Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Isospectrality and $3$-manifold groups


Author: Daniel Ruberman
Journal: Proc. Amer. Math. Soc. 129 (2001), 2467-2471
MSC (2000): Primary 57M10, 58J28
DOI: https://doi.org/10.1090/S0002-9939-00-05820-2
Published electronically: December 7, 2000
MathSciNet review: 1823933
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract:

The Chern-Simons and $\eta$-invariants are closely related invariants of a Riemannian $3$-manifold. A difference in their behavior under taking a finite covering space is exploited to give an obstruction to a group being the fundamental group of a closed $3$-dimensional manifold.


References [Enhancements On Off] (What's this?)

  • [APS75a] M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry: I, Math. Proc. Camb. Phil. Soc. 77 (1975), 43-69. MR 53:1655a
  • [APS75b] M.F. Atiyah, V.K. Patodi, and I.M. Singer, Spectral asymmetry and Riemannian geometry: II, Math. Proc. Camb. Phil. Soc. 78 (1975), 405-432. MR 53:1655b
  • [Bér92] Pierre Bérard, Transplantation et isospectralité. I, Math. Ann. 292 (1992), 547-559. MR 93a:58168
  • [CS74] S. Chern and J. Simons, Characteristic forms and geometric invariants, Annals of Math. 99 (1974), 48-69. MR 50:5811
  • [Dav83] James F. Davis, The surgery semicharacteristic, Proc. London Math. Soc. (3) 47 (1983), 411-428. MR 86a:57027
  • [DM85] J. F. Davis and R. J. Milgram, A survey of the spherical space form problem, Mathematical Reports, vol. 2, Harwood Academic Publishers, Chur, 1985. MR 87e:57001
  • [HS64] Jr. Hall, Marshall and James K. Senior, The groups of order $2\sp{n}\,(n\leq 6)$, The Macmillan Co., New York, 1964. MR 29:5889
  • [Lee73] R. Lee, Semicharacteristic classes, Topology 12 (1973), 183-199. MR 50:14809
  • [Mil57] John Milnor, Groups which act on ${S}\sp n$ without fixed points, Amer. J. Math. 79 (1957), 623-630. MR 19:761d
  • [S$^+$93] Martin Schönert et al., GAP - Groups, Algorithms, and Programming, Lehrstuhl D für Mathematik, Rheinisch Westfälische Technische Hochschule, Aachen, Germany, third ed., 1993.
  • [Sun85] T. Sunada, Riemannian coverings and isospectral manifolds, Annals of Math. 121 (1985), 169-186. MR 86h:58141
  • [Tho86] C.B. Thomas, Elliptic structures on $3$-spheres, London Math. Soc. Lecture Note Ser., vol. 104, Cambridge Univ. Press, Cambridge, 1986. MR 87m:57015

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 57M10, 58J28

Retrieve articles in all journals with MSC (2000): 57M10, 58J28


Additional Information

Daniel Ruberman
Affiliation: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02454
Email: ruberman@brandeis.edu

DOI: https://doi.org/10.1090/S0002-9939-00-05820-2
Received by editor(s): March 16, 1998
Received by editor(s) in revised form: December 6, 1999
Published electronically: December 7, 2000
Additional Notes: The author was partially supported by NSF Grant 4-50645
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society