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# Crossed Modules

Communicated by *Notices* Associate Editor Steven Sam

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.Footnote^{1} 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 group -(operator) (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 with respect to conjugation, and let -group be a homomorphism of The triple -groups. 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 , on -action 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

2as a *crossed extension of -fold 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 homomorphism -equivariant 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 of the images in -conjugates of the members of generate that is, (with a slight abuse of the notation , the family ,) generates as an group: With the notation -operator ( , 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 group arises. Heuristically, an -operator*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

4is 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:

5The 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 with one of the triangles removed is -sphereBH82, 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 group on -operator 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 whence the -subgroup, on -action passes to an on -action 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 , on -action factors through a structure, and -module 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 such that -groups 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 module -crossed and a set map there is a unique homomorphism , of such that -groups is a morphism of crossed modules. A standard construction shows that this free 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 -crossed just constructed from a presentation of a group is plainly the free module on the injection -crossed The resulting extension . of groups then displays the -operator module -crossed as the quotient of a free module and thereby yields structural insight. It is also common to refer to -crossed 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

6Given 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 , structure on -kernel is equivalent to a structure on -module 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 . module -crossed 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

7(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. Footnote^{2}

^{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.

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.Footnote^{3} 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 in the group cohomology group -cocycle is zero. Indeed, once a choice of and has been made, the vanishing of that is equivalent to the unlabeled -cocycle morphism -module in 7 admitting an extension to a homomorphism of and setting -groups, and substituting for we obtain a diagram of the kind ,7 having zero.

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 . on -action induces a on the abelianized group -action 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 is still injective. For -modules and using the fact that , is a family of free of -generators 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 When we take -modules. 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 epimorphism -module defined on the fourth term of the standard resolution with the above unlabeled arrow verbatim recovers the in -cocycleEM47, 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 , .

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 -moduleWhi41. 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 the crossed -cells, -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 CW complex with a single zero cell -dimensional having , in bijection with -cells and in bijection with -cells in such a way that the fundamental group of the amounts to the free group on -skeleton and that the attaching maps of the define, via the boundary map -cells 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 module -crossed is free on the attaching maps of the and hence canonically isomorphic to the -cells module written above as -crossed The . -skeleton of the universal covering space

Thus the group of “essential identities” among the relations

The paper Pei49 arose out of a combinatorial study of

with exact rows displays how the *lens space* when *sphere theorem*, the second homotopy group of a general

A *aspherical* when its degree