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

[math]\displaystyle{ \operatorname{Li}_2(z) = \sum_{k=1}^\infty {z^k \over k^2}. }[/math]

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

[math]\displaystyle{ \operatorname{Li}_2 (z) = -\int_0^z{\log (1-t) \over t} \,\mathrm{d}t. }[/math]

The Bloch–Wigner function is related to dilogarithm function by

[math]\displaystyle{ \operatorname{D}_2 (z) = \operatorname{Im} (\operatorname{Li}_2 (z) )+\arg(1-z)\log|z| }[/math], if [math]\displaystyle{ z \in \mathbb{C} \setminus \{0, 1\}. }[/math]

This function enjoys several remarkable properties, e.g.

• [math]\displaystyle{ \operatorname{D}_2 (z) }[/math] is real analytic on [math]\displaystyle{ \mathbb{C} \setminus \{0, 1\}. }[/math]
• [math]\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). }[/math]
• [math]\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. }[/math]

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

Definition

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

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

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

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

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

[math]\displaystyle{ \operatorname{B}_2(K) = \operatorname{H}^1(\operatorname{Spec}(K), \operatorname{B}^\bullet) }[/math]

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

[math]\displaystyle{ 0 \longrightarrow \operatorname{B}_2(K) \longrightarrow A(K) \stackrel{d}{\longrightarrow} \wedge^2 K^* \longrightarrow \operatorname{K}_2(K) \longrightarrow 0 }[/math]

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

Relations between K₃ and the Bloch group

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

[math]\displaystyle{ \operatorname{coker}(\pi_3(\operatorname{BGM}(K)^+) \rightarrow \operatorname{K}_3(K)) = \operatorname{B}_2(K)/2c }[/math]

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

[math]\displaystyle{ 0 \rightarrow \operatorname{Tor}(K^*, K^*)^{\sim} \rightarrow \operatorname{K}_3(K)_{ind} \rightarrow \operatorname{B}_2(K) \rightarrow 0 }[/math]

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

[math]\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)\ . }[/math]

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

[math]\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\ . }[/math]

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

[math]\displaystyle{ X=\bigcup^n_{j=1}\Delta(z_j) }[/math]

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

[math]\displaystyle{ vol(X)=\sum^n_{j=1} D_{2}(z) }[/math]

by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with [math]\displaystyle{ \text{Im}\ z_j\gt 0 }[/math] for all [math]\displaystyle{ j }[/math] .

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 [math]\displaystyle{ \scriptstyle \psi x = x+ \frac{x^2}{2^2}+ \frac{x^3}{3^2}+ \cdots+ \frac{x^n}{n^2}+ \cdots }[/math]". 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). "[math]\displaystyle{ \operatorname{K}_3 }[/math] 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.