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

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

MathSciNet review:
1415359

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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]**C. Adams, Thrice-punctured spheres in hyperbolic 3-manifolds, Trans. Amer. Math. Soc. 287 (1985), 645-656. MR**86k:57008****[A2]**C. Adams, Augmented alternating link complements are hyperbolic, London Math. Soc. Lecture Notes 112 (D. B. A. Epstein, ed.), 115-130. MR**89f:57003****[M]**R. Meyerhoff, Density of the Chern-Simons invariant for hyperbolic 3-manifolds, in Low-dimensional topology and Kleinian groups, London Math. Soc. Lect. Notes 112, D. B. A. Epstein, editor, Cambridge University Press, (1987), 217-240. MR**88k:57033a****[MO]**R. Meyerhoff and M. Ouyang, The -invariant of cusped hyperbolic 3-manifolds, to appear in Canadian Math. Bull.**[MR]**R. Meyerhoff and D. Ruberman, Mutation and the -invariant, J. Differential Geom. 31 (1990), 101-130. MR**91j:57017****[N]**W. Neumann, Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-manifolds, in Topology'90, Proceedings of the Research Semester on Low Dimensional Topology, de Gruyter Verlag, 1992. MR**93i:57020****[Y]**T. Yoshida, The -invariant of hyperbolic 3-manifolds, Invent. Math. 81 (1985), 473-514. MR**87f:58153**

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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:
https://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