Hilbert's 3rd problem,number theory, motives, cyclic homology,...

Question

This talk by Jinhyun Park connects a lot of interesting themes, making me curious to read more about that. Do you know where?

Answer 1

This circle of topics is certainly one of my favourite surprising connections in mathematics. I will try to outline what little I understand of the big picture. Apologies for the length.

Hilbert's 3rd problem and Dehn complexes: As is well-known, Hilbert's 3rd problem asked for examples of tetrahedra of equal volume which are not scissors congruent, and Dehn solved it using the invariant now named after him. However, this only gave rise to further questions: what about higher-dimensional euclidean spaces, and what about the other classical geometries? By now, I guess the proper objects to consider are the Sah algebra and Dehn complexes I will outline.

To study scissors congruences, it is best to consider the scissors congruence groups for all dimensions at the same time. Join of simplices induces a product, dimension a grading. In the spherical case, the Dehn invariants provide a coproduct, and the duality of spherical simplices gives an involution, making the total spherical scissors congruences into a graded Hopf algebra. This object is called Sah algebra.

The coproduct statement does not work in the other geometries since the dihedral-angle part of the Dehn invariant always introduces a spherical geometry component. However, in the other geometries, we still get comodule structures over the spherical scissors congruences. It is also possible (and interesting) to consider the graded cobar complexes for these modules over the coalgebra of spherical scissors congruences. These complexes were introduced by Goncharov who called them Dehn complexes (in the JAMS paper discussed below).

The Hopf algebra and comodule picture is discussed in the following book (not easy to get but very much worth reading):

• C.-H. Sah. Hilbert's third problem: scissors congruence. Research notes in mathematics 33, Pitman, 1969.

Another book concerning scissors congruences (also very much worth reading but does not discuss the Hopf algebra view) is:

• J.-L. Dupont. Scissors congruences, group homology and characteristic classes. Nankai tracts in mathematics 1, World Scientific, 2001.

A very general form of Hilbert's 3rd problem can then be formulated for all the classical geometries: are Dehn invariant and volume sufficient to completely characterize the scissors congruence class of a polytope? This is only known for euclidean space in dimensions $\leq 4$, for hyperbolic and spherical not even in dimension $3$. Even more general, one can wonder about the exact computation of the cohomology of the Dehn complexes.

Goncharov's conjectures then relate the above Dehn complexes to K-theoretic stuff. In the spherical and hyperbolic cases, Goncharov conjectures that the Dehn complexes compute the ($+1$ and $-1$-eigenspaces of complex conjugation on the) weight-graded pieces of algebraic K-theory of $\mathbb{C}$. This is formulated and discussed in

• A. Goncharov. Volumes of hyperbolic manifolds and mixed Tate motives. J. Amer. Math. Soc. 12 (1999), 569-618.

Maybe it would also make sense to expect an explicit quasi-isomorphism between Bloch's cycle complexes (computing motivic cohomology) and the Dehn complexes. I am not sure if a formulation like this exists in the literature, but probably the above conjecture says something like ``the Sah algebra should be isomorphic to the motivic Galois group of mixed Tate motives over $\mathbb{C}$'' (only that we do not have the latter).

Anyway, the conjecture is known in weight 2 due to the work of Dupont, Sah, Bloch, Suslin (this goes under the name of motivic weight $2$ complex, dilogarithm complex etc). The trilogarithm case has been worked out by Goncharov, see e.g.

• A. Goncharov. Geometry of configurations, polylogarithms and motivic cohomology. Adv. Math. 144 (1995), 197-318.

Regulators, volumes and number theory Goncharov's conjecture translates the regulator on K-theory to the volume of hyperbolic simplices (with Dehn invariant $0$). From this point of view, Goncharov's conjecture is a far-reaching generalization of the fact that volumes of hyperbolic 3-manifolds can be expressed in terms of dilogarithms. Under Goncharov's conjecture, the generalized version of Hilbert's 3rd problem for hyperbolic spaces and spheres translates into an injectivity conjecture for K-theoretic regulators due to Ramakrishnan. Currently, it is not even known if $K_3(\mathbb{C})$ (which is relevant for scissors congruences in $\mathbb{H}^3$) is bigger than $K_3(\overline{\mathbb{Q}})$ (which we understand in terms of Borel regulators). This is the number theory appearing in the title (and might be related to other conjectures on the description of the period algebra which I know nothing about).

Probably the consequences relating regulators and volumes actually motivated the conjecture in the first place. Certainly the relation between K-theory and polylogarithms is part of the motivation for Bloch's definition of mixed Tate motives as modules over a suitable Lie algebra.

Euclidean geometry and cyclic homology: So far, we have talked about the relation between the noneuclidean geometries and algebraic K-theory. Now, infinitesimally, hyperbolic space looks like euclidean space. As a consequence, euclidean Dehn complexes are the ``tangent spaces'' to hyperbolic Dehn complexes. Again, Goncharov has a precise conjecture how the euclidean Dehn complex should be related to mixed Tate motives over the dual numbers $\mathbb{C}[\epsilon]/(\epsilon^2)$, cf.

• A. Goncharov. Euclidean scissors congruence groups and mixed Tate motives over dual numbers. Math. Res. Lett. 11 (2004), 771-784.

This is how we get a relation to cyclic homology, because cyclic homology is related to the tangent space of algebraic K-theory. This way, the euclidean part of Hilbert's 3rd problem involves talking about additive Chow groups (about which you can learn by reading e.g. the papers of Park, but also Cathelineau, Bloch, Esnault,...).

You see the appearance of groups related to cyclic homology by reformulating the classical Dehn-Sydler theorem as an exact sequence: $$ 0\to \mathbb{R}\to\operatorname{SC}(\mathbb{E}^3)\stackrel{D}{\longrightarrow}\mathbb{R}\otimes S^1\to\Omega^1_{\mathbb{R}/\mathbb{Q}}\to 0, $$ where the map $D$ is the Dehn invariant from 3-dimensional scissors congruences (mapping a three-simplex to the sum of edge lengths tensor dihedral angles). The kernel of the Dehn invariant is detected by the volume map to $\mathbb{R}$, and the cokernel of the Dehn invariant is a group of Kähler differentials.

Further reading: check the works of Sah, Dupont, Cathelineau, Goncharov,... There is too much literature to mention in this already oversized answer.

Answer 2

"What is motivic measure?" by Hales. http://www.ams.org/journals/bull/2005-42-02/S0273-0979-05-01053-0/home.html