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

:$\operatorname_2(z) = \sum_^\infty .$

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

:$\operatorname_2 (z) = -\int_0^z \,\mathrmt.$

The Bloch–Wigner function is related to dilogarithm function by

:$\operatorname_2 (z) = \operatorname (\operatorname_2 (z) )+\arg(1-z)\log, z,$, if $z \in \mathbb \setminus \.$

This function enjoys several remarkable properties, e.g.

*$\operatorname_2 (z)$ is real analytic on $\mathbb \setminus \.$
*$\operatorname_2 (z) = \operatorname_2 \left(1-\frac\right) = \operatorname_2 \left(\frac\right) = - \operatorname_2 \left(\frac\right) = -\operatorname_2 (1-z) = -\operatorname_2 \left(\frac\right).$
*$\operatorname_2 (x) + \operatorname_2 (y) + \operatorname_2 \left(\frac\right) + \operatorname_2 (1-xy) + \operatorname_2 \left(\frac\right) = 0.$

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

Definition

Let ''K'' be a field and define \mathbb (K) = \mathbb \setminus \/math> as the free abelian group generated by symbols 'x'' Abel's functional equation implies that ''D''₂ vanishes on the subgroup ''D''(''K'') of ''Z''(''K'') generated by elements

:
+ + \left frac\right+ -xy+ \left frac\right

Denote by ''A'' (''K'') the quotient of $\mathbb (K)$ by the subgroup ''D''(''K''). The Bloch-Suslin complex is defined as the following cochain complex, concentrated in degrees one and two
:$\operatorname^\bullet: A(K) \stackrel \wedge^2 K^*$, where d = x \wedge (1-x),

then the Bloch group was defined by Bloch

:$\operatorname_2(K) = \operatorname^1(\operatorname(K), \operatorname^\bullet)$

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

:$0 \longrightarrow \operatorname_2(K) \longrightarrow A(K) \stackrel \wedge^2 K^* \longrightarrow \operatorname_2(K) \longrightarrow 0$

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

Relations between K₃ and the Bloch group

If ''c'' denotes the element + -x\in \operatorname_2(K) and the field is infinite, Suslin proved the element ''c'' does not depend on the choice of ''x'', and

:$\operatorname(\pi_3(\operatorname(K)^+) \rightarrow \operatorname_3(K)) = \operatorname_2(K)/2c$

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

:$0 \rightarrow \operatorname(K^*, K^*)^ \rightarrow \operatorname_3(K)_ \rightarrow \operatorname_2(K) \rightarrow 0$

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

:$\left\langle p_,p_,p_,p_\right\rangle =D_\left(\frac\right)\ .$

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

:$\left\langle \partial(p_,p_,p_,p_,p_)\right\rangle =\sum_^(-1)^\left\langle p_,..,\hat_,..,p_\right\rangle =0\ .$

In addition, given a hyperbolic manifold $X=\mathbb^/\Gamma$ , one can decompose
:$X=\bigcup^n_\Delta(z_j)$
where the $\Delta(z_j)$ are ''ideal tetrahedra''. whose all vertices are at infinity on $\partial\mathbb^3$ . Here the $z_j$ are certain complex numbers with $\text\ z>0$ . Each ideal tetrahedron is isometric to one with its vertices at $0, 1, z, \infty$ for some $z$ with $\text\ z>0$ . Here $z$ is the cross-ratio of the vertices of the tetrahedron. Thus the volume of the tetrahedron depends only one single parameter $z$ . showed that for ideal tetrahedron $\Delta$ , $vol(\Delta(z))=D_(z)$ where $D_(z)$ is the Bloch-Wigner dilogarithm. For general hyperbolic 3-manifold one obtains
:$vol(X)=\sum^n_ D_(z)$
by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with $\text\ z_j>0$ for all $j$ .

Generalizations

Via substituting dilogarithm by trilogarithm or even higher polylogarithms, the notion of Bloch group was extended by Goncharov and Zagier . 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 .

References

* (this 1826 manuscript was only published posthumously.)
*
*
*
*
*
* {{cite book , last=Zagier , first= D. , contribution= Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields , title= Arithmetic Algebraic Geometry , editor1-last= van der Geer , editor1-first= G. , editor2-last= Oort , editor2-first= F. , editor3-last= Steenbrink , editor3-first= J , location= Boston , publisher= Birkhäuser , year= 1990 , pages= 391–430

Algebraic topology