A note on the Chern-Simons invariant

of hyperbolic 3-manifolds

Author:
Mingqing Ouyang

Journal:
Proc. Amer. Math. Soc. **125** (1997), 1845-1851

MSC (1991):
Primary 57N10; Secondary 57M25

MathSciNet review:
1415359

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Abstract | References | Similar Articles | Additional Information

Abstract: In this note we study how the Chern-Simons invariant behaves when two hyperbolic 3-manifolds are glued together along incompressible

thrice-punctured spheres. Such an operation produces many hyperbolic 3-manifolds with different numbers of cusps sharing the same volume and the same Chern-Simons invariant. The results in this note, combined with those of Meyerhoff and Ruberman, give an algorithm for determining the unknown constant in Neumann's simplicial formula for the Chern-Simons invariant of hyperbolic 3-manifolds.

**[A1]**Colin C. Adams,*Thrice-punctured spheres in hyperbolic 3-manifolds*, Trans. Amer. Math. Soc.**287**(1985), no. 2, 645–656. MR**768730**, 10.1090/S0002-9947-1985-0768730-6**[A2]**Colin C. Adams,*Augmented alternating link complements are hyperbolic*, Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), London Math. Soc. Lecture Note Ser., vol. 112, Cambridge Univ. Press, Cambridge, 1986, pp. 115–130. MR**903861****[M]**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****[MO]**R. Meyerhoff and M. Ouyang, The -invariant of cusped hyperbolic 3-manifolds, to appear in Canadian Math. Bull.**[MR]**Robert Meyerhoff and Daniel Ruberman,*Mutation and the 𝜂-invariant*, J. Differential Geom.**31**(1990), no. 1, 101–130. MR**1030667****[N]**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****[Y]**Tomoyoshi Yoshida,*The 𝜂-invariant of hyperbolic 3-manifolds*, Invent. Math.**81**(1985), no. 3, 473–514. MR**807069**, 10.1007/BF01388583

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Additional Information

**Mingqing Ouyang**

Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Email:
mouyang@math.lsa.umich.edu

DOI:
http://dx.doi.org/10.1090/S0002-9939-97-04022-7

Received by editor(s):
December 5, 1995

Communicated by:
Ronald Stern

Article copyright:
© Copyright 1997
American Mathematical Society