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

$\operatorname {Li} _{2}(z)=\sum _{k=1}^{\infty }{z^{k} \over k^{2}}.$

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

$\operatorname {Li} _{2}(z)=-\int _{0}^{z}{\log(1-t) \over t}\,\mathrm {d} t.$

The Bloch–Wigner function is related to dilogarithm function by

$\operatorname {D} _{2}(z)=\operatorname {Im} (\operatorname {Li} _{2}(z))+\arg(1-z)\log |z|$, if $z\in \mathbb {C} \setminus \{0,1\}.$

This function enjoys several remarkable properties, e.g.

• $\operatorname {D} _{2}(z)$ is real analytic on $\mathbb {C} \setminus \{0,1\}.$
• $\operatorname {D} _{2}(z)=\operatorname {D} _{2}\left(1-{\frac {1}{z}}\right)=\operatorname {D} _{2}\left({\frac {1}{1-z}}\right)=-\operatorname {D} _{2}\left({\frac {1}{z}}\right)=-\operatorname {D} _{2}(1-z)=-\operatorname {D} _{2}\left({\frac {-z}{1-z}}\right).$
• $\operatorname {D} _{2}(x)+\operatorname {D} _{2}(y)+\operatorname {D} _{2}\left({\frac {1-x}{1-xy}}\right)+\operatorname {D} _{2}(1-xy)+\operatorname {D} _{2}\left({\frac {1-y}{1-xy}}\right)=0.$

The last equation is a variance of Abel's functional equation for the dilogarithm Template:Harv.

Definition

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

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

then the Bloch group was defined by Bloch Template:Harv

$\operatorname {B} _{2}(K)=\operatorname {H} ^{1}(\operatorname {Spec} (K),\operatorname {B} ^{\bullet })$

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

$0\longrightarrow \operatorname {B} _{2}(K)\longrightarrow A(K){\stackrel {d}{\longrightarrow }}\wedge ^{2}K^{*}\longrightarrow \operatorname {K} _{2}(K)\longrightarrow 0$

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

Relations between K₃ and the Bloch group

If c denotes the element $[x]+[1-x]\in \operatorname {B} _{2}(K)$ and the field is infinite, Suslin proved Template:Harv the element c does not depend on the choice of x, and

$\operatorname {coker} (\pi _{3}(\operatorname {BGM} (K)^{+})\rightarrow \operatorname {K} _{3}(K))=\operatorname {B} _{2}(K)/2c$

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

$0\rightarrow \operatorname {Tor} (K^{*},K^{*})^{\sim }\rightarrow \operatorname {K} _{3}(K)_{ind}\rightarrow \operatorname {B} _{2}(K)\rightarrow 0$

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.

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