The Bloch group $\mathcal{B}(\mathbb{C})$ is the quotient of the degree-3 group homology $H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z})$ of the complex projective linear group $PSL(2,\mathbb{C})$ (with discrete topology) by its torsion subgroup, which is $\mathbb{Q}/\mathbb{Z}$.
Accordingly $H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z})$ itself is also called the extended Bloch group (Neumann 04).
The fundamental class of a hyperbolic 3-manifold $X$ canonically gives an element in the Bloch group: let $X = \mathbb{H}^3/\Gamma$ for $\Gamma\hookrightarrow PSL(2,\mathbb{C})$, then the homotopy type of $X$ is that of $B\Gamma$ and hence the fundamental class maps forward under
(e.g. Neumann 11, section 2.2)
There is a group homomorphism
such that when applied to a fundamental class $[X]$ of a hyperbolic 3-manifold according to the above, then it yields its complex volume
This is called the Cheeger-Simons class.
(e.g. Neumann 11, section 2.3)
Up to torsion subgroups, the extended Bloch group of a is isomorphic to the third algebraic K-theory group $K_3^{ind}(\mathbb{C})$ (Suslin90, Zickert 09).
For $k$ an algebraic number field and $\{\sigma_i \colon k \hookrightarrow \mathbb{C}\}$ its complex embeddings, write
for the induced volume measures, using the Cheeger-Simons class from above. The direct product of these volumes is called the Borel regulator
(e.g. Zickert 09, (1.2))
Now the point is that restricted to the Bloch group proper the Borel regulator is injective.
(e.g. Neumann 11, theorem 2.4)
The relation to algebraic K-theory and the Borel regulator is due to
Chih-Han Sah, Homology of classical Lie groups made discrete. III., J. Pure Appl. Algebra, 56(3):269–312, 1989.
Andrei Suslin. $K_3$ of a field, and the Bloch group. Trudy Mat. Inst. Steklov., 183:180–199, 229, 1990. Translated in Proc. Steklov Inst. Math. 1991, no. 4, 217–239, Galois theory, rings, algebraic groups and their applications (Russian).
Christian Zickert, The extended Bloch group and algebraic K-theory (arXiv:0910.4005)
Review is in
Last revised on August 7, 2015 at 05:10:33. See the history of this page for a list of all contributions to it.