Celebratio Mathematica — Neumann

A celebration of Walter Neumann

Works connected to Jun Yang

Filter the Bibliography List

year	title	people

			clear

W. D. Neumann and J. Yang: “Rationality problems for \( K \)-theory and Chern–Simons invariants of hyperbolic 3-manifolds,” Enseign. Math. (2) 41 : 3–4 (1995), pp. 281–296. MR 1365848 Zbl 0861.57022 article

W. D. Neumann and J. Yang: “Bloch invariants of hyperbolic 3-manifolds,” Duke Math. J. 96 : 1 (1999), pp. 29–59. MR 1663915 Zbl 0943.57008 ArXiv math/9712224 article

We define an invariant \( \beta(M) \) of a finite volume hyperbolic 3-manifold \( M \) in the Bloch group \( \mathscr{B}(\mathbb{C}) \) and show it is determined by the simplex parameters of any degree one ideal triangulation of \( M \). We show \( \beta(M) \) lies in a subgroup of \( \mathscr{B}(\mathbb{C}) \) of finite \( \mathbb{Q} \)-rank determined by the invariant trace field of \( M \). Moreover, the Chern–Simons invariant of \( M \) is determined modulo rationals by \( \beta(M) \). This leads to a simplicial formula and rationality results for the Chern–Simons invariant which appear elsewhere.

Generalizations of \( \beta(M) \) are also described, as well as several interesting examples. An appendix describes a scissors congruence interpretation of \( \mathscr{B}(\mathbb{C}) \).