Computing arithmetic invariants of 3-manifolds

Experimental Mathematics

Volumn 9, Issue 1, 2000, Pages 127-152

  Coulson, David ^a   Goodman, Oliver A ^a   Hodgson, Craig D ^a   Neumann, Walter D ^b  

^a UNIVERSITY OF MELBOURNE   (Australia)

^b Och Spine at New York Presbyterian Hospitals   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0034348618     PISSN: 10586458     EISSN: None     Source Type: Journal    
DOI: 10.1080/10586458.2000.10504641     Document Type: Article

References (45)

1. [Atiyah et al. 1975] M. F. Atiyah, V. K. Patodi, and I. M. Singer, "Spectral asymmetry and Riemannian geometry, II", Math. Proc. Cambridge Philos. Soc. 78:3 (1975), 405-432.
2. [Batut et al. 1998] C. Batut, D. Bernardi, H. Cohen, and M. Olivier, User's Guide to Pari-GP, version 2.0, Université de Bordeaux, 1998. See ftp:// megrez.math.u-bordeaux.fr/pub/pari.
3. 1", Kobe J. Math. 3:2 (1987), 117-147.
4. [Callahan and Reid 1998] P. J. Callahan and A. W. Reid, "Hyperbolic structures on knot complements", Chaos Solitons Fractals 9:4-5 (1998), 705-738.
5. [Cheeger and Simons 1985] J. Cheeger and J. Simons, "Differential characters and geometric invariants", pp. 50-80 in Geometry and topology (College Park, MD, 1983/84), Lecture Notes in Math. 1167, Springer, Berlin, 1985.
6. [Chern and Simons 1971] S.-s. Chern and J. 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.
7. [Cohen 1993] H. Cohen, A course in computational algebraic number theory, Graduate Texts in Math. 138, Springer, Berlin, 1993.
8. [Dupont and Kamber 1993] J. L. Dupont and F. W. Kamber, "Cheeger-Chern-Simons classes of transversally symmetric foliations: dependence relations and eta-invariants", Math. Ann. 295:3 (1993), 449-468.
9. [Dupont and Sah 1982] J. L. Dupont and C. H. Sah, "Scissors congruences, II", J. Pure Appl. Algebra 25:2 (1982), 159-195. Corrigendum in 30:2 (1983), 217.
10. [Epstein and Penner 1988] D. B. A. Epstein and R. C. Penner, "Euclidean decompositions of noncompact hyperbolic manifolds", J. Differential Geom. 27:1 (1988), 67-80.
11. [Gehring et al. 1997] F. W. Gehring, C. Maclachlan, G. J. Martin, and A. W. Reid, "Arithmeticity, discreteness and volume", Trans. Amer. Math. Soc. 349:9 (1997), 3611-3643.
12. [Goodman et al. 1998] O. A. Goodman, C. D. Hodgson, and W. D. Neumann, "Snap Home Page", 1998. See http://www.ms.unimelb.edu.au/˜snap. Includes source distribution and extensive tables of results of Snap computations.
13. [Hildebrand and Weeks 1989] M. Hildebrand and J. Weeks, "A computer generated census of cusped hyperbolic 3-manifolds", pp. 53-59 in Computers and mathematics (Cambridge, MA, 1989), edited by E. Kaltofen and S. Watt, Springer, New York, 1989.
14. [Hilden et al. 1992] H. M. Hilden, M. T. Lozano, and J. M. Montesinos-Amilibia, "A characterization of arithmetic subgroups of SL(2, ℝ) and SL(2, ℂ)", Math. Nachr. 159 (1992), 245-270.
15. [Hodgson and Weeks ≥ 2000] C. D. Hodgson and J. Weeks, "A census of closed hyperbolic 3-manifolds". In preparation. See also ftp://ftp.northnet.org/pub/ weeks/SnapPea/ClosedCensus.
16. [Kirby 1997] R. Kirby, "Problems in low-dimensional topology", pp. 35-473 in Geometric topology (Athens, GA, 1993), vol. 2, edited by W. H. Kazez, AMS/IP Stud. Adv. Math. 2, Amer. Math. Soc., Providence, RI, 1997.
17. [Lang 1986] S. Lang, Algebraic number theory, Second ed., Graduate Texts in Math. 110, Springer, New York, 1986.
18. [Lenstra et al. 1982] A. K. Lenstra, H. W. Lenstra Jr., and L. Lovász, "Factoring polynomials with rational coefficients", Math. Ann. 261:4 (1982), 515-534.
19. [Macbeath 1983] A. M. Macbeath, "Commensurability of co-compact three-dimensional hyperbolic groups", Duke Math. J. 50:4 (1983), 1245-1253. Erratum in 56:1, (1988), 219. FMT
20. [Magnus 1980] W. Magnus, "Rings of Fricke characters and automorphism groups of free groups", Math. Z. 170:1 (1980), 91-103.
21. [Meyerhoff 1986] R. Meyerhoff, "Hyperbolic 3-manifolds with equal volumes but different Chem-Simons invariants", pp. 209-215 in Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984), edited by D. B. A. Epstein, London Math. Soc. lecture notes series 112, Cambridge Univ. Press, Cambridge, 1986.
22. [Meyerhoff and Neumann 1992] R. Meyerhoff and W. D. Neumann, "An asymptotic formula for the eta invariants of hyperbolic 3-manifolds", Comment. Math. Helv. 67:1 (1992), 28-46.
23. [Meyerhoff and Ouyang 1997] R. Meyerhoff and M. Ouyang, "The η-invariants of cusped hyperbolic 3-manifolds", Canad. Math. Bull. 40:2 (1997), 204-213.
24. [Milnor and Stasheff 1974] J. W. Milnor and J. D. Stasheff, Characteristic classes, Annals of Math. Studies 76, Princeton University Press, Princeton, N. J., 1974.
25. [Mostow 1973] G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Math. Studies 78, Princeton University Press, Princeton, N.J., 1973.
26. [Neumann 1978] W. D. Neumann, "Homotopy invariance of Atiyah invariants", pp. 181-188 in Algebraic and geometric topology (Stanford, 1976), vol. 2, edited by R. J. Milgram, Proc. Sympos. Pure Math. 32, Amer. Math. Soc., Providence, 1978.
27. [Neumann 1992] W. D. Neumann, "Combinatorics of triangulations and the Chern-Simons invariant for hyperbolic 3-manifolds", pp. 243-271 in Topology '90 (Columbus, OH, 1990), edited by B. Apanasov et al., de Gruyter, Berlin, 1992.
28. [Neumann 1998] W. D. Neumann, "Hilbert's 3rd problem and invariants of 3-manifolds", pp. 383-411 in The Epstein birthday schrift, edited by I. Rivin et al., Geom. and Top. Monog. 1, International Press and Warwick University, Coventry, UK, 1998. See http:// www.maths.warwick.ac.uk/gt.
29. [Neumann ≥ 2000] W. D. Neumann, "The extended Bloch group and the Chern-Simons invariant". In preparation. Preliminary version at http://www.math. columbia.edu/˜neumann/preprints.
30. [Neumann and Reid 1992a] W. D. Neumann and A. W. Reid, "Arithmetic of hyperbolic manifolds", pp. 273-310 in Topology '90 (Columbus, OH, 1990), edited by B. Apanasov et al., de Gruyter, Berlin, 1992.
31. [Neumann and Reid 1992b] W. D. Neumann and A. W. Reid, "Notes on Adams' small volume orbifolds", pp. 311-314 in Topology '90 (Columbus, OH, 1990), edited by B. Apanasov et al., de Gruyter, Berlin, 1992.
32. [Neumann and Yang 1995a] W. D. Neumann and J. Yang, "Invariants from triangulations of hyperbolic 3-manifolds", Electron. Res. Announc. Amer. Math. Soc. 1:2 (1995), 72-79.
33. [Neumann and Yang 1995b] 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), 281-296.
34. [Neumann and Yang 1999] W. D. Neumann and J. Yang, "Bloch invariants of hyperbolic 3-manifolds", Duke Math. J. 96:1 (1999), 29-59.
35. [Neumann and Zagier 1985] W. D. Neumann and D. Zagier, "Volumes of hyperbolic three-manifolds", Topology 24:3 (1985), 307-332.
36. [Ouyang 1997] M. Ouyang, "A simplicial formula for the η-invariant of hyperbolic 3-manifolds", Topology 36:2 (1997), 411-421.
37. [Pohst 1993] M. E. Pohst, Computational algebraic number theory, DMV Seminar 21, Birkhäuser, Basel, 1993.
38. [Reid 1987] A. W. Reid, Arithmetic Kleinian groups and their Fuchsian subgroups, Ph.D. thesis, Univ. of Aberdeen, 1987.
39. [Reid 1990] A. W. Reid, "A note on trace-fields of Kleinian groups", Bull. London Math. Soc. 22:4 (1990), 349-352.
40. [Schönhage 1987] A. Schönhage, "Equation solving in terms of computational complexity", pp. 131-153 in Proceedings of the International Congress of Mathematicians (Berkeley, CA, 1986), vol. 1, edited by A. M. Gleason, Amer. Math. Soc., Providence, RI, 1987.
41. [Takeuchi 1975] K. Takeuchi, "A characterization of arithmetic Fuchsian groups", J. Math. Soc. Japan 27:4 (1975), 600-612.
42. [Thurston 1979] W. P. Thurston, "The geometry and topology of 3-manifolds", Technical report, Princeton University, 1979. Available at http://www.msri.org/ publications/books/gt3m.
43. [Vignéras 1980] M.-F. Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Math. 800, Springer, Berlin, 1980.
44. [Weeks 1993] J. R. Weeks, "SnapPea", 1993. See http:// www.northnet.org/weeks/index/SnapPea.html.
45. [Yoshida 1985] T. Yoshida, "The η-invariant of hyperbolic 3-manifolds", Invent. Math. 81:3 (1985), 473-514.