Abel's identity for the dilogarithm (see the wikipedia page about polylogarithms)
plays a role in web geometry as it is one of the abelian relations of the
first example of exceptional web (Bol's 5-web) to appear in the literature. 


I have heard it is important in other domains (cohomology of SL(3,C), algebraic K-theory, motives ). 
I would like to learn more about it. 


I am asking for:


1. Insights on why Abel's identity is relevant in this or that field;
2. References where it plays a role.




---


**Edit.**
I have just learned from this [blog](http://lamington.wordpress.com/) about [Bridgeman's orthospectrum identity](http://lamington.wordpress.com/2009/10/24/bridgemans-orthospectrum-identity/).
Those interest in the question above might want to take a look at it.



Abel's equation for the dilog
One basic answer is given by hyperbolic geometry.


Ideal tetrahedra in hyperbolic 3-space $\mathbb{H}^3$ are equivalent (under the action of the automorphism group $PGL\_2(C))$ to tetrahedra with vertices $\{0,1,\infty,z\}$, and their volume is given by $D(z)$, where $D(z)$ is the Bloch-Wigner dilogarithm, which is a slightly modified version of the dilogarithm. This amounts to writing down the hyperbolic metric and evaluating an integral, which turns out to be (very close to) $Li\_2(z)$ (although it is real valued for complex $z$).


The tetrahedron $\{0,1,\infty,z\}$ is equivalent under $PSL\_2(\mathbb C)$ to $\{0,1,\infty,1/(1-z)\}$ and $\{0,1,\infty,1-1/z\}$, and so we get formulae:


$$D(z) = D(1/(1-z)) = D(1 - 1/z).$$


The tetrahedron $\{0,1,\infty,z\}$ is also equivalent to $\{0,1,\infty,1/z\}$, except with an odd permutation of the vertices, and thus: $D(z) = - D(1/z).$


Finally, choose a random point $y$ in the boundary $\mathbb P^1(\mathbb C)$ of $\mathbb H^3$. If we take the tetrahedron $\{0,1,\infty,y\}$, we can break it off into $\{0,1,\infty,x\}$ and three other tetrahedra (just like in Euclidean space). Transforming the coordinates of the other three tetrahedra into the standard form gives the 5-term relation:


$$D(x) - D(y) + D\left(\dfrac yx\right) - D\left(\dfrac {1-x^{-1}}{1-y^{-1}}\right) + D\left(\dfrac {1-x}{1-y}\right) = 0,$$


which gives a proof of Abel's equation.


Let's think some more about a closed hyperbolic 3-manifold $M$. By definition, $M = \mathbb H^3/\Gamma$ for a lattice $\Gamma$ in $PSL\_2(\mathbb{C})$. Since $\mathbb{H}^3$ is contractible, $M$ is a $K(\pi,1)$ space, and so there is a canonical isomorphism $H\_\*(M, \mathbb{Z}) = H\_\*(\Gamma, \mathbb{Z})$, comparing simplicial homology with the group homology of $\Gamma$. Now $M$ has a fundamental class $[M]$ in $H\_3(M, \mathbb{Z})$, which gives an element in $H\_3(\Gamma, \mathbb{Z})$ and hence also a class in $H\_3(PSL\_2(\mathbb{C}), \mathbb{Z})$.
On the other hand, $[M]$ can be decomposed ("triangulated") into ideal tetrehedra with parameters $z\_i$. The set of parameters $[z\_i]$ is not unique, however, the only real "move" is the subdivision of tetrahedra, and so associated to $M$ we get an element of the group generated by $[z\_i]$ for $z\_i$ in $P^1(\mathbb{C})$ and with relations exactly of the form satisfied by $D$ above. This is essentially the definition of the Bloch group. $D$ is a function this group, and this decomposition gives a map from $H\_3(PSL\_2(\mathbb{C}), \mathbb{Z})$ to the Bloch group.


Note that it is not obvious that the $z\_i$ can be taken inside some field $\mathbb{F}$, this is a consequence of Mostow Rigidity. It turns out that if we take the Bloch group $B(\mathbb{F})$ generated by elements of $\mathbb{F}$, this is, by work of Suslin, essentially equal to $K\_3(\mathbb{F})$.


To summarize, the connection between the identity, the cohomology of $PSL\_2(\mathbb{C})$, and the Bloch group is well understood, see some papers by Walter Neumann. For the connection between the Bloch group $B(\mathbb{F})$ and $K\_3(\mathbb{F})$, see papers of Suslin. The connection with motives is more speculative, but here you should look at some papers of Goncharov.


(There are some generalizations/connections to higher regulators for K-groups, but this is a very nice example to understand, being both somewhat accessible yet still very interesting.)


