybkova concern the structure of net subgroups in linear and hyperbolic unitary groups. These papers are related to that of A. Bak and A. Stepanov [28] which develops general procedures to determine when classifying sandwiches in classical-like groups are nilpotent. The results here include applications to net subgroups. These applications overlap with results of Golubchik [23] (in the Moscow State University references).

All the members of this group have made research visits to Bielefeld and three have or will be spending a year or longer as Humboldt Fellows. Below is a summary of the results achieved by the group.

The 30 year old problem to find a Grothendieck K_{0}-construction
of K_{1} of an exact category is solved in two papers
[25] and [26] (of the Bielefeld references)
of A. Nenashev who is currently finishing his stay in Bielefeld as a Humboldt
Fellow. The paper [27] of Nenashev makes
a start at solving the same problem for K_{2} of an exact category
and the paper [28] applies the results in
[25] and [26] to
l-operations on K_{1}.

A. Suslin and coworkers have made significant advances to the cohomology of group schemes, the K-theory and cohomology of sheaves, Chow groups, and the homology, cohomology, and K-theory of GL and related functors. The results on group schemes are contained in [1] - [3]. Work on Chow groups and the K-theory and cohomology of sheaves are contained in the articles [6] and [7] which will appear in an upcoming monograph in the Annals of Mathematics Studies. Two very important articles [8] and [9] concerning the cohomology of GL and related functors are appearing in the Annals of Mathematics. The article [8] builds on an earlier success [2] of Suslin and Friedlander solving the long standing conjecture that the cohomology algebra of a finite group scheme is finitely generated, by extending significantly the scope for making Ext-group calculations. Results here include a complete determination of all Ext-groups between classical functors in the category of strict polynomial functors of finite degree. Methods and results developed in [8] are used in [9]. Further articles on the homology, cohomology, and K-theory of GL are found under [28] - [32] and [34].

A. Merkurjev and coworkers have carried out extensive research on the K-theory of algebraic groups. Several of their articles have appeared in K-theory. Definitive results on R-equivalence and index theory for algebraic groups are contained in the articles [16] - [19] and [23] - [27]. Further aspects of the K-theory of algebraic groups are found in [20], [21], [26] and [27] and a survey is published in [22].

Problems concerning isotropy and splitting for quadratic forms over fields are studied and solved in [10] - [13] and [36].

The article [25] (of the St. Petersburg State University references) handles certain stable range questions for affine algebras and the paper [5] has results on a conjecture of Grothendieck for Azumaya algebras.

We see the results achieved as fulfilling the essence of the Objectives of the Work Programme, and as properly taking up new opportunities which arose in the course of the work.

Here we mention Bak's new methods of *Global actions*, giving a purely
algebraic version of a topological space, and whose homotopy groups give
higher algebraic K-groups. This has also been developed in a number of directions
with Bak's direct collaborators, particularly with the groups of Suslin and
of Vavilov. A recent development is the link with algebraic homotopy expertise
from Bangor and nonabelian homological algebra expertise from Tiblisi.

Another stream which comes in via the Tbilisi-Bangor connection is the
generalisation of Grothendieck's Galois Theory by Janelidze. Links with the
theory of descent are already clear. In terms of the *impact of INTAS*,
we mention that the work at Tbilisi has been considerably influenced by the
work at Bangor on crossed modules and related topics, such as non-abelian
tensor product of groups. The late development of input of Lawson's work
on inverse semigroups to the programme is a direct result of INTAS.

Other very significant work includes the full solution of the Riesz-Radon
problem (1908) obtained by Mikhalev and coworkers at Moscow State University,
results of Suslin and coworkers on the K-theory and cohomology of sheaves,
chow groups and GL, results of Merkurjev on R-equivalence and the rationality
problem for semisimple adjoint classical groups and on index reduction formulas
of twisted flag varieties of semisimple algebraic groups, the solution by
Nenashev of the classical problem of finding a Grothendieck
K_{0}-construction for K_{1} of an exact category, and a
generalization of Milnor's conjecture for quadratic forms together with
supporting evidence by Bak and Vavilov.

This is just a mention of considerable work given in more detail in the individual reports and in the lists of papers. Of course the INTAS support is one part of a thriving range of connections which makes it difficult to quantify exactly how much this support is responsible for the overall progress. This support has clearly significant effect, both in terms of the actual help given to the participating NIS teams, and in terms of the cooperation which it has encouraged and will continue to encourage.

Scientific Output |
published | in press/accepted | submitted | in preparation |

Paper in an International Journal | 135 | 19 | 24 | 42 |

Paper in a National Journal *) | 3 | 3 | 4 | 6 |

Abstract in proceedings of a conference | 1 | |||

Book, Monograph *) | 2 | 0 | 0 | 0 |

Internal Report **) | ||||

Thesis (MSc, PhD, etc.) *) | 11 | |||

Patent | 0 | 0 | 0 | 0 |

Oral Presentation, Public Lecture | Many! |

In general this project has worked as part of a multifunded project. Bangor and Bielefeld have kept in excellent contact but using funding from British Council/ARC and the SOCRATES Programme. Other funds have supported a seminar in Tbilisi for NIS participants. Bak took part in the Pontrjagin 80th Birthday meeting in Moscow in the Summer of 1998, and gave two talks. In fact this meeting was partly sponsored by INTAS, but Bak's funding was from the DFG. Many NIS visitors to Bielefeld have been supported by funding from the DFG and Humboldt Foundation.

The Bielefeld group has cooperated closely with members of the groups at St. Petersburg State University, Steklov Institute, Moscow State University, and Bangor University and is beginning a cooperation with the Mathematical Institute of the Georgian Academy of Sciences. Members from all of the universities and institutes in the INTAS project have made research visits to Bielefeld and 5 members (Izhboldin, Merkurjev, Nenashev, Panin, and Vavilov) have been or are currently Humboldt Fellows in Bielefeld.

There is considerable contact and collaboration between the two groups in St. Petersburg.

The INTAS funds at Bangor largely supported visits of Brown to Utrecht to a PSSL meeting where he discussed with Janelidze; to Dunkerque to meet Janelidze who was visiting there; a visit of Dr R. Vilanueva (Valencia) to discuss the book project on crossed modules; a visit of Prof Buchberger (Linz) an expert on Grobner bases, indeed the founder of the algorithms in the area, to give advice on our developing work in the area; a visit of Bak to Bangor; a two month stay of Dr \.I. Içen (Inonu) to work with Brown on new methods in holonomy - this work is in press or in preparation.

Members from all of the universities and institutes in the INTAS project have made research visits to Bielefeld and 5 members (Izhboldin, Merkurjev, Nenashev, Panin, and Vavilov) have been or are currently Humboldt Fellows in Bielefeld. A. Nenashev from the Steklov group has been for over a year in Bielefeld. INTAS funding extended this support conveniently.

Janelidze from Tbilisi has been travelling in the West throughout this period with other support, and communication with him and Bangor has taken place at various meetings. Inassaridze also visited Bielefeld and Bangor with support from an INTAS Fellowship in Dec 1999-Jan 2000.

Mikhalev and Artamonov visited Bangor for a week in Jan, 2000.

The following visits took place under INTAS funding.

*To Bangor: *

Tbilisi group: Inassaridze: 1 month October 1997

Patchkoria: 1 month Jan-Feb 2000

Moscow State University: Artamonov 1 week Jan 2000

Mikhalev 1 week Jan 2000

*To Bielefeld*

St Petersburg State University: Vavilov: 2 weeks Nov 1997, 3 weeks Jan -
Feb 2000

Stepanov: 5 weeks May - June 1998

Sivatski: 3 weeks May - June 1998

Dybkhova: 3 weeks Nov - Dec 1998

Mischenko: 2 weeks Nov 1997

Moscow State University: Artamonov 2 weeks Jan 1999

Mischenko: 2 weeks Nov 1997

Steklov Institute: Pushin 3 months April - June 1999

Joukovitski 2 months April - May 1999

Yagounov 2-3 weeks

Nenashev 2 weeks

*From Bangor:*

Brown visited Utrecht, Dunkerque to meet Janelidze (4 days):

*From Bielefeld: *

Bak visited Moscow for one week.

Funds at Bangor were used to support other visitors relevant to the work
programme:

Içen (Inonu) 2 months subsistence (for work on holonomy and local
subgroupoids)

Sivera (Valencia) 2 weeks subsistence (for work on a planned book on higher
dimensional group theoretic methods in topology and algebra)

Buchberger (Linz) 1 week travel and subsistence for discussions on Gröbner
bases (May 1998).

Visit of Bak (1 week, Jan 1998). Other strong contacts Bangor-Bielefeld were
supported by an ARC/British Council grant.

Funds at Bielefeld were used also to support 3 workshops of 1 week to 10
days, two in 1998 and one in 1999.

Intensity of Collaboration |
high | rather high | rather low | low |

West Û East | * | |||

WestÛ West | * | |||

East Û East | * |

The East-East collaboration is largely at St Petersburg.

*Time Planning*

The results achieved are completely consonant with the overall thrust of the Work Programme.

Problems encountered |
major | minor | none | not applicable |

Co-operation of team members | * | |||

Transfer of funds | * | |||

Telecommunication | * | |||

Transfer of goods | * | |||

Other | * |

No action required.

The spending has been in accordance with the work programme allowing for the final 10%. The funds on salaries and travel are according to the work programme allowing for the final 10%.

This has been a multifunded project and the various support has contributed greatly to the success of the project.

1. A British Council/ARC Grant supported Bangor/Bielefeld visits for the
collaboration (workshops in Bielfeld and in Bangor). 2. Janelidize was widely
supported for work in the West in the period of the grant and meetings of
Brown and Janelidze in the West took place as a result.

3. Inassaridze was partially supported by an INTAS Young Scientist Grant,
and this included visits to Bielefeld and Bangor Dec 1999-Jan 2000.

4. A DFG/RFBR Grant supported Bielefeld/St. Petersburg and Bielefeld/Moscow
visits for collaboration.

5. Each of five NIS members Izhboldin, Merkurjev, Nenashev, Panin, and Vavilov
were supported as Humboldt Fellows for a year in Bielefeld.

6. Members of all teams (except Bielefeld) were supported for stays in Bielefeld
of various lengths by the SFB 343.

Work by participants has led to the publication of some spectacular results.

These include a solution of the long standing conjecture that the cohomology algebra of a finite group scheme is finitely generated and a further development of methods here to achieve a complete determination of all Ext-groups between classical functors in the category of strict polynomial functors of finite degree.

A solution of the long standing problem of finding a Grothendieck
K_{0}-construction for K_{1} of an exact category is obtained.

A generalization of Milnor's conjecture giving criteria when sums of n-fold Pfister classes in the Witt ring are trivial is made and evidence in support of this criteria is proved.

The group of R-equivalence classes for all adjoint semisimple classical groups is computed and index reduction formulas are determined for the twisted flag varieties of any semisimple algebraic group.

A full solution of the Riezz-Radon problem for integral representations of a Radon measure on an arbitrary Hausdorff space is obtained.

A dimension theory for categories is developed and applied to determining structural properties of certain classical-like groups.

Foundations for global actions have been laid and a model category description of the homotopy theory of global actions is developed.

Work on homotopy coherence and on computational methods has confirmed the fundamental rôle of crossed complexes as an extension of chain complex methods which can deal with non simply connected spaces. These yield new methods of computing modules of identities among relations for presentations of groups. Rewriting methods have been vastly extended from presentations of monoids to presentations of induced actions of categories, extending the field to computational category theory.

A wide range of applications of the Brown-Loday non abelian tensor product have been found, including new forms of non abelian homology and of homology mod q.

The main results in category theory continue the programme of Categorical Galois Theory, which is shown to have a wide range of applications, involving Descent Theory, internal groupoids and commutator theory, and to link with the homotopical algebra methods developed by the Bangor group.

The project involved about 30 participants, from two EU and four NIS centres.

**Some key papers**

E. Friedlander and A. Suslin, Cohomology of finite group schemes over a field,
*Inventiones Math.* **127** (1997), pp. 209-270

V. Franjou, E. Friedlander, A. Scorichenko and A. Suslin, General Linear
and Functor Cohomology over Finite Fields - * Annals of Math*
**150** (1999), 663 - 728

A. Bak, Global Actions: The algebraic counterpart of a topological space,
invited paper for the 100'th anniversary of P.S. Alexandroff, *Uspekhi
Mat. Nauk* **52**:5 (1997), 71 - 112, English translation: *Russian
Math. Surveys* **52**:5 (1997), 955 - 996

A. V. Mikhalev, V. K. Zakharov, Integral representation for Radon measures
on an arbitrary Hausdorff space, *Fundamental and Applied Mathematics*
**3** (1997), N 4, 1135-1172.

A. Merkurjev and I.A. Panin and A.R. Wadsworth, Index reduction
formulas for twisted flag varieties. II. * K-Theory* **14** (1998),
101-196.

A. Nenashev, Double short exact sequences and K_{1} of an exact category,
*K-Theory* **14** (1998), no. 1, 23-41.

G. Janelidze and R.H. Street, Galois theory in Symmetric Monoidal Categories,
*J. Algebra* **220** (1999) 174-187.

Brown, R., Golasinski, M., Porter, T. and Tonks, A., ``On function spaces of equivariant maps and the equivariant homotopy theory of crossed complexes'', Indag. Math. 8 (1997) 157-172; `II : the general topological group case'' K-theory (to appear).

**Coordinator's home page**

http://www.bangor.ac.uk/~mas010

The extensive article on `Higher dimensional group theory' should also be noticed. There is also a link to a workshop at Bangor in January, 2000.

`The project' is very broad and many aspects would have been started and continued whatever the funding. None the less, the scope of the project and the actual and potential interconnections have been enormously increased by the existence of the project, and I would strongly rate the success of the support in scientific terms, both already realised and for the future.

Role of INTAS |
Definitely yes | rather yes | rather not | definitely not |

Would the project have been started | ||||

without funding from INTAS? | * | |||

Would the project have been carried out | ||||

without funding from INTAS? | * |

In the above it should be emphasised that the project carried out was multifaceted and multifunded. None the less, I consider that the INTAS funding gave an extension of the overall work plan of the participants leading to many results which would have not been conceived without this funding.

Main achievement of the project |
Very important | quite important | less important | not important |

Exciting science | * | |||

new international contacts | * | |||

additional prestige for my lab | * | |||

additional funds for my lab | * | |||

helping scientists in the NIS | * | |||

other: The opening out of a broader range of interactions
than originally envisaged, |
||||

and so the development of new prospects. |

The project will continue in its multifaceted mode.

A further INTAS proposal with these and some further partner(s) is being planned.

**General**

No recommendations.

The relevant publication lists of the participants are accessible as
follows:

Bangor

Bielefeld

Tbilisi

Moscow State University

St Petersburg State University

Steklov Institutue, St Petersburg

File translated from T

On 6 Jan 2001, 23:25.