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Algebra & Algebraic Topology

Introduction.

The research in algebra at Bangor has to a large extent been motivated by problems in algebraic topology and homological algebra. The recent spate of new and exciting concepts (crossed modules, crossed n-cubes, nonabelian tensor products, etc.) originating in those two areas has opened out many algebraic aspects of the theory and applications which are waiting to be investigated.

Many final year pure mathematics courses have some Algebraic Topology in them. Typically the fundamental group and/or the homology groups are defined, studied and applied to various problems such as the existence of certain types of continuous maps, the classification of knots, and other problems of classification usually in low dimensions. These problems usually involve two basic questions:

In both cases the method of algebraic topology is to model certain important aspects of each space by some algebraic gadget, perhaps a group or a set of interrelated groups, use these to translate the problem to an algebraic context; try to solve that algebraic problem and finally to reinterpret the results back in terms of spaces.

Homology theory was originally a part of algebraic topology, concerned with finding numerical invariants of spaces. Its methods grew out of an analysis of the behaviour of path integrals in complex analysis, but in the 1920’s and 30’s it became clear that the invariants involved were better viewed as numbers associated to certain Abelian groups.

In the 1940's the theory of the "homology groups" of a space was developed further and Eilenberg and Mac Lane applied these tools to certain spaces associated to groups; they linked up the groups that resulted with various other invariants that had been developed earlier by group theorists for tackling purely algebraic problems. Combining the two outlooks it became clear that these methods, suitably adapted, gave significant tools in many other areas of algebra.

The connections between algebraic topology, homological algebra and areas of "pure" algebra, number theory and algebraic geometry have remained very strong. In fact the spin-off from homological algebra in these other areas has been central in their development over the last 30 years.

At Bangor our interest in homological algebra continues in this same tradition. New ideas, developed initially for applications in topology, have turned out to shed new light on old established problems in algebra. We have started to develop these ideas for their own sake and already there are significant results appearing even though the area is still relatively young. Some of this work has been done by research students working at Bangor.

Algebra

A. Crossed modules of groups:

A crossed module (of groups) consists of two groups G, C, an action of G on C and a homomorphism, d : C -> G, satisfying:

d(g^c) = gcg-1

and

(dc)^c´ = c c´ c^{-1} .

(We have written g^c for the element that results when the element g in G acts of c in C)

The theory of crossed modules of groups started with their occurrence in algebraic topology, but they also occur naturally in the study of automorphisms of groups and in combinatorial group theory with the free crossed module generated by a presentation of a group. This latter construction is closely linked with problems of 2-dimensional homotopy theory and with the determination of identities among relations.

Link diagram illustrating the reduction of identities in a presentation of the trivial group

Another new idea that needs further study is that of the nonabelian tensor product of groups. This would seem to have significance in the theory of extensions of groups as well as being a fascinating construction in itself. There are clearly a lot of interesting and challenging group theoretic problems to be solved in this new area.

Recently with the arrival of new powerful computing packages, in particular GAP and MAGMA, it has become feasible to begin the classification of finite crossed modules by adapting existing algorithms used for groups. These computational activities would not be feasible without the excellent research computer laboratory recently set up with EPSRC and university funds.

B. Rings, algebras and modules:

1. Crossed modules of algebras.

A crossed module of algebras has a structure in between that of a module and an ideal. Comparatively little is known about this structure even in the commutative case, but these crossed modules are only the bottom of a tower of related structures, crossed n-cubes, and here our knowledge is best described as being non-existent.

2. Simplicial commutative algebras.

These objects occur in studying various problems in algebraic geometry, and the cohomology of commutative rings as well as leading naturally to crossed modules of algebras. Their structure is being studied in its own right as an example of higher dimensional algebra, and exciting new idea that permeates much of the research in algebra at Bangor.

C. Monoids.

Trying to adapt the above ideas to handle problems in monoid theory is non-trivial. Kernals play an important part in the development of the ideas both in group theory and for algebras but kernals are not available for monoids. The analogous theory therefore needs careful development, yet would seem to be important for links with language theory and theoretical computer science. (See also research in the related topic of Semigroup and Automata Theory

ALGEBRAIC TOPOLOGY AND HOMOLOGICAL ALGEBRA

A. Algebraic Models for Homotopy Types.

1. Groups v. groupoids.

One of the simplest algebraic models for a space (with base-point) is the fundamental group. One of the earliest homotopy invariants to be considered, it was derived from loops at a point in the space. An important theorem on the fundamental group is the Van Kampen Theorem, which shows how the fundamental group of a union is related to the fundamental groups of the parts, provided that any intersection of the parts is connected. The fundamental groupoid derives from all paths between points of a set of base points. It allows for an elegant and powerful reformulation of most aspects of the fundamental group, including the van Kampen Theorem, covering spaces, and orbit spaces.

2. Double Groupoids and Multiple Groupoids.

The development of homotopy theory has been held up by a concentration on group theory, since the obvious definition of a "double group" yields only an abelian group. However, double groupoids are much more complicated than groups, and are appropriate for 2-dimensional homotopy theory. The theory and applications of the higher homotopy groupoids of a filtered space was developed in an extensive Bangor-Durham collaboration, involving 5 research students. Double groupoids have a rich algebraic structure as well as geometric and combinatorial aspects. The double and multiple groupoids or a filtered space satisfy a van Kampen theorem.

3. Crossed complexes.

The Bangor-Durham collaboration went on to explore higher dimensional analogues of these double groupoids. The resulting theory gave seven equivalent descriptions of the gadgets that resulted. Each description blends differing amounts of geometric, algebraic and combinatorial structure.

How can one imagine these gadgets? The simplest way is to think of groupoids as arising by mapping intervals into a space, double groupoids arising from mapping in squares, and crossed complexes/omega-groupoids arise from mapping higher dimensional cubes. There is much work still to be done in this area.

4. Cat^n-groups.

Special types of double groupoids are equivalent to crossed modules. Loday at Strasbourg proved that a higher dimensional version of crossed modules gave a complete model for homotopy types in much higher dimensions than was previously possible. Loday continued work in this area in collaboration with Professor Brown here at Bangor, and this theory has now been pushed further to include a Van Kampen Theorem. Even the initial applications give striking results in homotopy theory, homological algebra and group theory.

5. Non-Abelian homological and homotopical algebra.

The classical areas of homological algebra developed from homology theory so gave "Abelian" information only. The new methods recently developed at Bangor, Strasbourg and other centres have their origins in homotopy theory and hence are better suited to giving "non-Abelian" information. The term Homotopical Algebra has been used for this general area, although this term is also used for describing a larger related area of ideas.

R.Brown and P.J.Higgins have developed the category, Crs, of crossed complexes as a tool in algebraic topology and homotopy theory. The recently developed tensor product of crossed complexes allows for a notion of crossed differential algebra, and even of Hopf algebras. Gadgets of this type seem appropriate for applications in algebraic topology where previous methods are able to deal only with simply connected spaces. There is a very rich area here to investigate.

B. Cohomology of algebraic structures.

1. Group (co)homology.

Although giving Abelian information, the homology groups and cohomology groups of a group G are closely linked with the new tools mentioned earlier. A new generalisation of tensor constructions, classically available for vecvtor spaces and modules, has been developed at Bangor and Strasbourg to apply to groups, and this allows a new description to be given of some of the homology groups of a group. This is but one of the several applications found for these new tools in the last few years.

2. Non-Abelian cohomology.

Given groups G and H, can one find all groups E that contain a copy of H as a normal subgroup in such a way that the quotient group E/H is isomorphic to G? This "extension problem" is analysed by the cohomology groups of G if H is Abelian. If H is not Abelian, the problem is very much more tricky. One needs non-Abelian cohomology, and the coefficients are a crossed module describing the "symmetry" of the group H. Higher order non-Abelian cohomology needs "higher order crossed modules", which are themselves related to multiple groupoids.


Related notions of non-Abelian cohomology of spaces have also geometric interests, and there are analogous problems in commutative algebra and other areas. It is helpful to think of these non-Abelian methods as being non-linear, which suggests a large variety of potential areas for applications of this theory.

Recent Publications:

2000, 1999, 1998, 1997, 1996

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