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\begin{document}

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\section*{identity among the relations}

\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

We consider a presentation, $\mathcal{P} = (X : R)$, of a group $G$. We thus have a short exact sequence,

\begin{displaymath}
1\to N \to F \to G \to 1,
\end{displaymath}
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 $R$:

\begin{displaymath}
c = {}^{u_1}(r_1^{\varepsilon_1}){}^{u_2}(r_2^{\varepsilon_2})\ldots 
{}^{u_n}(r_n^{\varepsilon_n}),
\end{displaymath}
often says such elements are \emph{consequences} of $R$. Heuristically, an \textbf{identity among the relations} of $\mathcal{P}$ is such an element, $c$, which equals 1.

The problem of what this actually \emph{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 $a^3 = 1$! Doesn't that say the presentation is $( a : 1)$? Why is this then different from the situation with the `presentation', $( a : a = 1)$?

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.

\hypertarget{example_proper_powers}{}\subsection*{{Example (Proper powers)}}\label{example_proper_powers}

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

\begin{displaymath}
{}^s r r^{-1} = 1
\end{displaymath}
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 \emph{root of} $r$ and if $q \gt 1$, $r$ is called a \emph{proper power}.

\hypertarget{example_standard_presentation_of_}{}\subsection*{{Example (Standard presentation of $S_3$)}}\label{example_standard_presentation_of_}

Here we take up the example from [[Cayley graph]]:

Consider one of the standard presentations of $S_3$,

\begin{displaymath}
(a,b  :  a^3, b^2, (ab)^2).
\end{displaymath}
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 \emph{slight} redundancy, so there must be some identities among the relations!

Note that the Cayley graph is [[planar graph|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 `outer' and `outside' mean `round the back face'.) 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).)

\hypertarget{example_free_abelian_group_}{}\subsection*{{Example (Free Abelian group $\mathbb{Z}^3$)}}\label{example_free_abelian_group_}

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 \emph{presented group},

\begin{displaymath}
G=\mathbb{Z}^{3}=\langle x,y,z\mid [y,z],[z,x],[x,y]\rangle
\end{displaymath}
then label the relators $r_x= [y,z]$, $r_y=[z,x]$, $r_z= [x,y]$.

There is an identity:

\begin{displaymath}
[[x,y],{}^y z].[[y,z],{}^z x].[[z,x],{}^x y] =1
\end{displaymath}
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 \emph{Hall-Witt identity}, the \emph{Jacobi-Hall-Witt identity}, the \emph{Jacobi identity}, and various variants of these. There is thus a word,

\begin{displaymath}
[r_z,{}^y z].[r_x,{}^z x].[r_y,{}^x y]
\end{displaymath}
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 \emph{identity among the relations} for this presented group.

Note the [[Cayley quiver]] of this presentation is infinite.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

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 $\partial : C(\mathcal{P}) \to F(X)$.

\begin{defn}
\label{module}\hypertarget{module}{}
The \emph{module of identities} , $\kappa(\mathcal{P})$, of $\mathcal{P}$ is $Ker\, \partial$.

\end{defn}
An \emph{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:

\begin{displaymath}
0\to \kappa(\mathcal{P}) \to C(\mathcal{P}) \xrightarrow{\partial} F(X)\to G\to 1.
\end{displaymath}
\hypertarget{related_entries}{}\subsection*{{Related entries}}\label{related_entries}

\begin{itemize}%
\item [[Cayley graph]]


\item [[homotopical syzygy]]


\item [[homological syzygy]]



\end{itemize}
\hypertarget{references}{}\subsection*{{References:}}\label{references}

\begin{itemize}%
\item [[R. Brown]] and [[J. Huebschmann]], \emph{Identities among relations}, in R.Brown and T.L.Thickstun, eds., Low Dimensional Topology, London Math. Soc Lecture Notes, Cambridge University Press, (1982).


\item [[M. Kapranov]] and [[M. Saito]], 1999, \emph{Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions}, in \emph{Higher homotopy structure in topology and mathematical physics} (Poughkeepsie, N.Y. 1996), volume 227 of Contemporary Mathematics, 191 – 225, AMS


\item [[J.-L. Loday]], 2000, \emph{Homotopical Syzygies}, in \emph{Une dégustation topologique: Homotopy theory in the Swiss Alps}, volume 265 of Contemporary Mathematics, 99 – 127, AMS.



\end{itemize}
[[!redirects identities among relations]] [[!redirects identities among the relations]]



\end{document}