Invariants from triangulations of hyperbolic 3-manifolds

Neumann, Walter; Yang, Jun

doi:10.1090/S1079-6762-95-02003-8

Invariants from triangulations of hyperbolic 3-manifolds

Authors: Walter D. Neumann and Jun Yang
Journal: Electron. Res. Announc. Amer. Math. Soc. 1 (1995), 72-79
MSC (1991): Primary 57M50, 30F40, 19E99, 22E40, 57R20
DOI: https://doi.org/10.1090/S1079-6762-95-02003-8
MathSciNet review: 1350682
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Abstract: For any finite volume hyperbolic 3-manifold $M$ we use ideal triangulation to define an invariant $\beta (M)$ in the Bloch group $\mathcal {B}(\mathbb {C})$. It actually lies in the subgroup of $\mathcal {B}(\mathbb {C})$ determined by the invariant trace field of $M$. The Chern-Simons invariant of $M$ is determined modulo rationals by $\beta (M)$. This implies rationality and — assuming the Ramakrishnan conjecture — irrationality results for Chern Simons invariants.

S. Bloch: Higher regulators, algebraic $K$-theory, and zeta functions of elliptic curves, Lecture notes U.C. Irvine (1978).

Armand Borel, Cohomologie de ${\rm SL}_{n}$ et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977), no. 4, 613–636 (French). MR 506168

Shiing-shen Chern and James Simons, Some cohomology classes in principal fiber bundles and their application to riemannian geometry, Proc. Nat. Acad. Sci. U.S.A. 68 (1971), 791–794. MR 279732, DOI https://doi.org/10.1073/pnas.68.4.791

Warren Dicks and M. J. Dunwoody, Groups acting on graphs, Cambridge Studies in Advanced Mathematics, vol. 17, Cambridge University Press, Cambridge, 1989. MR 1001965

Johan L. Dupont, The dilogarithm as a characteristic class for flat bundles, Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985), 1987, pp. 137–164. MR 885101, DOI https://doi.org/10.1016/0022-4049%2887%2990021-1

Chih Han Sah, Scissors congruences. I. The Gauss-Bonnet map, Math. Scand. 49 (1981), no. 2, 181–210 (1982). MR 661890, DOI https://doi.org/10.7146/math.scand.a-11930

D. B. A. Epstein and R. C. Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988), no. 1, 67–80. MR 918457

A.B. Goncharov: Volumes of hyperbolic manifolds and mixed Tate motives, (in preparation).

Richard M. Hain, Classical polylogarithms, Motives (Seattle, WA, 1991) Proc. Sympos. Pure Math., vol. 55, Amer. Math. Soc., Providence, RI, 1994, pp. 3–42. MR 1265550, DOI https://doi.org/10.1007/bf01446290

B. Gross: On the values of Artin $L$-functions, preprint (Brown, 1980).

Robert Meyerhoff, Density of the Chern-Simons invariant for hyperbolic $3$-manifolds, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984) London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 217–239. MR 903867

Walter D. Neumann, Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic $3$-manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 243–271. MR 1184415

Walter D. Neumann and Alan W. Reid, Arithmetic of hyperbolic manifolds, Topology ’90 (Columbus, OH, 1990) Ohio State Univ. Math. Res. Inst. Publ., vol. 1, de Gruyter, Berlin, 1992, pp. 273–310. MR 1184416

Walter D. Neumann and Alan W. Reid, Amalgamation and the invariant trace field of a Kleinian group, Math. Proc. Cambridge Philos. Soc. 109 (1991), no. 3, 509–515. MR 1094749, DOI https://doi.org/10.1017/S0305004100069942

W. D. Neumann, J. Yang: Rationality problems for $K$-theory and Chern-Simons invariants of hyperbolic 3-manifolds, Enseignement Mathématique (to appear).

W. D. Neumann, J. Yang: Bloch invariants of hyperbolic 3-manifolds, in preparation.

Walter D. Neumann and Don Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985), no. 3, 307–332. MR 815482, DOI https://doi.org/10.1016/0040-9383%2885%2990004-7

Dinakar Ramakrishnan, Regulators, algebraic cycles, and values of $L$-functions, Algebraic $K$-theory and algebraic number theory (Honolulu, HI, 1987) Contemp. Math., vol. 83, Amer. Math. Soc., Providence, RI, 1989, pp. 183–310. MR 991982, DOI https://doi.org/10.1090/conm/083/991982

Alan W. Reid, A note on trace-fields of Kleinian groups, Bull. London Math. Soc. 22 (1990), no. 4, 349–352. MR 1058310, DOI https://doi.org/10.1112/blms/22.4.349

A. A. Suslin, Algebraic $K$-theory of fields, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986) Amer. Math. Soc., Providence, RI, 1987, pp. 222–244. MR 934225

William P. Thurston, Hyperbolic structures on $3$-manifolds. I. Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986), no. 2, 203–246. MR 855294, DOI https://doi.org/10.2307/1971277

William P. Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982), no. 3, 357–381. MR 648524, DOI https://doi.org/10.1090/S0273-0979-1982-15003-0

J. Weeks: Snappea, Software for hyperbolic 3-manifolds, available atftp://ftp.geom.umn.edu/pub/software/snappea

Don Zagier, Polylogarithms, Dedekind zeta functions and the algebraic $K$-theory of fields, Arithmetic algebraic geometry (Texel, 1989) Progr. Math., vol. 89, Birkhäuser Boston, Boston, MA, 1991, pp. 391–430. MR 1085270