In mathematics, the Bloch group is a cohomology group of the BlochâÂÂSuslin complex, named after Spencer Bloch and Andrei Suslin. It is closely related to polylogarithm, hyperbolic geometry and algebraic K-theory.
The dilogarithm function is the function defined by the power series
It can be extended by analytic continuation, where the path of integration avoids the cut from 1 to +âÂÂ
The BlochâÂÂWigner function is related to dilogarithm function by
This function enjoys several remarkable properties, e.g.
The last equation is a variant of Abel's functional equation for the dilogarithm .
Let K be a field and define as the free abelian group generated by symbols [x]. Abel's functional equation implies that D<sub>2</sub> vanishes on the subgroup D(K) of Z(K) generated by elements
Denote by A (K) the quotient of by the subgroup D(K). The Bloch-Suslin complex is defined as the following cochain complex, concentrated in degrees one and two
then the Bloch group was defined by Bloch
The BlochâÂÂSuslin complex can be extended to be an exact sequence
This assertion is due to the Matsumoto theorem on K<sub>2</sub> for fields.
If c denotes the element and the field is infinite, Suslin proved the element c does not depend on the choice of x, and
where GM(K) is the subgroup of GL(K), consisting of monomial matrices, and BGM(K)<sup>+</sup> is the Quillen's plus-construction. Moreover, let K<sub>3</sub><sup>M</sup> denote the Milnor's K-group, then there exists an exact sequence
where K<sub>3</sub>(K)<sub>ind</sub> = coker(K<sub>3</sub><sup>M</sup>(K) â K<sub>3</sub>(K)) and Tor(K<sup>*</sup>, K<sup>*</sup>)<sup>~</sup> is the unique nontrivial extension of Tor(K<sup>*</sup>, K<sup>*</sup>) by means of Z/2.
The Bloch-Wigner function , which is defined on , has the following meaning: Let be 3-dimensional hyperbolic space and its half space model. One can regard elements of as points at infinity on . A tetrahedron, all of whose vertices are at infinity, is called an ideal tetrahedron. We denote such a tetrahedron by and its (signed) volume by where are the vertices. Then under the appropriate metric up to constants we can obtain its cross-ratio:
In particular, . Due to the five terms relation of , the volume of the boundary of non-degenerate ideal tetrahedron equals 0 if and only if
In addition, given a hyperbolic manifold , one can decompose
where the are ideal tetrahedra. whose all vertices are at infinity on . Here the are certain complex numbers with . Each ideal tetrahedron is isometric to one with its vertices at for some with . Here is the cross-ratio of the vertices of the tetrahedron. Thus the volume of the tetrahedron depends only one single parameter . showed that for ideal tetrahedron , where is the Bloch-Wigner dilogarithm. For general hyperbolic 3-manifold one obtains
by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with for all .
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<sub>n</sub> 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 .