Bloch group

In the mathematical subfield of algebra , the Bloch group is an approach to the explicit description of the 3rd algebraic K-theory of bodies . It is also of importance in the investigation of dilogarithms , in the formalization of Hilbert's 3rd problem and in the topology of 3-dimensional hyperbolic manifolds .

definition

Let it be a body and ${\displaystyle K}$

${\displaystyle \mathbb {Z} \left[K\setminus \left\{0,1\right\}\right]}$

the free Abelian group generated by formally . We denote with the corresponding element of . ${\displaystyle K\setminus \left\{0,1\right\}}$ ${\displaystyle \left[x\right]}$ ${\displaystyle x\in K\setminus \left\{0,1\right\}}$ ${\displaystyle \mathbb {Z} \left[K\setminus \left\{0,1\right\}\right]}$

The pre-Bloch group is the quotient of modulo that of all "5-term relations" ${\displaystyle {\mathfrak {p}}(K)}$ ${\displaystyle \mathbb {Z} \left[K\setminus \left\{0,1\right\}\right]}$

${\displaystyle \left[x\right]-\left[y\right]+\left[{\frac {y}{x}}\right]-\left[{\frac {1-x^{-1}}{1-y^{-1}}}\right]+\left[{\frac {1-x}{1-y}}\right],\quad x,y\in K\setminus \left\{0,1\right\}}$

created subgroup .

A homomorphism

${\displaystyle \phi \colon \mathbb {Z} \left[K\setminus \left\{0,1\right\}\right]\to K^{*}\otimes _{\mathbb {Z} }K^{*}}$

is defined by

${\displaystyle \phi \left(\sum _{i}\lambda _{i}z_{i}\right)=\sum _{i}\lambda _{i}z_{i}\otimes (1-z_{i})}$

for . One calculates that a well-defined homomorphism ${\displaystyle \lambda _{i}\in \mathbb {Z} ,z_{i}\in K\setminus \left\{0,1\right\}}$ ${\displaystyle \phi }$

${\displaystyle D\colon {\mathfrak {p}}(K)\to (K^{*}\otimes K^{*})/\langle x\otimes y-y\otimes x\colon x,y\in K^{*}\rangle }$

induced. This homomorphism is because of the relationship to Hilbert's third problem as Dehn invariant called.

The Bloch Group is defined as the core of . ${\displaystyle \operatorname {B} (K)\subset {\mathfrak {p}}(K)}$ ${\displaystyle D}$

From the definition of the Bloch group and the Matsumoto theorem it follows that the Bloch group is part of an exact sequence

${\displaystyle 0\longrightarrow \operatorname {B} (K)\longrightarrow {\mathfrak {p}}(K){\stackrel {D}{\longrightarrow }}\wedge ^{2}K^{*}\longrightarrow \operatorname {K} _{2}(K)\longrightarrow 0}$

is. This sequence is called the Bloch-Suslin complex and is sometimes used as the definition of the Bloch group.

Geometric interpretation

Let it be the projective straight line above the body and ${\displaystyle P^{1}K}$ ${\displaystyle K}$

${\displaystyle (C_{*}(P^{1}K),d_{*})}$

the chain complex whose -th group is that of the -Tuples ${\displaystyle i}$ ${\displaystyle C_{i}(P^{1}K)}$ ${\displaystyle (i+1)}$

${\displaystyle (x_{0},\ldots ,x_{i})}$

pairwise different points formally generated free Abelian group and its differential by the formula ${\displaystyle x_{k}\in P^{1}K}$ ${\displaystyle d_{i}\colon C_{i}(P^{1}K)\to C_{i-1}(P^{1}K)}$

${\displaystyle d_{i}(x_{0},\ldots ,x_{i})=\sum _{k=0}^{i}(-1)^{k}(x_{0},\ldots ,x_{k-1},x_{k+1},\ldots ,x_{i})}$

given is. Then

${\displaystyle {\mathfrak {p}}(K)=H_{3}(C_{*}(P^{1}K)\otimes _{\mathbb {Z} G}\mathbb {Z} ,d\otimes id)}$

for the effect of on . ${\displaystyle G:=PGL(2,K)}$ ${\displaystyle P^{1}K}$

In particular, one has a canonical homomorphism

${\displaystyle H_{3}(PGL(2,K)\to {\mathfrak {p}}(K)}$,

the from the through

${\displaystyle g\to g.\infty }$

given figure

${\displaystyle PGL(2,K)\to P^{1}(K)}$

is induced. (The choice of as the base point is arbitrary, choosing a different base point would also induce a homomorphism.) The image of this homomorphism is even in . ${\displaystyle \infty \in P^{1}K}$ ${\displaystyle B(K)}$

Under isomorphism , a 4-tuple of elements corresponds to its double ratio . The homomorphism forms accordingly ${\displaystyle H_{3}(C_{*}(P^{1}K)\otimes _{\mathbb {Z} G}\mathbb {Z} ,d\otimes id)\to {\mathfrak {p}}(K)}$ ${\displaystyle P^{1}K}$

${\displaystyle H_{3}(PGL(2,K)\to {\mathfrak {p}}(K)}$

a 4-tuple on the double ratio of the 4 points . ${\displaystyle (g_{0},g_{1},g_{2},g_{3})}$ ${\displaystyle g_{0}\infty ,g_{1}\infty ,g_{2}\infty ,g_{3}\infty }$

Bloch-Wigner series

There is an exact sequence for algebraically closed bodies ${\displaystyle K}$

${\displaystyle 0\longrightarrow \mu _{K}\longrightarrow H_{3}(SL(2,K))\longrightarrow {\mathfrak {p}}(K)\longrightarrow \wedge ^{2}(K^{*}/\mu _{K})\longrightarrow H_{2}(SL(2,K))\longrightarrow 0}$,

wherein the roots of unity in designated. ${\displaystyle \mu _{K}}$ ${\displaystyle K}$

An immediate consequence is the exact sequence

${\displaystyle 0\longrightarrow \mu _{K}\longrightarrow H_{3}(SL(2,K))\longrightarrow B(K)\longrightarrow 0}$.

For you get the exact sequence ${\displaystyle K=\mathbb {C} }$

${\displaystyle 0\longrightarrow \mathbb {Q} /\mathbb {Z} \longrightarrow H_{3}(SL(2,\mathbb {C} ))\longrightarrow B(\mathbb {C} )\longrightarrow 0}$.

In order to integrate the -Summanden, W. Neumann defined for the expanded Bloch group . This is isomorphic to . ${\displaystyle \mathbb {Q} /\mathbb {Z} }$ ${\displaystyle K=\mathbb {C} }$ ${\displaystyle {\widehat {B}}(\mathbb {C} )}$ ${\displaystyle H_{3}(SL(2,\mathbb {C} ))}$

Bloch group and Bloch-Wigner dilogarithm

The Bloch-Wigner dilogarithm defined for ${\displaystyle z\in \mathbb {C} \setminus \left\{0,1\right\}}$

${\displaystyle \operatorname {D} _{2}(z)=\operatorname {Im} (\operatorname {Li} _{2}(z))+\arg(1-z)\log |z|}$

satisfies the functional equation

${\displaystyle \operatorname {D} _{2}(x)+\operatorname {D} _{2}(y)+\operatorname {D} _{2}\left({\frac {1-x}{1-xy}}\right)+\operatorname {D} _{2}(1-xy)+\operatorname {D} _{2}\left({\frac {1-y}{1-xy}}\right)=0}$

and therefore defines a well-defined mapping

${\displaystyle \operatorname {D} _{2}(z)\colon {\mathcal {B}}(\mathbb {C} )\to \mathbb {C} }$.

The Bloch-Wigner dilogarithm is the only measurable map that contains the functional equation ${\displaystyle F\colon \mathbb {C} \setminus \left\{0,1\right\}\to \mathbb {R} }$

${\displaystyle F(x)-F(y)+F\left({\frac {y}{x}}\right)-F\left({\frac {1-x^{-1}}{1-y^{-1}}}\right)+F\left({\frac {1-x}{1-y}}\right)}$

fulfilled for all . The definition of the Bloch group can therefore also be interpreted as the minimal group on which the Bloch-Wigner dilogarithm is well defined. Generalizations of this approach to higher polylogarithms lead to definitions of higher Bloch groups. ${\displaystyle x,y\in \mathbb {C} \setminus \left\{0,1\right\}}$

Algebraic properties

If is infinite, then the element hangs ${\displaystyle K}$

${\displaystyle [z]+[1-z]\in \operatorname {B} (K)}$

not from from. It is designated with and fulfills the relation . ${\displaystyle z\in K\setminus \left\{0,1\right\}}$ ${\displaystyle c_{K}}$ ${\displaystyle 6c_{K}=0\in \operatorname {B} (K)}$

If algebraically closed then is a divisible group . Then also apply to the relations ${\displaystyle K}$ ${\displaystyle \operatorname {B} (K)}$ ${\displaystyle z\in K\setminus \left\{0,1\right\}}$

${\displaystyle \left[z\right]+\left[1-z\right]=0}$

${\displaystyle \left[z\right]+\left[{\frac {1}{z}}\right]=0}$

and one can introduce symbols with which all 5-term relations remain valid. ${\displaystyle \left[0\right]=\left[1\right]=\left[\infty \right]=0}$

In particular applies to algebraically closed, infinite fields. From the above relations it then follows for all z. ${\displaystyle c_{K}=0}$ ${\displaystyle \left[z\right]=\left[1-{\frac {1}{z}}\right]=\left[{\frac {1}{1-z}}\right]}$

Applications

Bloch group and homology of the linear group

Application of the canonical homomorphism defined by the action of on the projective line (see the geometric interpretation above) yields an isomorphism ${\displaystyle GL(2,K)}$ ${\displaystyle P^{1}K}$ ${\displaystyle C_{*}(GL(2,K))\to C_{*}(P^{1}K)}$

${\displaystyle H_{3}(GL(2,K))/H_{3}(GM(2,K))\cong B(K)}$,

where denotes the group of monomial matrices . ${\displaystyle GM(2,K)\subset GL(2,K)}$

An isomorphism is obtained for larger ones ${\displaystyle n}$

${\displaystyle H_{3}(GL(n,K))/H_{3}(GM(n,K))\cong B(K)/2c_{K}}$

for the element of the order defined above, a maximum of 6. ${\displaystyle c_{K}:=\left[x\right]+\left[1-x\right]\in B(K)}$

The extended Bloch group defined by Neumann provides an explicit realization of . ${\displaystyle H_{3}(SL(2,\mathbb {C} ))}$ ${\displaystyle {\widehat {\operatorname {B} }}(\mathbb {C} )}$

Bloch group and K theory

The same mapping induces an isomorphism

${\displaystyle \operatorname {coker} (\pi _{3}(\operatorname {BGM} (K)^{+})\rightarrow \operatorname {K} _{3}(K))\cong \operatorname {B} (K)/2c_{K}}$

where the application of the plus construction to the classifying space denotes. ${\displaystyle BGM(K)^{+}}$ ${\displaystyle BGM(K)}$

Denote the Milnorsche K theory , then after we have an exact sequence Suslin ${\displaystyle K_{3}^{M}}$

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

with K ₃ ( K ) _ind = coker (K ₃ ^M ( K ) → K ₃ ( K )) and Tor ( K ^* , K ^* ) ^~ the unambiguous nontrivial extension of Tor ( K ^* , K ^* ) with Z / 2 , or equivalent

${\displaystyle 0\rightarrow {\widetilde {\mu _{K}}}\rightarrow \operatorname {K} _{3}(K)_{ind}\rightarrow \operatorname {B} (K)\rightarrow 0}$,

where the group of roots of unity of K and the nontrivial extension of with (or in characteristic 2 :) denotes. ${\displaystyle \mu _{K}}$ ${\displaystyle {\widetilde {\mu _{K}}}}$ ${\displaystyle \mu _{K}}$ ${\displaystyle \mathbb {Z} /2\mathbb {Z} }$ ${\displaystyle {\widetilde {\mu _{K}}}=\mu _{K}}$

Bloch group and hyperbolic geometry

For is the Abelian group freely generated by the non-degenerate ideal hyperbolic simplices . That a simplex under the isomorphism ${\displaystyle K=\mathbb {C} }$ ${\displaystyle C_{3}(P^{1}\mathbb {C} )}$

${\displaystyle H_{3}(C_{*}(P^{1}\mathbb {C} )\otimes _{\mathbb {Z} G}\mathbb {Z} )\cong {\mathfrak {p}}(\mathbb {C} )}$

The corresponding element is the double ratio of the 4 corners, the Bloch-Wigner dilogarithm gives the volume of the ideal simplex. ${\displaystyle z}$ ${\displaystyle D_{2}(z)}$

One can use this to define an invariant of hyperbolic 3-manifolds. Let be a hyperbolic 3-manifold with an ideal triangulation and be the double ratios of the simplices, then is ${\displaystyle M}$ ${\displaystyle z_{1},\ldots ,z_{r}}$

${\displaystyle \left[z_{1}\right]+\ldots +\left[z_{r}\right]}$

an element of (the Dehn invariant is zero) and defines an invariant of the manifold, from which one can calculate the hyperbolic volume of the manifold by using the Bloch-Wigner dilogarithm, among other things . ${\displaystyle B(\mathbb {C} )}$

Bloch group and secondary characteristic classes

Using the Bloch group and the Rogers dilogarithm one can specify explicit formulas for the secondary characteristic classes and , whereby one needs the extended Rogers dilogarithm and the extended Bloch group for the real part . ${\displaystyle {\hat {p}}_{1}}$ ${\displaystyle {\hat {c}}_{2}}$ ${\displaystyle {\hat {c}}_{2}}$

literature

• Spencer Bloch : Higher regulators, algebraic K-theory, and zeta functions of elliptic curves. CRM Monograph Series, 11th American Mathematical Society, Providence, RI, 2000. ISBN 0-8218-2114-8
• Johan Dupont , Chi Han Sah : Scissors congruences. II. J. Pure Appl. Algebra 25 (1982) no. 2, 159-195.
• Andrei Suslin : K ₃ of a field, and the Bloch group. (Russian, translated into English in: Proc. Steklov Inst. Math. 1991, no. 4, 217-239.) Galois theory, rings, algebraic groups and their applications (Russian). Trudy Mat. Inst. Steklov. 183: 180-199, 229 (1990).
• Johan Dupont: Scissors congruences, group homology and characteristic classes. Nankai Tracts in Mathematics, 1st World Scientific Publishing Co., Inc., River Edge, NJ, 2001. ISBN 981-02-4507-6 ; 981-02-4508-4

Individual evidence

1. ↑ Suslin, op.cit., Lemma 2.2
2. ↑ The consequence is a reformulation of an unpublished result by Bloch and Wigner, a proof can be found in Dupont-Sah, op.cit., See also Dupont, op.cit., Theorem 8.19
3. ↑ Suslin, op.cit., Lemma 1.3
4. ↑ Suslin, op.cit., Theorem 2.1
5. ↑ Suslin, op.cit., Theorem 4.1