Volume 1 (1998)
Download this article
For printing
Recent Volumes
Volume 1, 1998
Volume 2, 1999
Volume 3, 2000
Volume 4, 2002
Volume 5, 2002
Volume 6, 2003
Volume 7, 2004
Volume 8, 2006
Volume 9, 2006
Volume 10, 2007
Volume 11, 2007
Volume 12, 2007
Volume 13, 2008
Volume 14, 2008
Volume 15, 2008
Volume 16, 2009
Volume 17, 2011
Volume 18, 2012
Volume 19, 2015
The Series
All Volumes
 
About this Series
Ethics Statement
Purchase Printed Copies
Author Index
ISSN 1464-8997 (online)
ISSN 1464-8989 (print)
 
MSP Books and Monographs
Other MSP Publications

Hilbert's 3rd Problem and Invariants of 3–manifolds

Walter D Neumann

Geometry & Topology Monographs 1 (1998) 383–411

DOI: 10.2140/gtm.1998.1.383
Bibliography
1 C Batut, D Bernardi, H Cohen, M Olivier, Pari-GP
2 S Bloch, Higher regulators, algebraic K–theory, and zeta functions of elliptic curves, lecture notes, University of California, Irvine (1978)
3 M Bökstedt, M Brun, J Dupont, Homology of O(n) and O¹(1,n) made discrete: an application of edgewise subdivision, J. Pure Appl. Algebra 123 (1998) 131–152 MR1492898
4 A Borel, Cohomologie de SLn et valeurs de fonctions zeta aux points entiers, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 4 (1977) 613–636 MR0506168
5 J Cheeger, J Simons, Differential characters and geometric invariants, from: "Geometry and topology (College Park, Md., 1983/84)", Lecture Notes in Math. 1167, Springer (1985) 50–80 MR827262
6 S s Chern, 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 MR0279732
7 M Dehn, Ueber den Rauminhalt, Math. Ann. 55 (1901) 465–478 MR1511157
8 J L Dupont, The dilogarithm as a characteristic class for flat bundles, from: "Proceedings of the Northwestern conference on cohomology of groups (Evanston, Ill., 1985)" (1987) 137–164 MR885101
9 J L Dupont, Algebra of polytopes and homology of flag complexes, Osaka J. Math. 19 (1982) 599–641 MR676240
10 J L Dupont, F W Kamber, Cheeger–Chern–Simons classes of transversally symmetric foliations: dependence relations and eta-invariants, Math. Ann. 295 (1993) 449–468 MR1204831
11 J L Dupont, W Parry, C H Sah, Homology of classical Lie groups made discrete II: H 2,H3, and relations with scissors congruences, J. Algebra 113 (1988) 215–260 MR928063
12 J L Dupont, C H Sah, Scissors congruences II, J. Pure Appl. Algebra 25 (1982) 159–195 MR662760
13 J L Dupont, C H Sah, Homology of Euclidean groups of motions made discrete and Euclidean scissors congruences, Acta Math. 164 (1990) 1–27 MR1037596
14 J L Dupont, C H Sah, Three questions about simplices in spherical and hyperbolic 3–space, from: "The Gelfand Mathematical Seminars, 1996–1999", Gelfand Math. Sem., Birkhäuser (2000) 49–76 MR1731633
15 D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80 MR918457
16 A Goncharov, Volumes of hyperbolic manifolds and mixed Tate motives, J. Amer. Math. Soc. 12 (1999) 569–618 MR1649192
17 O Goodman, Snap
18 D Hilbert, Mathematical problems, from: "Mathematical developments arising from Hilbert's problems", Proc. Symp. Pure Math. 28, Amer. Math. Soc. (1976)
19 B Jessen, The algebra of polyhedra and the Dehn–Sydler theorem, Math. Scand. 22 (1968) 241–256 MR0251633
20 B Jessen, Zur Algebra der Polytope, Nachr. Akad. Wiss. Göttingen Math.-Phys. Kl. II (1972) 47–53 MR0353150
21 R Meyerhoff, Hyperbolic 3–manifolds with equal volumes but different Chern–Simons invariants, from: "Low-dimensional topology and Kleinian groups (Coventry/Durham, 1984)", London Math. Soc. Lecture Note Ser. 112, Cambridge Univ. Press (1986) 209–215 MR903866
22 J Milnor, Hyperbolic geometry: the first 150 years, Bull. Amer. Math. Soc. (N.S.) 6 (1982) 9–24 MR634431
23 W D Neumann, Combinatorics of triangulations and the Chern–Simons invariant for hyperbolic 3–manifolds, from: "Topology '90 (Columbus, OH, 1990)", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 243–271 MR1184415
24 W D Neumann, Extended Bloch group and the Cheeger–Chern–Simons class, Geom. Topol. 8 (2004) 413–474 MR2033484
25 R Meyerhoff, W D Neumann, An asymptotic formula for the eta invariants of hyperbolic 3–manifolds, Comment. Math. Helv. 67 (1992) 28–46 MR1144612
26 W D Neumann, A W Reid, Arithmetic of hyperbolic manifolds, from: "Topology '90 (Columbus, OH, 1990)", Ohio State Univ. Math. Res. Inst. Publ. 1, de Gruyter (1992) 273–310 MR1184416
27 W D Neumann, A W Reid, Amalgamation and the invariant trace field of a Kleinian group, Math. Proc. Cambridge Philos. Soc. 109 (1991) 509–515 MR1094749
28 W D Neumann, J Yang, Rationality problems for K–theory and Chern–Simons invariants of hyperbolic 3–manifolds, Enseign. Math. (2) 41 (1995) 281–296 MR1365848
29 W D Neumann, J Yang, Invariants from triangulations of hyperbolic 3–manifolds, Electron. Res. Announc. Amer. Math. Soc. 1 (1995) 72–79 MR1350682
30 W D Neumann, J Yang, Bloch invariants of hyperbolic 3–manifolds, Duke Math. J. 96 (1999) 29–59 MR1663915
31 W D Neumann, D Zagier, Volumes of hyperbolic three-manifolds, Topology 24 (1985) 307–332 MR815482
32 M Ouyang, A simplicial formula for the η–invariant of hyperbolic 3–manifolds, Topology 36 (1997) 411–421 MR1415596
33 D Ramakrishnan, Regulators, algebraic cycles, and values of L–functions, from: "Algebraic K–theory and algebraic number theory (Honolulu, HI, 1987)", Contemp. Math. 83, Amer. Math. Soc. (1989) 183–310 MR991982
34 A W Reid, A note on trace-fields of Kleinian groups, Bull. London Math. Soc. 22 (1990) 349–352 MR1058310
35 A Reznikov, Rationality of secondary classes, J. Differential Geom. 43 (1996) 674–692 MR1412681
36 C H Sah, Hilbert's third problem: scissors congruence, Research Notes in Mathematics 33, Pitman (Advanced Publishing Program) (1979) MR554756
37 C H Sah, Scissors congruences I: The Gauss–Bonnet map, Math. Scand. 49 (1981) 181–210 MR661890
38 A A Suslin, Algebraic K–theory of fields, from: "Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Berkeley, Calif., 1986)", Amer. Math. Soc. (1987) 222–244 MR934225
39 A A Suslin, K3 of a field, and the Bloch group, Trudy Mat. Inst. Steklov. 183 (1990) 180–199 MR1092031 Translated in Proc. Steklov Inst. Math. \bf1991, no. 4, 217–239, Galois theory, rings, algebraic groups and their applications (Russian)
40 J P Sydler, Conditions nécessaires et suffisantes pour l'équivalence des polyèdres de l'espace euclidien à trois dimensions, Comment. Math. Helv. 40 (1965) 43–80 MR0192407
41 W P Thurston, The geometry and topology of 3–manifolds, lecture notes, Princeton University (1977)
42 W P Thurston, Hyperbolic structures on 3–manifolds I: Deformation of acylindrical manifolds, Ann. of Math. (2) 124 (1986) 203–246 MR855294
43 W P Thurston, Three-dimensional manifolds, Kleinian groups and hyperbolic geometry, Bull. Amer. Math. Soc. (N.S.) 6 (1982) 357–381 MR648524
44 J Weeks, Snappea
45 T Yoshida, The η–invariant of hyperbolic 3–manifolds, Invent. Math. 81 (1985) 473–514 MR807069
46 D Zagier, The Bloch–Wigner–Ramakrishnan polylogarithm function, Math. Ann. 286 (1990) 613–624 MR1032949
47 D Zagier, Polylogarithms, Dedekind zeta functions and the algebraic K–theory of fields, from: "Arithmetic algebraic geometry (Texel, 1989)", Progr. Math. 89, Birkhäuser (1991) 391–430 MR1085270
48 V B Zylev, On the equi-dissectability of two equi-completable polytopes, Dokl. Akad. Nauk SSSR 161 (1965) 515–516 MR0177347
49 V B Zylev, G–composedness and G–complementability, Dokl. Akad. Nauk SSSR 179 (1968) 529–530 MR0226480