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.
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Bloch–Wigner function
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 +∞
The Bloch–Wigner function is related to dilogarithm function by
This function enjoys several remarkable properties, e.g.
The last equation is a variance of Abel's functional equation for the dilogarithm (Abel 1881).
Definition
Let K be a field and define
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
then the Bloch group was defined by Bloch (Bloch 1978)
The Bloch–Suslin complex can be extended to be an exact sequence
This assertion is due to the Matsumoto theorem on K2 for fields.
Relations between K3 and the Bloch group
If c denotes the element
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
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
In particular,
In addition, given a hyperbolic manifold
where the
by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with
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).