Bloch group

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

${\mathrm{Li}}_2(z)=\sum _{k=1}^{\mathrm{\infty }}\frac{z^k}{k^2}.$

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

${\mathrm{Li}}_2(z)=-\underset{0}{\overset{z}{\int }}\frac{\log (1-t)}{t}\mathrm{d}t.$

The Bloch–Wigner function is related to dilogarithm function by

${\mathrm{D}}_2(z)=\mathrm{Im}({\mathrm{Li}}_2(z))+\arg (1-z)\log |z|$, if $z\in \mathbb{ℂ}\setminus \{0,1\}.$

This function enjoys several remarkable properties, e.g.

• ${\mathrm{D}}_2(z)$ is real analytic on $\mathbb{ℂ}\setminus \{0,1\}.$
• ${\mathrm{D}}_2(z)={\mathrm{D}}_2(1-\frac{1}{z})={\mathrm{D}}_2\left(\frac{1}{1-z}\right)=-{\mathrm{D}}_2\left(\frac{1}{z}\right)=-{\mathrm{D}}_2(1-z)=-{\mathrm{D}}_2\left(\frac{-z}{1-z}\right).$
• ${\mathrm{D}}_2(x)+{\mathrm{D}}_2(y)+{\mathrm{D}}_2\left(\frac{1-x}{1-xy}\right)+{\mathrm{D}}_2(1-xy)+{\mathrm{D}}_2\left(\frac{1-y}{1-xy}\right)=0.$

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

Definition

Let K be a field and define $\mathbb{\mathbb{Z}}(K)=\mathbb{\mathbb{Z}}[K\setminus \{0,1\}]$ 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

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

Denote by A (K) the quotient of $\mathbb{\mathbb{Z}}(K)$ by the subgroup D(K). The Bloch-Suslin complex is defined as the following cochain complex, concentrated in degrees one and two

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

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

${\mathrm{B}}_2(K)={\mathrm{H}}^1(\mathrm{Spec}(K),{\mathrm{B}}^{\bullet })$

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

$0⟶{\mathrm{B}}_2(K)⟶A(K)\stackrel{d}{⟶}{\wedge }^2{K}^{*}⟶{\mathrm{K}}_2(K)⟶0$

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

Relations between K₃ and the Bloch group

If c denotes the element $[x]+[1-x]\in {\mathrm{B}}_2(K)$ and the field is infinite, Suslin proved (Suslin 1990) the element c does not depend on the choice of x, and

$\mathrm{coker}({\pi }_3(\mathrm{BGM}(K{)}^{+})\to {\mathrm{K}}_3(K))={\mathrm{B}}_2(K)/2c$

where GM(K) is the subgroup of GL(K), consisting of monomial matrices, and BGM(K)⁺ is the Quillen's plus-construction. Moreover, let K₃^M denote the Milnor's K-group, then there exists an exact sequence

$0\to \mathrm{Tor}(K^{*},K^{*}{)}^{\sim }\to {\mathrm{K}}_3(K{)}_{ind}\to {\mathrm{B}}_2(K)\to 0$

where K₃(K)_ind = coker(K₃^M(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 $D_2(z)$ , which is defined on $\mathbb{ℂ}\setminus \{0,1\}=\mathbb{ℂ}{P}^1\setminus \{0,1,\mathrm{\infty }\}$ , has the following meaning: Let ${\mathbb{ℍ}}^3$ be 3-dimensional hyperbolic space and ${\mathbb{ℍ}}^3=\mathbb{ℂ}\times {\mathbb{ℝ}}_{>0}$ its half space model. One can regard elements of $\mathbb{ℂ}\cup \{\mathrm{\infty }\}=\mathbb{ℂ}{P}^1$ as points at infinity on ${\mathbb{ℍ}}^3$. A tetrahedron, all of whose vertices are at infinity, is called an ideal tetrahedron. We denote such a tetrahedron by $(p_0,p_1,p_2,p_3)$ and its (signed) volume by $⟨p_0,p_1,p_2,p_3⟩$ where $p_1,\dots ,p_3\in \mathbb{ℂ}{P}^1$ are the vertices. Then under the appropriate metric up to constants we can obtain its cross-ratio:

$⟨p_0,p_1,p_2,p_3⟩=D_2\left(\frac{(p_0-p_2)(p_1-p_3)}{(p_0-p_1)(p_2-p_3)}\right).$

In particular, $D_2(z)=⟨0,1,z,\mathrm{\infty }⟩$ . Due to the five terms relation of $D_2(z)$ , the volume of the boundary of non-degenerate ideal tetrahedron $(p_0,p_1,p_2,p_3,p_4)$ equals 0 if and only if

$⟨\partial (p_0,p_1,p_2,p_3,p_4)⟩=\sum _{i=0}^4(-1{)}^i⟨p_0,..,{\hat{p}}_i,..,p_4⟩=0.$

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

$X=\bigcup _{j=1}^n\mathrm{\Delta }(z_j)$

where the $\mathrm{\Delta }(z_j)$ are ideal tetrahedra. whose all vertices are at infinity on $\partial {\mathbb{ℍ}}^3$ . Here the $z_j$ are certain complex numbers with $\text{Im}z>0$ . Each ideal tetrahedron is isometric to one with its vertices at $0,1,z,\mathrm{\infty }$ for some $z$ with $\text{Im}z>0$ . Here $z$ is the cross-ratio of the vertices of the tetrahedron. Thus the volume of the tetrahedron depends only one single parameter $z$ . (Neumann Zagier) showed that for ideal tetrahedron $\mathrm{\Delta }$ , $vol(\mathrm{\Delta }(z))=D_2(z)$ where $D_2(z)$ is the Bloch-Wigner dilogarithm. For general hyperbolic 3-manifold one obtains

$vol(X)=\sum _{j=1}^n{D}_2(z)$

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

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 B_n 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).

References

• Abel, N.H. (1881). "Note sur la fonction $\psi x=x+\frac{x^2}{2^2}+\frac{x^3}{3^2}+\cdots +\frac{x^n}{n^2}+\cdots$". in Sylow, L.; Lie, S. (in French). Œuvres complètes de Niels Henrik Abel − Nouvelle édition, Tome II. Christiania [Oslo]: Grøndahl & Søn. pp. 189–193. http://www.abelprisen.no/verker/oeuvres_1881_del2/oeuvres_completes_de_abel_nouv_ed_2_kap14_opt.pdf.  (this 1826 manuscript was only published posthumously.)
• Bloch, S. (1978). "Applications of the dilogarithm function in algebraic K-theory and algebraic geometry". in Nagata, M. Proc. Int. Symp. on Alg. Geometry. Tokyo: Kinokuniya. pp. 103–114. 
• Goncharov, A.B. (1991). "The classical trilogarithm, algebraic K-theory of fields, and Dedekind zeta-functions". Bull. AMS. pp. 155–162. https://www.ams.org/journals/bull/1991-24-01/S0273-0979-1991-15975-6/S0273-0979-1991-15975-6.pdf. 
• Neumann, W.D. (2004). "Extended Bloch group and the Cheeger-Chern-Simons class". Extended Bloch group and the Cheeger–Chern–Simons class. 8. pp. 413–474. doi:10.2140/gt.2004.8.413. Bibcode: 2003math......7092N. 
• Neumann, W.D.; Zagier, D. (1985). "Volumes of hyperbolic three-manifolds". Topology 24 (3): 307–332. doi:10.1016/0040-9383(85)90004-7. 
• Suslin, A.A. (1990). "${\mathrm{K}}_3$ of a field, and the Bloch group" (in Russian). Trudy Mat. Inst. Steklov. pp. 180–199. http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=1914&option_lang=eng. 
• Zagier, D. (1990). "Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields". in van der Geer, G.; Oort, F.; Steenbrink, J. Arithmetic Algebraic Geometry. Boston: Birkhäuser. pp. 391–430.