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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 about Bridgeman's orthospectrum identity. Those interest in the question above might want to take a look at it.

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up vote 11 down vote accepted

One basic answer is given by hyperbolic geometry.

Ideal tetrahedra in hyperbolic 3-space H^3 are equivalent (under the action of the automorphism group PGL_2(C)) to tetrahedra with vertices {0,1,oo,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,oo,z} is equivalent under PSL_(2)(C) to {0,1,oo,1/(1-z)} and {0,1,oo,1-1/z}, and so we get formulae:

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

The tetrahedron {0,1,oo,z} is also equivalent to {0,1,oo,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 P^1(C) of H^3. If we take the tetrahedron {0,1,oo,y}, we can break it off into {0,1,oo,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(y/x) - D((1-1/x)/(1-1/y)) + D((1-x)/(1-y)) = 0,which gives a proof of Abel's equation.

Let's think some more about a closed hyperbolic 3-manifold M. By definition, M = H^3/Gamma for a lattice Gamma in PSL_2(C). Since H^3 is contractible, M is a K(Pi,1) space, and so there is a canonical isomorphism H_*(M,Z) = H_*(Gamma,Z), comparing simplicial homology with the group homology of Gamma. Now M has a fundamental class [M] in H_3(M,Z), which gives an element in H_3(Gamma,Z) and hence also a class in H_3(PSL_2(C),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(C) and with relation s 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(C),Z) to the Bloch group.

Note that it is not obvious that the z_i can be taken inside some field F, this is a consequence of Mostow Rigidity. It turns out that if we take the Bloch group B(F) generated by elements of F, this is, by work of Suslin, essentially equal to K_3(F).

To summarize, the connection between the identity, the cohomology of PSL_2(C), and the Bloch group is well understood, see some papers by Walter Neumann. For the connection between the Bloch group B(F) and K_3(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.)

If I could, I would add a flag to this post "hyperbolic geometry".

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Thanks. Very nice answer! I will try to add the hyperbolic-geometry tag to the question. – Jorge Vitório Pereira Oct 25 '09 at 4:35
added a hyperbolic-geometry tag to the question – Greg Stevenson Oct 25 '09 at 4:44

There is a remarkable article, "The remarkable dilogarithm," J. Math. Phys. Sci. 22 (1988), 131--145, by Don Zagier, which was recently reprinted and updated as "The dilogarithm function" (63 pages!) in one of the collections by Springer Verlag.

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For the relation to motives (and K-theory), I'd suggest the first several article of Motives, volume 2 (the proceedings of the Seattle conference link text). I don't really know this stuff, but it is apparently believed that the polylogarithms are related to the "higher regulators" from K-theory to Deligne cohomology. These regulators are supposed to help explain the values of L-functions of motives at integers. Apparently the usual logarithm occurs in the first chern class of a variety (and the regulators are thought of as generalizations of chern class, or something). Good luck.

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There is a little book by Bloch, called Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. It was published quite recently but is based on a famous lecture series from the late 70s or so. He treats the dilog specifically, rather than the more general polylog framework referred to by Rob. See in particular chapter 6.

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