Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold? - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-24T11:03:03Z http://mathoverflow.net/feeds/question/105931 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/105931/is-compact-flat-manifold-cusp-cross-sections-of-a-complete-finite-volume-hyperbol Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold? J. GE 2012-08-30T11:56:45Z 2012-08-30T14:21:34Z <p>Let \$M^{n-1}\$ be a closed flat manifold. Is it true that there exists a hyperbolic manifold \$N^n\$ with finite volume such that \$M\$ is a cusp cross-section of \$N\$?</p> <p>It was proved in "On the geometric boundaries of hyperbolic 4-manifolds" by Long and Reid in Geom. Topol. 4 (2000), 171–178 that there are 3-manifolds are not cusp cross-sections of any complete finite volume one-cusped hyperbolic 4-manifold, due to the obstructions of eta-invariant. They constructed a 3-dim flat manifold with eta invariant = -4/3. if it was the cusp of a hyperbolic manifold then it has to be integer according to Aatiyah-Patodi-Singer's theorem.</p> <p>My question is can it always a cross-section of several-cusped hyperbolic 4-manifold?</p> http://mathoverflow.net/questions/105931/is-compact-flat-manifold-cusp-cross-sections-of-a-complete-finite-volume-hyperbol/105944#105944 Answer by Igor Rivin for Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold? Igor Rivin 2012-08-30T13:20:53Z 2012-08-30T13:20:53Z <p>Since there are countably many finite-volume \$3\$-manifolds, and uncountably many flat tori, the answer is NO for \$n=3,\$ at least if your question is geometric, as opposed to topological.</p> http://mathoverflow.net/questions/105931/is-compact-flat-manifold-cusp-cross-sections-of-a-complete-finite-volume-hyperbol/105948#105948 Answer by Richard Kent for Is compact flat manifold cusp cross-sections of a complete finite volume hyperbolic manifold? Richard Kent 2012-08-30T14:21:34Z 2012-08-30T14:21:34Z <p>Yes. Every (compact) flat \$n\$-manifold is diffeomorphic to a cusp cross section of a hyperbolic \$(n+1)\$-manifold. This is a theorem of McReynolds, Controlling manifold covers of orbifolds, Math. Res. Lett. 16 (2009), 651-662.</p> <p>In your case, \$n=3\$, this is a theorem of Nimershiem. B. E. Nimershiem, All ﬂat three-manifolds appear as cusps of hyperbolic four-manifolds, Topology Appl. 90 (1998), 109–133.</p>