We consider a presentation, $\mathcal{P} = (X : R)$, of a group $G$. We thus have a short exact sequence,
where $F = F(X)$, the free group on the set $X$, $R$ is a subset of $F$ and $N = N(R)$ is the normal closure in $F$ of the set $R$. The group, $F$, acts on $N$ by conjugation: ${}^u c = u c u^{-1}$, $c\in N$, $u \in F$, and the elements of $N$ are words in the conjugates of the elements of $R$:
often says such elements are consequences of $R$. Heuristically, an identity among the relations of $\mathcal{P}$ is such an element, $c$, which equals 1.
The problem of what this actually means is analogous to that of working with a relation, since, for example, in the presentation, $( a : a^3)$, of $C_3$, the cyclic group of order 3, if $a$ is thought of as being an element of $C_3$, then $a^3 = 1$! Doesnβt that say the presentation is $( a : 1)$? Why is this then different from the situation with the βpresentationβ, $( a : a = 1)$?
To get around that mess, the free group on the generators, $F(X)$, was introduced and, of course, in $F(\{a\})$, $a^3$ is not 1. The relation is thus thought of as being a piece of data that is giving the instruction to rewrite that element to 1.
An analogous free algebraic device, namely a free crossed module on the presentation can be introduced to handle the identities.
Before giving the formal definition we will look at some examples.
Suppose $r\in R$, but it is a power of some element $s\in F$, i.e., $r = s^m$. Of course, $r s = s r$ and
so ${}^s r . r^{-1}$ is an identity. In fact, there will be a unique $z\in F$ with $r = z^q$, $q$ maximal with this property. This $z$ is called the root of $r$ and if $q \gt 1$, $r$ is called a proper power.
Here we take up the example from Cayley graph:
Consider one of the standard presentations of $S_3$,
Write $r = a^3$, $s = b^2$, $t = (ab)^2$. Here the presentation leads to $F= F(a,b)$, free of rank 2, and $N(R) \subset F$, so it must be free as well, by the Nielsen-Schreier theorem. Its rank will be 7, given by the Schreier index formula or, geometrically, it will be the fundamental group of the Cayley quiver, also called the Cayley graph, of the presentation. This group is free on generators corresponding to edges outside a maximal tree.
The set of normal generators of $N(R)$ has 3 elements; $N(R)$ is free on 7 elements (corresponding to the edges not in the tree), yet it is specified as consisting of products of conjugates of $r$, $s$ and $t$, and there are infinitely many of these conjugates. Clearly there must be some slight redundancy, so there must be some identities among the relations!
Note that the Cayley graph is planar.
A path around the first triangle, $1 \to a \to a^2$, corresponds to the relation $r$; each other region corresponds to a conjugate of one of $r$, $s$ or $t$. (It may help in what follows to think of the graph being embedded on a 2-sphere, so βouterβ and βoutsideβ mean βround the back faceβ.) Consider a loop around a region.
Pick a path to a start vertex of the loop, a path starting at 1. For instance the path that leaves 1 and goes along edges labelled $a$, $b$ and then goes around $a a a$ before returning by $b^{-1} a^{-1}$ gives $a b r b^{-1} a^{-1}$. Now the path around the outside can be written as a product of paths around the inner parts of the graph, e.g. $(a b a b) b^{-1} a^{-1} b^{-1} (b b) (b^{-1}a^{-1} b^{-1} a^{-1}) \ldots$ and so on, thus $r$ can be written in a non-trivial way as a product of conjugates of $r$, $s$ and $t$. (An explicit identity constructed like this is given in the paper by Brown and Huebschmann (see below).)
In a presentation of the free Abelian group on 3 generators, one would expect the commutators, $[x,y]$, $[x,z]$ and $[y,z]$. There is a well-known identity, that expands out to give an identity among these relations (again see Brown-Huebschman (below), p.154, or Loday, (again below) for a complete treatment.)
In a bit more detail, look at this as a presented group,
then label the relators $r_x= [y,z]$, $r_y=[z,x]$, $r_z= [x,y]$.
There is an identity:
which is easily checked to hold just by expanding it out and cancelling terms. This is one of the (many) forms of the well-known identity known variously as the Hall-Witt identity, the Jacobi-Hall-Witt identity, the Jacobi identity, and various variants of these. There is thus a word,
which can be written out as a product of conjugates of the three relations, $r_x$, $r_y$ and $r_z$, and which evaluates to 1 when considered as an element of $F(x,y,z)$. It is thus another case of an identity among the relations for this presented group.
Note the Cayley quiver of this presentation is infinite.
We can now formally define the module of identities of a presentation, $\mathcal{P} = (X : R)$.
We first form the free crossed module on the mapping $R\to F(X)$, which we will denote by $\partial : C(\mathcal{P}) \to F(X)$.
The module of identities , $\kappa(\mathcal{P})$, of $\mathcal{P}$ is $Ker\, \partial$.
An identity among the relations of $\mathcal{P}$ is then an element of $\kappa(\mathcal{P})$.
By construction, the group presented by $\mathcal{P}$ is $G \cong F(X)/Im\, \partial$, where $Im \,\partial$ is just the normal closure of the set, $R$, of relations and we know that $Ker\, \partial$ is a $G$-module. In fact we have an exact sequence:
R. Brown and J. Huebschmann, Identities among relations, in R.Brown and T.L.Thickstun, eds., Low Dimensional Topology, London Math. Soc Lecture Notes, Cambridge University Press, (1982).
M. Kapranov and M. Saito, 1999, Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions, in Higher homotopy structure in topology and mathematical physics (Poughkeepsie, N.Y. 1996), volume 227 of Contemporary Mathematics, 191 β 225, AMS
J.-L. Loday, 2000, Homotopical Syzygies, in Une dΓ©gustation topologique: Homotopy theory in the Swiss Alps, volume 265 of Contemporary Mathematics, 99 β 127, AMS.