Geodesic
On the geometric boundaries of hyperbolic 4–manifolds
Long, Darren D1 ; Reid, Alan W2
1 Department of Mathematics, University of California, Santa Barbara, California 93106, USA
2 Department of Mathematics, University of Texas, Austin, Texas 78712, USA
Geometry & topology, Tome 4 (2000) no. 1, pp. 171-178.

Voir la notice de l'article provenant de la source Mathematical Sciences Publishers

We provide, for hyperbolic and flat 3–manifolds, obstructions to bounding hyperbolic 4–manifolds, thus resolving in the negative a question of Farrell and Zdravkovska.

DOI : 10.2140/gt.2000.4.171
Keywords: hyperbolic 3–manifold, flat manifold, totally geodesic, $\eta$–invariant

Long, Darren D 1 ; Reid, Alan W 2

1 Department of Mathematics, University of California, Santa Barbara, California 93106, USA
2 Department of Mathematics, University of Texas, Austin, Texas 78712, USA 
@article{GT_2000_4_1_a4,
     author = {Long, Darren D and Reid, Alan W},
     title = {On the geometric boundaries of hyperbolic 4{\textendash}manifolds},
     journal = {Geometry & topology},
     pages = {171--178},
     publisher = {mathdoc},
     volume = {4},
     number = {1},
     year = {2000},
     doi = {10.2140/gt.2000.4.171},
     url = {http://geodesic.mathdoc.fr/articles/10.2140/gt.2000.4.171/}
}
TY  - JOUR
AU  - Long, Darren D
AU  - Reid, Alan W
TI  - On the geometric boundaries of hyperbolic 4–manifolds
JO  - Geometry & topology
PY  - 2000
SP  - 171
EP  - 178
VL  - 4
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/articles/10.2140/gt.2000.4.171/
DO  - 10.2140/gt.2000.4.171
ID  - GT_2000_4_1_a4
ER  - 
%0 Journal Article
%A Long, Darren D
%A Reid, Alan W
%T On the geometric boundaries of hyperbolic 4–manifolds
%J Geometry & topology
%D 2000
%P 171-178
%V 4
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/articles/10.2140/gt.2000.4.171/
%R 10.2140/gt.2000.4.171
%F GT_2000_4_1_a4
Long, Darren D; Reid, Alan W. On the geometric boundaries of hyperbolic 4–manifolds. Geometry & topology, Tome 4 (2000) no. 1, pp. 171-178. doi : 10.2140/gt.2000.4.171. http://geodesic.mathdoc.fr/articles/10.2140/gt.2000.4.171/

[1] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry, Bull. London Math. Soc. 5 (1973) 229

[2] M F Atiyah, V K Patodi, I M Singer, Spectral asymmetry and Riemannian geometry I, Math. Proc. Cambridge Philos. Soc. 77 (1975) 43

[3] A Basmajian, Tubular neighborhoods of totally geodesic hypersurfaces in hyperbolic manifolds, Invent. Math. 117 (1994) 207

[4] R L Bishop, R J Crittenden, Geometry of manifolds, Pure and Applied Mathematics XV, Academic Press (1964)

[5] T Eguchi, P B Gilkey, A J Hanson, Gravitation, gauge theories and differential geometry, Phys. Rep. 66 (1980) 213

[6] C M Gordon, Knots, homology spheres, and contractible 4–manifolds, Topology 14 (1975) 151

[7] G W Gibbons, Tunnelling with a negative cosmological constant, Nuclear Phys. B 472 (1996) 683

[8] P B Gilkey, The boundary integrand in the formula for the signature and Euler characteristic of a Riemannian manifold with boundary, Advances in Math. 15 (1975) 334

[9] M Gromov, Volume and bounded cohomology, Inst. Hautes Études Sci. Publ. Math. (1982)

[10] G C Hamrick, D C Royster, Flat Riemannian manifolds are boundaries, Invent. Math. 66 (1982) 405

[11] F T Farrell, S Zdravkovska, Do almost flat manifolds bound?, Michigan Math. J. 30 (1983) 199

[12] D D Long, A W Reid, All flat manifolds are cusps of hyperbolic orbifolds, Algebr. Geom. Topol. 2 (2002) 285

[13] R Meyerhoff, W D Neumann, An asymptotic formula for the eta invariants of hyperbolic 3–manifolds, Comment. Math. Helv. 67 (1992) 28

[14] B E Nimershiem, All flat three-manifolds appear as cusps of hyperbolic four-manifolds, Topology Appl. 90 (1998) 109

[15] M Q Ouyang, Geometric invariants for Seifert fibred 3–manifolds, Trans. Amer. Math. Soc. 346 (1994) 641

[16] J G Ratcliffe, S T Tschantz, Gravitational instantons of constant curvature, Classical Quantum Gravity 15 (1998) 2613

[17] J G Ratcliffe, S T Tschantz, The volume spectrum of hyperbolic 4–manifolds, Experiment. Math. 9 (2000) 101

[18] W P Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press (1997)

[19] H C Wang, Topics on totally discontinuous groups, from: "Symmetric spaces (Short Courses, Washington Univ., St. Louis, Mo., 1969–1970)", Pure and Appl. Math. 8, Dekker (1972) 459

[20] J Weeks, Topological questions in cosmology, from: "Proceedings of the Oklahoma conference on computational topology", to appear

Cité par Sources :