The Extended Bloch Group and the Cheeger–Chern–Simons Class

Goette, Sebastian; Zickert, Christian K.

doi:10.2140/gt.2007.11.1623

Cited by 22 publications

(42 citation statements)

References 8 publications

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“…The formula generalizes the formula in Goette-Zickert [16] for n = 2. Recall that a homology class can be represented by a formal sum of tuples (g 0 , .…”

Section: Remark 12supporting

confidence: 51%

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

Garoufalidis¹,

Thurston²,

Zickert³

2015

Duke Math. J.

186

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Abstract. 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 π1(M ) into SL(n, C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmüller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann's extended Bloch group B(C), 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.

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“…The formula generalizes the formula in Goette-Zickert [16] for n = 2. Recall that a homology class can be represented by a formal sum of tuples (g 0 , .…”

Section: Remark 12supporting

confidence: 51%

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

Garoufalidis¹,

Thurston²,

Zickert³

2015

Duke Math. J.

186

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show abstract

“…As we mentioned before, there is a close analogy between our paper and Neumann's work on the extended Bloch group B(C); see [Ne,DZ,GZ,Ga1]. Let us summarize this into a table, which enhances the table of [BE,Sec.5].…”

Section: Theorem 1 (A)mentioning

confidence: 56%

An extended version of additive K-theory

Garoufalidis¹

2008

J K-Theor

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Abstract. There are two infinitesimal (i.e., additive) versions of the K-theory of a field F : one introduced by Cathelineau, which is an F -module, and another one introduced by Bloch-Esnault, which is an F * -module. Both versions are equipped with a regulator map, when F is the field of complex numbers.In our short paper we will introduce an extended version of Cathelineau's group, and a complex-valued regulator map given by the entropy. We will also give a comparison map between our extended version and Cathelineau's group.Our results were motivated by two unrelated sources: Neumann's work on the extended Bloch group (which is isomorphic to indecomposable K 3 of the complex numbers), and the study of singularities of generating series of hypergeometric multisums.

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“…φ K determines K and conversely is determined by K via Galois invariance. [35,50]. The next conjecture was formulated by Zagier.…”

Section: The Modularity Conjecturementioning

confidence: 93%

Quantum knot invariants

Garoufalidis

2018

Res Math Sci

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This is a survey talk on one of the best known quantum knot invariants, the colored Jones polynomial of a knot, and its relation to the algebraic/geometric topology and hyperbolic geometry of the knot complement. We review several aspects of the colored Jones polynomial, emphasizing modularity, stability and effective computations. The talk was given in the Mathematische Arbeitstagung June 24-July 1, 2011.

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The Extended Bloch Group and the Cheeger–Chern–Simons Class

Abstract: Abstract. We present a formula for the full Cheeger-Chern-Simons class of the tautological flat complex vector bundle of rank 2 over BSL(2, C δ ). Our formula improves the formula in [DZ], where the class is only computed modulo 2-torsion.

Cited by 22 publications

References 8 publications

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

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

An extended version of additive K-theory

Quantum knot invariants

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