nLab Bloch group

Idea

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).

Properties

Relation to hyperbolic 3-manifolds

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

$$H_3(X,\mathbb{Z}) \simeq H_3(B \Gamma, \mathbb{Z}) \simeq H_3^{grp}(\Gamma, \mathbb{Z}) \longrightarrow H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z}) \longrightarrow H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z})/(\mathbb{Q}/\mathbb{Z}) \,.$$

(e.g. Neumann 11, section 2.2)

Cheeger-Simons class and complex volume

There is a group homomorphism

$$\hat c \colon H_3(\flat PSL(2,\mathbb{C}), \mathbb{Z}) \longrightarrow \mathbb{C}/\pi^2 \mathbb{Z}$$

such that when applied to a fundamental class $[X]$ of a hyperbolic 3-manifold according to the above, then it yields its complex volume

$$\hat c([X]) = cs(X) + i vol(X) \,.$$

This is called the Cheeger-Simons class.

(e.g. Neumann 11, section 2.3)

Relation to algebraic K-theory

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).

Relation to the Borel regulator

For $k$ an algebraic number field and $\{\sigma_i \colon k \hookrightarrow \mathbb{C}\}$ its complex embeddings, write

$$vol_i \coloneqq vol \circ (\sigma_i)_\ast \colon H_3(PSL(2,k),\mathbb{Z}) \to \mathbb{R}$$

for the induced volume measures, using the Cheeger-Simons class from above. The direct product of these volumes is called the Borel regulator

$$Borel \coloneqq (vol_1, \cdots, vol_{r_2}) \colon H_3(PSL(2,\mathbb{C}), \mathbb{Z}) \longrightarrow \mathbb{R}^{r_2} \,.$$

(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)

Related concepts

• hyperbolic manifold

• Cheeger-Simons class, complex volume

• Borel regulator

References

• Walter Neumann, Extended Bloch group and the Cheeger-Chern-Simons class, Geom. Topol. 8 (2004) 413-474 (arXiv:math/0307092)

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

• Walter Neumann, section 2.4 Realizing arithmetic invariants of hyperbolic 3-manifolds, Contemporary Math 541 (Amer. Math. Soc. 2011), 233–246 (arXiv:1108.0062)