The extended Bloch group and algebraic K-theory

We define an extended Bloch group for an arbitrary field F, and show that this group is canonically isomorphic to K_3^ind(F) if F is a number field. This gives an explicit description of K_3^ind(F) in terms of generators and relations. We give a concrete formula for the regulator, and derive concrete symbol expressions generating the torsion. As an application, we show that a hyperbolic 3-manifold with finite volume and invariant trace field k has a fundamental class in K_3^ind(k) tensor Z[1/2].Comment: 32 pages, 5 figure

Holomorphic polylogarithms and Bloch complexes

For an integer n>2 we define a polylogarithm, which is a holomorphic function on the universal abelian cover of C-{0,1} defined modulo (2 pi i)^n/(n-1)!. We use the formal properties of its functional relations to define groups lifting Goncharov's Bloch groups of a field F, and show that they fit into a complex lifting Goncharov's Bloch complex. When F=C we show that the imaginary part (when n is even) or real part (when n is odd) of the holomorphic polylogarithm agrees with Goncharov's real valued polylogarithm on the first cohomology group of the lifted Bloch complex. When n=2, this group is Neumann's extended Bloch group. Goncharov's complex conjecturally computes the rational motivic cohomology of F, and one may speculate whether the lifted complex computes the integral motivic cohomology. Finally, we construct a lift of Goncharov's regulator on the 5th homology of SL(3,C) to a complex regulator. This use polylogarithm relations arising from the cluster ensemble structure on the Grassmannians Gr(3,6) and Gr(3,7).Comment: 31 page

The complex volume of SL(n,C)-representations of 3-manifolds

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann's extended Bloch group, and use this to obtain an efficient formula for the Cheeger-Chern-Simons invariant, and in particular for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes, suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.Comment: 44 pages, 11 figure

The Lie coalgebra of multiple polylogarithms

We use Goncharov's coproduct of multiple polylogarithms to define a Lie coalgebra over an arbitrary field. It is generated by symbols subject to inductively defined relations, which we think of as functional relations for multiple polylogarithms. In particular, we have inversion relations and shuffle relations. We relate our definition to Goncharov's Bloch groups, and to the concrete model in weight less than 5 by Goncharov and Rudenko.Comment: 14 page

Results of four years of digital urban monitoring of Rattus norvegicus with RatMap in Hamburg including data on infestation near the surface and in underground sewers

Plenge-Bönig, A., Zickert, A., Baumgardt, K., Sammann, A