Bloch group



The Bloch group ()\mathcal{B}(\mathbb{C}) is the quotient of the degree-3 group homology H 3 grp(PSL(2,),)H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z}) of the complex projective linear group PSL(2,)PSL(2,\mathbb{C}) (with discrete topology) by its torsion subgroup, which is /\mathbb{Q}/\mathbb{Z}.

Accordingly H 3 grp(PSL(2,),)H_3^{grp}(\flat PSL(2,\mathbb{C}), \mathbb{Z}) itself is also called the extended Bloch group (Neumann 04).


Relation to hyperbolic 3-manifolds

The fundamental class of a hyperbolic 3-manifold XX canonically gives an element in the Bloch group: let X= 3/ΓX = \mathbb{H}^3/\Gamma for ΓPSL(2,)\Gamma\hookrightarrow PSL(2,\mathbb{C}), then the homotopy type of XX is that of BΓB\Gamma and hence the fundamental class maps forward under

H 3(X,)H 3(BΓ,)H 3 grp(Γ,)H 3 grp(PSL(2,),)H 3 grp(PSL(2,),)/(/). 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

c^:H 3(PSL(2,),)/π 2 \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][X] of a hyperbolic 3-manifold according to the above, then it yields its complex volume

c^([X])=cs(X)+ivol(X). \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()K_3^{ind}(\mathbb{C}) (Suslin90, Zickert 09).

Relation to the Borel regulator

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

vol ivol(σ i) *:H 3(PSL(2,k),) 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(vol 1,,vol r 2):H 3(PSL(2,),) r 2. 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)


The relation to algebraic K-theory and the Borel regulator is due to

Review is in