Krein's condition

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

${\mathrm{D}}_2(z)=\mathrm{\Im }({\mathrm{Li}}_2(z))+\arg (1-z)\log \left|z\right|$, 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 variance of Abel's functional equation for the dilogarithm Template:Harv.

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 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

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

then the Bloch group was defined by Bloch Template:Harv

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

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 Template:Harv 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.

Generalizations

Via substituting dilogarithm by trilogarithm or even higher polylogarithms, the notion of Bloch group was extended by Goncharov Template:Harv and Zagier Template:Harv. It is widely conjectured that those generalized Bloch groups B_n should be related to algebraic K-theory or motivic cohomology. There also exist some generalizations of Bloch group in the other direction, for example, the extended Bloch group defined by Neumann Template:Harv.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534 (this 1826 manuscript was only published posthumously.)

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534