Bloch Group Research Articles

A refined Bloch-Wigner exact sequence in characteristic 2

Let A be a local domain of characteristic 2 such that its residue field has more than 64 elements. Then we find an exact relation between the third integral homology of the group SL2(A) and Hutchinson's refined Bloch group RB(A).

The Lie coalgebra of multiple polylogarithms

We use Goncharov's coproduct of multiple polylogarithms to define a Lie coalgebra over an arbitrary field. It is generated by symbols subject to inductively defined relations, which we think of as functional relations for multiple polylogarithms. In particular, we have inversion relations and shuffle relations. We relate our definition to Goncharov's Bloch groups, and to the concrete model for L(F)≤4 by Goncharov and Rudenko.

A refined scissors congruence group and the third homology of SL2

There is a natural connection between the third homology of SL2(A) and the refined Bloch group RB(A) of a commutative ring A. In this article we investigate this connection and as the main result we show that if A is a universal GE2-domain such that −1∈A×2, then we have the exact sequenceH3(SM2(A),Z)→H3(SL2(A),Z)→RB(A)→0, where SM2(A) is the group of monomial matrices in SL2(A). Moreover, we show that RP1(A), the refined scissors congruence group of A, is naturally isomorphic to the relative homology group H3(SL2(A),SM2(A);Z).

Holomorphic polylogarithms and Bloch complexes

Abstract For an integer n ≥ 2 {n\geq 2} we define a polylogarithm ℒ ^ n {\widehat{\mathcal{L}}_{n}} , which is a holomorphic function on the universal abelian cover of ℂ ∖ { 0 , 1 } {\mathbb{C}\setminus\{0,1\}} defined modulo ( 2 ⁢ π ⁢ i ) n / ( n - 1 ) ! {(2\pi i)^{n}/(n-1)!} . We use the formal properties of its functional relations to define groups ℬ ^ k ⁢ ( F ^ ) {\widehat{\mathcal{B}}_{k}(\widehat{F})} lifting Goncharov’s Bloch groups ℬ k ⁢ ( F ) {\mathcal{B}_{k}(F)} of a field F, and show that they fit into a complex Γ ^ ⁢ ( F , n ) {\widehat{\Gamma}(F,n)} lifting Goncharov’s Bloch complex Γ ⁢ ( F , n ) {\Gamma(F,n)} . When F = ℂ {F=\mathbb{C}} , we show that the imaginary part (when n is even) or real part (when n is odd) of ℒ ^ n {\widehat{\mathcal{L}}_{n}} agrees with a real single valued polylogarithm ℒ n {\mathcal{L}_{n}} on the group H 1 ⁢ ( Γ ^ ⁢ ( ℂ , n ) ) {H^{1}(\widehat{\Gamma}(\mathbb{C},n))} . When n = 2 {n=2} , this group is Neumann’s extended Bloch group. Goncharov’s complex conjecturally computes the rational motivic cohomology of F, and one may speculate whether the extended complex computes the integral motivic cohomology. Finally, we use ℒ ^ 3 {\widehat{\mathcal{L}}_{3}} to construct a lift of Goncharov’s map H 5 ⁢ ( SL ⁡ ( 3 , ℂ ) ) → ℝ {H_{5}(\operatorname{SL}(3,\mathbb{C}))\to\mathbb{R}} to a complex valued map whose real part agrees with that of Goncharov. The lift makes use of the cluster ensemble structure on the Grassmannian Gr ⁡ ( 3 , 6 ) {\operatorname{Gr}(3,6)} .

Bloch groups, algebraic $K$-theory, units, and Nahm's Conjecture

Given an element of the Bloch group of a number field~$F$ and a natural number~$n$, we construct an explicit unit in the field $F_n=F(e^{2 \pi i/n})$, well-defined up to $\nn$-th powers of nonzero elements of~$F_n$. The construction uses the cyclic quantum dilogarithm, and under the identification of the Bloch group of~$F$ with the $K$-group $K_3(F)$ gives \changed{(up to an unidentified invertible scalar)} a \changed{formula} for a certain abstract Chern class from~$K_3(F)$. The units we define are conjectured to coincide with numbers appearing in the quantum modularity conjecture for the Kashaev invariant of knots (which was the original motivation for our investigation), and also appear in the radial asymptotics of Nahm sums near roots of unity. This latter connection is used to prove Nahm's conjecture relating the modularity of certain $q$-hypergeometric series to the vanishing of the associated elements in the Bloch group of~$\overline{\mathbb Q}$.

The reduced Dijkgraaf–Witten invariant of double twist knots in the Bloch group of 𝔽p

In 2004, Neumann showed that the complex hyperbolic volume of a hyperbolic 3-manifold [Formula: see text] can be obtained as the image of the Dijkgraaf–Witten invariant of [Formula: see text] by a certain 3-cocycle. After that, Zickert gave an analogue of Neumann’s work for free fields containing finite fields. The author formulated a geometric method to calculate a weaker version of Zickert’s analogue, called the reduced Dijkgraaf–Witten invariant, for finite fields and gave a formula for twist knot complements and [Formula: see text] in his previous work. In this paper, we show concretely how to calculate the reduced Dijkgraaf–Witten invariants of double twist knot complements and [Formula: see text], and give a formula of them for [Formula: see text].

The reduced Dijkgraaf–Witten invariant of twist knots in the Bloch group of a finite field

Let [Formula: see text] be a closed oriented 3-manifold and let [Formula: see text] be a discrete group. We consider a representation [Formula: see text]. For a 3-cocycle [Formula: see text], the Dijkgraaf–Witten invariant is given by [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], and [Formula: see text] denotes the fundamental class of [Formula: see text]. Note that [Formula: see text], where [Formula: see text] is the map induced by [Formula: see text], we consider an equivalent invariant [Formula: see text], and we also regard it as the Dijkgraaf–Witten invariant. In 2004, Neumann described the complex hyperbolic volume of [Formula: see text] in terms of the image of the Dijkgraaf–Witten invariant for [Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text]. In this paper, by replacing [Formula: see text] with a finite field [Formula: see text], we calculate the reduced Dijkgraaf–Witten invariants of the complements of twist knots, where the reduced Dijkgraaf–Witten invariant is the image of the Dijkgraaf–Witten invariant for SL[Formula: see text] by the Bloch–Wigner map from [Formula: see text] to the Bloch group of [Formula: see text].

Hyperbolic tessellations and generators of for imaginary quadratic fields

AbstractWe develop methods for constructing explicit generators, modulo torsion, of the$K_3$-groups of imaginary quadratic number fields. These methods are based on either tessellations of hyperbolic$3$-space or on direct calculations in suitable pre-Bloch groups and lead to the very first proven examples of explicit generators, modulo torsion, of any infinite$K_3$-group of a number field. As part of this approach, we make several improvements to the theory of Bloch groups for$ K_3 $of any field, predict the precise power of$2$that should occur in the Lichtenbaum conjecture at$ -1 $and prove that this prediction is valid for all abelian number fields.

Asymptotics of Nahm sums at roots of unity

We give a formula for the radial asymptotics to all orders of the special q-hypergeometric series known as Nahm sums at complex roots of unity. This result is used in Calegari et al. (Bloch groups, algebraic K-theory, units and Nahm’s conjecture. arXiv:1712.04887, 2017) to prove Nahm’s conjecture relating the modularity of Nahm sums to the vanishing of a certain invariant in K-theory. The power series occurring in our asymptotic formula are identical to the conjectured asymptotics of the Kashaev invariant of a knot once we convert Neumann–Zagier data into Nahm data, suggesting a deep connection between asymptotics of quantum knot invariants and asymptotics of Nahm sums that will be discussed further in a subsequent publication.

Functional relations for elliptic polylogarithms

Numerous examples of functional relations for multiple polylogarithms are known. For elliptic polylogarithms, however, tools for the exploration of functional relations are available, but only very few relations are identified. Starting from an approach of Zagier and Gangl, which in turn is based on considerations about an elliptic version of the Bloch group, we explore functional relations between elliptic polylogarithms and link them to the relations which can be derived using the elliptic symbol formalism. The elliptic symbol formalism in turn allows for an alternative proof of the validity of the elliptic Bloch relation. While the five-term identity is the prime example of a functional identity for multiple polylogarithms and implies many dilogarithm identities, the situation in the elliptic setup is more involved: there is no simple elliptic analogue, but rather a whole class of elliptic identities.

Infinitesimal Bloch regulator

The aim of the paper is to define an infinitesimal analog of the Bloch regulator, which attaches to a pair of meromorphic functions on a Riemann surface, a line bundle with connection on the punctured surface. In the infinitesimal context, we consider a pair (X,X_) of schemes over a field of characteristic 0, such that the regular scheme X_ is defined in X by a square-zero sheaf of ideals which is locally free on X_. We propose a definition of the weight two motivic cohomology of X based on the Bloch group, which is defined in terms of the functional equation of the dilogarithm. The analog of the Bloch regulator is a map from a subspace of the infinitesimal part of HM2(X,Q(2)) to the first cohomology group of the Zariski sheaf associated to an André-Quillen homology group. Using Goodwillie's theorem, we deduce that this map is an isomorphism, which is an infinitesimal analog of the injectivity conjecture for the Bloch regulator.

Braids, 3-Manifolds, Elementary Particles: Number Theory and Symmetry in Particle Physics

In this paper, we will describe a topological model for elementary particles based on 3-manifolds. Here, we will use Thurston’s geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement C ( K ) = S 3 \ ( K × D 2 ) of a knot K carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic 3-manifolds have a close connection to number theory (Bloch group, algebraic K-theory, quaternionic trace fields), which will be used in the description of fermions. Here, we choose the description of 3-manifolds by branched covers. Every 3-manifold can be described by a 3-fold branched cover of S 3 branched along a knot. In case of knot complements, one will obtain a 3-fold branched cover of the 3-disk D 3 branched along a 3-braid or 3-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to 3-braids (Bilson–Thompson model).

Properties and applications of one dimensional quasiperiodic lattices

Quasiperiodic lattices have been widely studied in cold atoms, which make one study extended-Anderson localization transition in one-dimensional (1D) or two-dimensional (2D) systems. In 2008, Inguscio's group prepared one-dimensional quasiperiodic lattice in cold atomic system and observed Anderson localization, which promoted the theoretical and experimental studies of quasiperiodic systems. Later, Bloch's group observed the many body localization in their prepared 1D and 2D quasiperiodic lattices. Recently, they also successfully observed the mobility edge and many body localization in a system with mobility edge in quasiperiodic lattices. These cold atom experiments have promoted the researches of many body localization and mobility edge. Quasiperiodic lattices have become a platform and its effects on many physical phenomena are being extensively studied, which can be expectantly observed in cold atomic experiments. This paper is based on the authors' some related work and briefly review some recent studies on the 1D quasiperiodic lattices, including some important cold atom experiments, some important properties of the quasiperiodic lattices and their effects on some physical phenomena, such as topological states.

The extended Bloch groups of biquadratic and dihedral number fields

In this paper, we study the Galois action on the extended Bloch groups of biquadratic and dihedral number fields. We prove that if F is a biquadratic number field, then the index Q2(F) in Browkin and Gangl's formulas on the Brauer–Kuroda relation can only be 1 or 2. This is exactly what Browkin and Gangl predicted in their paper. Moreover we give the explicit criteria for Q2(F)=1 or 2 in terms of the Tate kernels. We also prove that Q2(F)=1 or p for any dihedral extension F/Q whose Galois group is the dihedral group of order 2p, where p is an odd prime.

The third homology of $$\mathrm {SL}_2$$ of local rings

We describe the third homology of \(SL_2\) of local rings, over \(\mathbb {Z}\left[ \tfrac{1}{2}\right] \), in terms of a refined Bloch group. We use this result to elucidate the relationship of this homology group to the indecomposable part of \(K_3\) of the ring, extending and generalizing recent results in the case of fields. In particular, we prove that if A is a local domain with sufficiently large (possibly finite) residue field then the natural map \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) \rightarrow {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] \) induces an isomorphism \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) _{A^\times }\cong {K^{\mathrm { ind}}_3(A)}\left[ \tfrac{1}{2}\right] \) on coinvariants for the natural action of units \(A^\times \). We prove that the action of \(A^\times \) on \(\mathrm {H}_{3}\left( {\text {SL}}_{2}\left( A\right) ,{\mathbb {Z}}\left[ \tfrac{1}{2}\right] \right) \) is trivial when A is a complete discrete valuation ring with finite residue field of odd characteristic, and we show by example that this action is non-trivial for certain complete discrete valuation rings with infinite residue field.

Fundamental classes of 3-manifold groups representations in SL(4,R)

We compute the fundamental class (in the extended Bloch group) for representations of fundamental groups of [Formula: see text]-manifolds to [Formula: see text] that factor over [Formula: see text], in particular for those factoring over the isomorphism [Formula: see text]. We also discuss consequences for the number of connected components of [Formula: see text]-character varieties, and we show that there are knots with arbitrarily many components of vanishing Chern–Simons invariant in their [Formula: see text]-character varieties.

Duality and invariants of representations of fundamental groups of 3-manifolds into PGL(3,C)

We determine the explicit transformation under duality of generic configurations of four flags in $\PGL(3,\bC)$ in cross-ratio coordinates. As an application we prove invariance under duality of an invariant in the Bloch group obtained from decorated triangulations of 3-manifolds.

Homological and Bloch invariants for $\mathbb Q$-rank one spaces and flag structures

We use group homology to define invariants in algebraic K-theory and in an analogue of the Bloch group for \mathbb Q -rank one lattices and for some other geometric structures. We also show that the Bloch invariants of CR structures and of ag structures can be recovered by a fundamental class construction.

The complex volume of SL(n,C)-representations of 3-manifolds

For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of the fundamental group of M into SL(n,C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmueller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann's extended Bloch group, and use this to obtain an efficient formula for the Cheeger-Chern-Simons invariant, and in particular for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes, suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.

Nahm sums, stability and the colored Jones polynomial

Nahm sums are q-series of a special hypergeometric type that appear in character formulas in the conformal field theory, and give rise to elements of the Bloch group, and have interesting modularity properties. In our paper, we show how Nahm sums arise naturally in the quantum knot theory - we prove the stability of the coefficients of the colored Jones polynomial of an alternating link and present a Nahm sum formula for the resulting power series, defined in terms of a reduced diagram of the alternating link. The Nahm sum formula comes with a computer implementation, illustrated in numerous examples of proven or conjectural identities among q-series. Primary 57N10; Secondary 57M25.