Bloch group explained

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 ^{inf ty} _{k =1} {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 ^{z{ log} _{0} ( 1- t) \over t} 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 C \setminus \{ 0 , 1 \}.

This function enjoys several remarkable properties, e.g.

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

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

Definition

Let K be a field and define

Z (K) = 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[ (1- x)/(1- xy) \right] + [ 1- xy] + \left[ (1- y)/(1- xy) \right]

Denote by A (K) the quotient 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

\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 the element c does not depend on the choice of x, and\operatorname{coker}(\pi ^{+ )} _{3 (\operatorname{BGM}(K)} → \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 → \operatorname{Tor}(K ^{*,} K ^{*)} ^{\sim} → \operatorname{K} _{3(K)} _{ind} → \operatorname{B} _{2(K)} → 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 _{2} (z) , which is defined on C \setminus\{ 0 , 1 \} = C P ^{1} \setminus\{ 0 , 1 , inf ty\} , has the following meaning: Let H ^{3} be 3-dimensional hyperbolic space and H ^{3} = C x R _{>0} its half space model. One can regard elements of C \cup\{ inf ty\} = C P ^{1} as points at infinity on H ^{3} . A tetrahedron, all of whose vertices are at infinity, is called an ideal tetrahedron. We denote such a tetrahedron by (p _{0} ,p _{1} ,p _{2} ,p _{3} ) and its (signed) volume by \left\langle p _{0} ,p _{1} ,p _{2} ,p _{3} \right\rangle where p _{1} ,\ldots,p _{3} \in C P ^{1} are the vertices. Then under the appropriate metric up to constants we can obtain its cross-ratio:\left\langle p _{0} ,p _{1} ,p _{2} ,p _{3} \right\rangle = D _{2} \left( ((p _{0} - p _{2} )(p _{1} - p _{3} ))/((p _{0} - p _{1} )(p _{2} - p _{3} )) \right) .

In particular,

D _{2} (z) = \left\langle 0 , 1 ,z, inf ty\right\rangle . Due to the five terms relation of D _{2} (z) , the volume of the boundary of non-degenerate ideal tetrahedron (p _{0} ,p _{1} ,p _{2} ,p _{3} ,p _{4} ) equals 0 if and only if\left\langle \partial(p _{0} ,p _{1} ,p _{2} ,p _{3} ,p _{4} )\right\rangle = \sum ^{4} _{i =0} ( -1 ) ^{i} \left\langle p _{0} ,..,\hat{p} _{i} ,..,p _{4} \right\rangle =0 .

In addition, given a hyperbolic manifold

X = H ^{3} /\Gamma , one can decomposeX = c up ^{n} _{j =1} \Delta(z _{j)} where the \Delta(z _{j)} are ideal tetrahedra. whose all vertices are at infinity on \partial H ^{3} . Here the z _{j} are certain complex numbers with Im z> 0 . Each ideal tetrahedron is isometric to one with its vertices at 0 , 1 , z, inf ty for some z with Im 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 _{2} (z) where D _{2} (z) is the Bloch-Wigner dilogarithm. For general hyperbolic 3-manifold one obtainsvol(X) = \sum ^{n} _{j =1} D _{2} (z) by gluing them. The Mostow rigidity theorem guarantees only single value of the volume with Im 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

• Book: Abel, N.H. . Niels Henrik Abel . Note sur la fonction \scriptstyle \psi x = x + (x ^{2})/(2 ^{2}) + (x ^{3})/(3 ^{2}) + … + (x ^{n})/(n ^{2}) + … . French . http://www.abelprisen.no/verker/oeuvres_1881_del2/oeuvres_completes_de_abel_nouv_ed_2_kap14_opt.pdf . Sylow . L. . Lie . S. . Œuvres complètes de Niels Henrik Abel − Nouvelle édition, Tome II . Christiania [Oslo] . Grøndahl & Søn . 1826 . 1881 . 189–193 . (this 1826 manuscript was only published posthumously.)
• Book: Bloch, S. . Applications of the dilogarithm function in algebraic K-theory and algebraic geometry . Proc. Int. Symp. on Alg. Geometry . Nagata . M . Tokyo . Kinokuniya . 1978 . 103–114 .
• Book: Goncharov, A.B. . The classical trilogarithm, algebraic K-theory of fields, and Dedekind zeta-functions . https://www.ams.org/journals/bull/1991-24-01/S0273-0979-1991-15975-6/S0273-0979-1991-15975-6.pdf . Bull. AMS . 1991 . 155–162 .
• Book: Neumann, W.D. . Extended Bloch group and the Cheeger-Chern-Simons class . math/0307092 . Extended Bloch group and the Cheeger–Chern–Simons class . 2004 . 8 . 413–474 . 10.2140/gt.2004.8.413 . 2003math......7092N . 9169851 .
• Neumann . W.D. . Zagier . D. . Volumes of hyperbolic three-manifolds . Topology . 24 . 1985 . 3 . 307–332 . 10.1016/0040-9383(85)90004-7. free .
• Book: Suslin, A.A. . \operatorname{K} _{3} of a field, and the Bloch group . Russian . http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=tm&paperid=1914&option_lang=eng . Trudy Mat. Inst. Steklov . 1990 . 180–199 .
• Book: Zagier, D. . Polylogarithms, Dedekind zeta functions, and the algebraic K-theory of fields . Arithmetic Algebraic Geometry . van der Geer . G. . Oort . F. . Steenbrink . J . Boston . Birkhäuser . 1990 . 391–430 .