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Crossed Modules

Johannes Huebschmann

Communicated by Notices Associate Editor Steven Sam

Article cover

Dedicated to the memory of Donald J. Collins

1. Introduction

Modern chatbot software poses a threat to the health of our field. A scholarly article had better pass through the heads of at least two parties, cf. Huf10, p. 47.⁠Footnote1 Here we undertake the endeavor of writing such an article about crossed modules with an eye toward the past, this past being ignored by the recent activity in this area.

1

“It appears that the reporter has passed along some words without inquiring what they mean, and you are expected to read them just as uncritically for the happy illusion they give you of having learned something. It is all too reminiscent of an old definition of the lecture method of classroom instruction: a process by which the contents of the textbook of the instructor are transferred to the notebook of the student without passing through the head of either party.”

In ancient cultures, symmetry arose as repetition of patterns. The human being perceives such symmetry as harmonious and beautiful proportion and balance (music, art, architecture, etc.). The variety of patterns is untold. Symmetry enables us to structure this variety by recognizing repetitions, as Eurynome’s dancing structured chaos. The symmetries of an object are encoded in transformations that leave the object invariant. In modern mathematics, abstracting from the formal properties of such transformations led to the idea of a group. A typical example is the group of symmetries of the solutions of an equation, in modern mathematics termed Galois group, or each of the 17 plane symmetry groups. The operations that form these groups were already known to the Greeks. Besides in mathematics, symmetry and groups play a major role in physics, chemistry, engineering, materials science, crystallography, meteorology, etc. A group itself admits symmetries: the automorphisms of a group constitute a group. The crossed module concept arises by abstracting from the formal properties which this pair enjoys. For two groups and , the technology of crossed modules allows for a complete classification of the groups having as a normal subgroup and quotient group isomorphic to . This settles the extension problem for groups, raised by Hoelder at the end of the nineteenth century. Schreier explored this problem in terms of factor sets, and Turing implicitly noticed that it admits a solution in terms of crossed modules. Given and as symmetry groups, interpreting an extension as a symmetry group is an interesting task, as is, given the extension group as a symmetry group, interpreting and as symmetry groups. Crossed modules occur in mathematics under various circumstances as a means to structure a collection of mathematical patterns. A special example of a crossed module arises from an ordinary module over a group. While the concept of an ordinary module is lingua franca in mathematics, this is not the case of crossed modules.

Beyond some basic algebra and algebraic topology, we assume the reader familiar with some elementary category theory (“categorically thinking” suffices). This article is addressed to the nonexpert. We keep sophisticated technology like group cohomology, homotopy theory, and algebraic number theory at a minimum. We write an injection as and a surjection as .

The opener image of this article displays the beginning of the first counterpoint (contrapunctus) of the original printed version of J. S. Bach’s “Kunst der Fuge”. The order of the counterpoints (of the second part thereof) had been lost and, 100 years ago, Wolfgang Graeser, before enrolling as a mathematics student at Berlin university, restored an order (perhaps the original sequence) by means of symmetry considerations. This order is the nowadays generally accepted performance practice.

2. Definition and Basic Examples

A crossed module arises by abstraction from a structure we are all familiar with when we run into a normal subgroup of a group or into the group of automorphisms of a group: Denote the identity element of a group by and, for a group and a -(operator) group (a group together with an action of on from the left by automorphisms of ) we write the action as , for and . Consider two groups and , an action of on from the left, view the group as a -group with respect to conjugation, and let be a homomorphism of -groups. The triple constitutes a crossed module if, furthermore, for every pair of members of ,

that is, the members and of coincide. For a crossed module , the image of in is a normal subgroup, the kernel of is a central subgroup of , and the -action on induces an action of the quotient group on turning into a module over this group, and it is common to refer to the resulting exact sequence

as a crossed -fold extension of by . A morphism of crossed modules is defined in the obvious way. Thus crossed modules constitute a category. The terminology “crossed module” goes back to Whi49. The identities 1 appear in Pei49 and have come to be known in the literature as Peiffer identities Lyn50 (beware: not “Peiffer identity” as some of the present day literature suggests). There is a prehistory, however, Bae34Tur38; in particular, the Peiffer identities occur already in Tur38. The reader may consult Hue21, Section 3 for details.

For a group and a -module , the trivial homomorphism from to is a crossed module structure and, more generally, so is any -equivariant homomorphism from to such that the image of in acts trivially on . Relative to conjugation, the injection of a normal subgroup into the ambient group is manifestly a crossed module structure. Also, it is immediate that the homomorphism from a group to its group of automorphisms which sends a member of to the inner automorphism it defines turns into a crossed module. It is common to refer to the quotient group as the group of outer automorphisms of .

3. Identities Among Relations

An “identity among relations” is for a presentation of a group what a “syzygy among relations”, as considered by Hilbert, is for a presentation of a module: Consider a presentation of a group . That is to say, is a set of generators of and a family of (reduced) words in and its inverses such that the canonical epimorphism from the free group on to has the normal closure of (the image of) in as its kernel. Then the -conjugates of the images in of the members of generate , that is, (with a slight abuse of the notation ,) the family generates as an -operator group: With the notation (, ), we can write any member of in the form

but does not determine such an expression uniquely. Thus the issue of understanding the structure of as an -operator group arises. Heuristically, an identity among relations is such a specified product that recovers the identity element of . For example, consider the presentation

of the symmetric group on three letters. Straightforward verification shows that

is an identity among the relations in 3. A standard procedure enables us to read off this identity and others from the following prism-shaped tesselated -sphere:

The reader will notice that reading along the boundaries of the faces recovers the relators in 3. The projection, to a plane, of this prism-shaped -sphere with one of the triangles removed is BH82, Fig. 2 p. 155. In Section 10 of BH82, the reader can find precise methods to obtain such identities from “pictures”, see in particular BH82, Fig. 12 p. 194 for the case at hand, and in CH82 from diagrams etc. The corresponding term in Pei49 is “Randwegaggregat”.

To develop a formal understanding of the situation, let be the free -operator group on . The kernel of the canonical epimorphism from the free group on the (disjoint) union to realizes . Let denote the canonical homomorphism. The members of the kernel of are the identities among the relations (or among the relators) for the presentation Pei49Rei49BH82; Turing refers to them as “relations between the relations” Tur38, Section 2.

The Peiffer elements , as and range over , are identities that are always present, independently of any particular presentation under discussion. Following Rei49, let be the quotient group of modulo the subgroup in , necessarily normal, which the Peiffer elements generate. The Peiffer subgroup is an -subgroup, whence the -action on passes to an -action on , and the canonical homorphism induces a homomorphism that turns into a crossed module. In particular, the kernel , being central in , is an abelian group, the -action on factors through a -module structure, and parametrizes equivalence classes of “essential” or nontrivial identities associated to the presentation of Rei49. We shall see below that there are interesting cases where is trivial, that is, the Peiffer identities generate all identities.

Given a group , a -crossed module is a -group together with a homomorphism of -groups such that is a crossed module. Given a set and a set map into a group , the free crossed -module on is the crossed module enjoying the following property: Given a -crossed module and a set map , there is a unique homomorphism of -groups such that is a morphism of crossed modules. A standard construction shows that this free -crossed module always exists. By the universal property, such a free -crossed module is unique up to isomorphism, whence it is appropriate to use the definite article here. The -crossed module just constructed from a presentation of a group is plainly the free -crossed module on the injection . The resulting extension of -operator groups then displays the -crossed module as the quotient of a free -crossed module and thereby yields structural insight. It is also common to refer to as the free crossed module on .

4. Group Extensions and Abstract Kernels

Given two groups and , the issue is to parametrize the family of groups that contain as a normal subgroup and have quotient isomorphic to , or equivalently, in categorical terms, the family of groups that fit into an exact sequence of groups of the kind

Given an extension of the kind 6, conjugation in induces a homomorphism from to . It is common to refer to a triple that consists of two groups and together with a homomorphism as an abstract kernel or to the pair as an abstract -kernel. In Bae34 the terminology is “Kollektivcharakter”. We have just seen that a group extension determines an abstract kernel. Given two groups and together with an abstract kernel structure , the extension problem consists in realizing the abstract kernel, that is, in parametrizing the extensions of the kind 6 having as its abstract kernel provided such an extension exists, and the abstract kernel is then said to be extendible.

When is abelian, the group of outer automorphisms of amounts to the group , an abstract -kernel structure on is equivalent to a -module structure on , and the semidirect product group shows that this abstract kernel is extendible. When is nonabelian, not every abstract -kernel is extendible, however, Bae34, p. 415. We shall shortly see that nonextendible abstract kernels abound in mathematical nature.

For two homomorphisms and , let denote the pullback group, that is, the subgroup of that consists of the pairs that have the same image in .

For a crossed module , the action of on plainly defines an abstract kernel structure . On the other hand, given the abstract kernel , the pullback group and the canonical homomorphism which the crossed module structure induces combine to the crossed module , and coincides with the center of . In fact, this correspondence is a bijection between abstract kernels and crossed modules having as the center of . The group occurs in Bae34 as the “Aufloesung des Kollektivcharakters” .

We will now explain the solution of the extension problem: Consider an abstract kernel , and let be its associated crossed module. Let be a presentation of the group and, exploiting the freeness of , choose the set maps and in such a way that (i) the composite of the induced homomorphism with the epimorphism from to coincides with the epimorphism from to and (ii) the composite of with the injection from to coincides with the composite of with . In view of the universal property of the free -crossed module , the set map induces a morphism of crossed modules defined on the free crossed module on . Let and display the resulting morphism of crossed modules as

(This is Diagram 8 in Hue21, Section 3). The following is a version of Tur38, Theorem 4 p. 356 in modern language and terminology; see Hue21, Section 3 for details. Crossed modules, in particular free ones, exact sequences, commutative diagrams, pullbacks, etc. were not at Turing’s disposal, however, and he expressed his ideas in other ways. ⁠Footnote2

2

A. Turing and J.H.C. Whitehead worked as WWII codebreakers at Bletchley Park (UK) but we shall never know whether they then discussed the idea of a crossed module.

Theorem.

The abstract kernel is extendible if and only if, once has been chosen, the set map can be chosen in such a way that, in Diagram 7, the restriction of to is zero.

Proof.

It is immediate that the condition is necessary. To establish the converse, suppose, having been chosen, there is a choice of with the asserted property. Let denote the normal closure of in . From 7, we deduce the morphism

of crossed modules. Since 8 is a morphism of crossed modules, in the semidirect product group , the members of the kind as ranges over constitute a subgroup, necessarily normal. The quotient group of by this normal subgroup yields the group extension of by we seek.

This modern proof of the theorem is in Hue80a, Section 10 (phrased in terms of the coequalizer of and ). The diligent reader will notice that the proof is complete, i.e., no detail is left to the reader. It is an instance of a common observation to the effect that mathematics consists in continuously improving notation and terminology. Even nowadays there are textbooks that struggle to explain the extension problem and its solution in terms of lengthy and unilluminating cocycle calculations, and it is hard to find the above theorem, only recently dug out as a result of Turing’s Hue21.⁠Footnote3 Eilenberg–MacLane (quoting Tur38 but apparently not understanding Turing’s reasoning) developed the obstruction for an abstract kernel to be extendible in terms of the vanishing of the class of a group cohomology 3-cocycle of with values in EM47, Theorem 8.1. The unlabeled arrow in Diagram 7 recovers this 3-cocycle, and the condition in the theorem says that the class of this -cocycle in the group cohomology group is zero. Indeed, once a choice of and has been made, the vanishing of that -cocycle is equivalent to the unlabeled -module morphism in 7 admitting an extension to a homomorphism of -groups, and setting and substituting for , we obtain a diagram of the kind 7 having zero.

3

Turing would have slipped through an evaluation system based on bibliographic metrics.

To put flesh on the bones of the last remark, recall the augmentation map of a group defined on the integral group ring of by for and write the augmentation ideal as . The -action on induces a -action on the abelianized group in such a way that the set constitutes a set of free generators; thus is canically isomorphic to the free -module freely generated by (with a slight abuse of the notation ). The induced morphism of -modules is still injective. For and , using the fact that is a family of free -generators of , define by the identity , and let denote the image. Defining by where ranges over and over , and by where denotes the image of , as ranges over , renders the sequence

exact, and the three middle terms thereof together with the corresponding arrows constitute the beginning of a free resolution of in the category of -modules. When we take to be the standard presentation having the set of all members of distinct from , we obtain the first three terms of the standard resolution, and the composite of the -module epimorphism defined on the fourth term of the standard resolution with the above unlabeled arrow verbatim recovers the -cocycle in EM47, Theorem 8.1.

A congruence between two group extensions of a group by a group is a commutative diagram

with exact rows in the category of groups. For a group with center , the multiplication map restricts to a homomorphism .

Complement.

Let be an extendible abstract kernel and realize it by the group extension . The assignment to an extension of by the center of realizing the induced -module structure on of the group extension which the commutative diagram

with exact rows in the category of groups characterizes induces a faithful and transitive action of on the congruence classes of extensions of by realizing the abstract kernel .

As for the proof we note that, as before, since the left-hand and middle vertical arrows in 11 constitute a morphism of crossed modules, in the group , the triples , as ranges over and over , form a normal subgroup, and the group arises as the quotient of by this normal subgroup. When is abelian, it coincides with its center, Diagram 11 displays the operation of “Baer sum” of two extensions with abelian kernel, and the complement recovers the fact that the congruence classes of abelian extensions of by constitute an abelian group, the group structure being induced by the operation of Baer sum Bae34, indeed, the group cohomology group .

5. Combinatorial Group Theory and Low-dimensional Topology

Consider a space and a subspace (satisfying suitable local properties, being a CW complex and a subcomplex would suffice), and let be a base point in . The standard action of the fundamental group (based homotopy classes of continuous maps from a circle to with a suitably defined composition law) on the second relative homotopy group (based homotopy classes of continuous maps from a disk to such that the boundary circle maps to with a suitably defined composition law) and the boundary map turn into a crossed -module Whi41. See BHS11 for a leisurely introduction to homotopy groups building on an algebra of composition of cubes. Suffice it to mention that the higher homotopy groups of a space acquire an action of the fundamental group. In Whi41, J.H.C. Whitehead in particular proved that, when the space arises from by attaching -cells, the crossed -module is free on the homotopy classes in of the attaching maps of the -cells.

The Cayley graph of a group with respect to a family of generators is the directed graph having as its set of vertices and, for each pair , an oriented edge joining to . The oriented graph that underlies 5 is the Cayley graph of the group with respect to the generators and in 3. The geometric realization of a presentation of a group is a -dimensional CW complex with a single zero cell , having -cells in bijection with and -cells in bijection with in such a way that the fundamental group of the -skeleton amounts to the free group on and that the attaching maps of the -cells define, via the boundary map , the members of . By construction, the fundamental group of is isomorphic to , the fundamental group is canonically isomorphic to the free group on the generators and, by Whitehead’s theorem, the -crossed module is free on the attaching maps of the -cells and hence canonically isomorphic to the -crossed module written above as . The -skeleton of the universal covering space of then has fundamental group isomorphic to the normal closure of the relators in and, together with the appropriate orientation of its edges, then amounts to the Cayley graph of with respect to . Every -dimensional CW complex with a single -cell is of this kind. Since the higher homotopy groups of are zero, the long exact homotopy sequence of the pair reduces to the crossed -fold extension

Thus the group of “essential identities” among the relations amounts to the second homotopy group of (based homotopy classes of continuous maps from a -sphere to , with a suitably defined group structure). For illustration, consider the -skeleton of the prism-shaped tesselated -sphere 5. For each -gon and each -gon, attach two copies of the disk bounding it and, likewise, for each triangle, attach three copies of the disk bounding it. Thus we obtain a prism-shaped -complex having, beyond the six vertices and twelve edges, six faces bounded by -gons, six faces bounded by -gons, and six faces bounded by triangles, and this -complex realizes the universal covering space of the geometric realization of 3. By its very construction, it has eleven “chambers” (combinatorial -spheres). Since is simply connected, we conclude its second homotopy group is free abelian of rank eleven. For the inexperienced reader we note that choosing a maximal tree in and deforming this tree to a point is a homotopy equivalence, and the result is a bunch of eleven -spheres, necessarily having second homology group free abelian of rank eleven. Since is simply connected, the Hurewicz map from to this homology group is an isomorphism, and so is the homomorphism which the covering projection induces. Hence is free abelian of rank eleven. Under the isomorphism of crossed -modules, the based homotopy class of the innermost chamber goes to the class of the identity 4 in , necessarily nontrivial, as are the identities that correspond to the other chambers. Another interesting piece of information we extract from 5 is that , being the fundamental group of the Cayley graph of is, as a group, freely generated by any seven among the eight constituents of 4.

The paper Pei49 arose out of a combinatorial study of -manifolds: The present discussion applies to the -skeleton of a cell decomposition of a -manifold , and attaching -cells to build the -manifold under discussion “kills” some of the essential identities associated with : The commutative diagram

with exact rows displays how the -crossed module arises from the free -crossed module . For these and related issues, see, e.g., BH82 and the literature there. For illustration, consider a finite subgroup of such as, e.g., , the cyclic group of order , or the quaternion group of order eight. The orbit space , a lens space when , is a -manifold having nontrivial but trivial since the -sphere has trivial second homotopy group. Via Papakyriakopoulos’s sphere theorem, the second homotopy group of a general -manifold being nontrivial is equivalent to the manifold being geometrically splittable, similarly to what we hint at for tame links below. Thus a -manifold is geometrically splittable if and only if attaching the -cells of a cell decomposition does not kill all the essential identities arising from its -skeleton.

A -complex is said to be aspherical when its degree homotopy groups are trivial. This is equivalent to the second homotopy group being trivial. Lyndon’s identity theorem Lyn50 implies that, when has a single -cell, the -complex is aspherical if and only if the relator arising as the boundary of the -cell is not a proper power in . Thus the -crossed module structure of the normal closure of in is free in this case. For example, a closed surface distinct from the -sphere is aspherical, but this is an immediate consequence of the universal covering being the -plane. In the same vein, the exterior of a tame link in the -sphere is homotopy equivalent to a -complex—the operation of “squeezing” the -cells of a cell decomposition achieves this—and the link is geometrically unsplittable if and only if the exterior and hence the