Bloch group

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In mathematics, the Bloch group is a cohomology group of the Bloch–Suslin complex, named after Spencer Bloch and Andrei Suslin. It is closely related to polylogarithm, hyperbolic geometry and algebraic K-theory.

Bloch–Wigner function

The dilogarithm function is the function defined by the power series

${\displaystyle \operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2}}.}$

It can be extended by analytic continuation, where the path of integration avoids the cut from 1 to +∞

${\displaystyle \operatorname {Li} _{2}(z)=-\int _{0}^{z}{\log(1-t) \over t}\,\mathrm {d} t.}$

The Bloch–Wigner function is related to dilogarithm function by

${\displaystyle \operatorname {D} _{2}(z)=\operatorname {Im} (\operatorname {Li} _{2}(z))+\arg(1-z)\log |z|}$, if ${\displaystyle z\in \mathbb {C} \setminus \{0,1\}.}$

This function enjoys several remarkable properties, e.g.

• ${\displaystyle \operatorname {D} _{2}(z)}$ is real analytic on ${\displaystyle \mathbb {C} \setminus \{0,1\}.}$
• ${\displaystyle \operatorname {D} _{2}(z)=\operatorname {D} _{2}\left(1-{\frac {1}{z}}\right)=\operatorname {D} _{2}\left({\frac {1}{1-z}}\right)=-\operatorname {D} _{2}\left({\frac {1}{z}}\right)=-\operatorname {D} _{2}(1-z)=-\operatorname {D} _{2}\left({\frac {-z}{1-z}}\right).}$
• ${\displaystyle \operatorname {D} _{2}(x)+\operatorname {D} _{2}(y)+\operatorname {D} _{2}\left({\frac {1-x}{1-xy}}\right)+\operatorname {D} _{2}(1-xy)+\operatorname {D} _{2}\left({\frac {1-y}{1-xy}}\right)=0.}$

The last equation is a variance of Abel's functional equation for the dilogarithm (Abel 1881).

Definition

Let K be a field and define ${\displaystyle \mathbb {Z} (K)=\mathbb {Z} [K\setminus \{0,1\}]}$ as the free abelian group generated by symbols [x]. Abel's functional equation implies that D2 vanishes on the subgroup D (K) of Z (K) generated by elements

${\displaystyle [x]+[y]+\left[{\frac {1-x}{1-xy}}\right]+[1-xy]+\left[{\frac {1-y}{1-xy}}\right]}$

Denote by A (K) the factor-group of Z (K) by the subgroup D(K). The Bloch-Suslin complex is defined as the following cochain complex, concentrated in degrees one and two

${\displaystyle \operatorname {B} ^{\bullet }:A(K){\stackrel {d}{\longrightarrow }}\wedge ^{2}K^{*}}$, where ${\displaystyle d[x]=x\wedge (1-x)}$,

then the Bloch group was defined by Bloch (Bloch 1978)

${\displaystyle \operatorname {B} _{2}(K)=\operatorname {H} ^{1}(\operatorname {Spec} (K),\operatorname {B} ^{\bullet })}$

The Bloch–Suslin complex can be extended to be an exact sequence

${\displaystyle 0\longrightarrow \operatorname {B} _{2}(K)\longrightarrow A(K){\stackrel {d}{\longrightarrow }}\wedge ^{2}K^{*}\longrightarrow \operatorname {K} _{2}(K)\longrightarrow 0}$

This assertion is due to the Matsumoto theorem on K2 for fields.

Relations between K3 and the Bloch group

If c denotes the element ${\displaystyle [x]+[1-x]\in \operatorname {B} _{2}(K)}$ and the field is infinite, Suslin proved (Suslin 1990) the element c does not depend on the choice of x, and

${\displaystyle \operatorname {coker} (\pi _{3}(\operatorname {BGM} (K)^{+})\rightarrow \operatorname {K} _{3}(K))=\operatorname {B} _{2}(K)/2c}$

where GM(K) is the subgroup of GL(K), consisting of monomial matrices, and BGM(K)+ is the Quillen's plus-construction. Moreover, let K3M denote the Milnor's K-group, then there exists an exact sequence

${\displaystyle 0\rightarrow \operatorname {Tor} (K^{*},K^{*})^{\sim }\rightarrow \operatorname {K} _{3}(K)_{ind}\rightarrow \operatorname {B} _{2}(K)\rightarrow 0}$

where K3(K)ind = coker(K3M(K) → K3(K)) and Tor(K*, K*)~ is the unique nontrivial extension of Tor(K*, K*) by means of Z/2.

Relations to hyperbolic geometry in three-dimensions

The Bloch-Wigner function ${\displaystyle D_{2}(z)}$ , which is defined on ${\displaystyle \mathbb {C} \setminus \{0,1\}=\mathbb {C} P^{1}\setminus \{0,1,\infty \}}$ , has the following meaning: Let ${\displaystyle \mathbb {H} ^{3}}$ be 3-dimensional hyperbolic space and ${\displaystyle \mathbb {H} ^{3}=\mathbb {C} \times \mathbb {R} _{>0}}$ its half space model. One can regard elements of ${\displaystyle \mathbb {C} \cup \{\infty \}=\mathbb {C} P^{1}}$ as points at infinity on ${\displaystyle \mathbb {H} ^{3}}$. A tetrahedron, all of whose vertices are at infinity, is called an ideal tetrahedron. We denote such a tetrahedron by ${\displaystyle (p_{0},p_{1},p_{2},p_{3})}$ and its (signed) volume by ${\displaystyle \left\langle p_{0},p_{1},p_{2},p_{3}\right\rangle }$ where ${\displaystyle p_{1},\ldots ,p_{3}\in \mathbb {C} P^{1}}$ are the vertices. Then under the appropriate metric up to constants we can obtain its cross-ratio:

${\displaystyle \left\langle p_{0},p_{1},p_{2},p_{3}\right\rangle =D_{2}\left({\frac {(p_{0}-p_{2})(p_{1}-p_{3})}{(p_{0}-p_{1})(p_{2}-p_{3})}}\right)\ .}$

In particular, ${\displaystyle D_{2}(z)=\left\langle 0,1,z,\infty \right\rangle }$ . Due to the five terms relation of ${\displaystyle D_{2}(z)}$ , the volume of the boundary of non-degenerate ideal tetrahedron ${\displaystyle (p_{0},p_{1},p_{2},p_{3},p_{4})}$ equals 0 if and only if

${\displaystyle \left\langle \partial (p_{0},p_{1},p_{2},p_{3},p_{4})\right\rangle =\sum _{i=0}^{4}(-1)^{i}\left\langle p_{0},..,{\hat {p}}_{i},..,p_{4}\right\rangle =0\ .}$

In addition, given a hyperbolic manifold ${\displaystyle X=\mathbb {H} ^{3}/\Gamma }$ , one can decompose

${\displaystyle X=\bigcup _{j=1}^{n}\Delta (z_{j})}$

where the ${\displaystyle \Delta (z_{j})}$ are ideal tetrahedra. whose all vertices are at infinity on ${\displaystyle \partial \mathbb {H} ^{3}}$ . Here the ${\displaystyle z_{j}}$ are certain complex numbers with ${\displaystyle {\text{Im}}\ z>0}$ . Each ideal tetrahedron is isometric to one with its vertices at ${\displaystyle 0,1,z,\infty }$ for some ${\displaystyle z}$ with ${\displaystyle {\text{Im}}\ z>0}$ . Here ${\displaystyle z}$ is the cross-ratio of the vertices of the tetrahedron. Thus the volume of the tetrahedron depends only one single parameter ${\displaystyle z}$ . (Neumann & Zagier 1985) showed that for ideal tetrahedron ${\displaystyle \Delta }$ , ${\displaystyle vol(\Delta (z))=D_{2}(z)}$ where ${\displaystyle D_{2}(z)}$ is the Bloch-Wigner dilogarithm. For general hyperbolic 3-manifold one obtains

${\displaystyle vol(X)=\sum _{j=1}^{n}D_{2}(z)}$

by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with ${\displaystyle {\text{Im}}\ z_{j}>0}$ for all ${\displaystyle j}$ .

Generalizations

Via substituting dilogarithm by trilogarithm or even higher polylogarithms, the notion of Bloch group was extended by Goncharov (Goncharov 1991) and Zagier (Zagier 1990). It is widely conjectured that those generalized Bloch groups Bn should be related to algebraic K-theory or motivic cohomology. There are also generalizations of the Bloch group in other directions, for example, the extended Bloch group defined by Neumann (Neumann 2004).