Ramified covers of 3-torus - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-19T15:26:04Z http://mathoverflow.net/feeds/question/5546 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/5546/ramified-covers-of-3-torus Ramified covers of 3-torus Dmitri 2009-11-14T18:33:44Z 2009-11-15T23:27:03Z <p>It is known that every orientable 3-manfiold can be obtained as a ramified cover of S<sup>3</sup> with a ramification (of some order) at a link in S<sup>3</sup>. I am curious if there is a reasonable characterization of 3-manifolds that cover 3-torus? </p> <p>Added. Notice that such a manifold is enlargeble, so it does not admit a metric of positive scalar curvature, so for example a connected sum of n copies of S<sup>2</sup> x S<sup>1</sup> does not admit a ramified cover of T<sup>3</sup> (as far as I understand).</p> http://mathoverflow.net/questions/5546/ramified-covers-of-3-torus/5644#5644 Answer by Allan Edmonds for Ramified covers of 3-torus Allan Edmonds 2009-11-15T20:47:18Z 2009-11-15T22:01:02Z <p>Note that a branched covering induces an injection of rational cohomology rings, by transfer considerations. Therefore the cohomology of a manifold that is a branched covering of $T^3$ must contain three classes of degree 1 whose triple cup product is nontrivial. </p> <p>This condition on a manifold $M^3$ implies the existence of a map $M^3\to T^3$ of nonzero degree. Passing to a covering space of $T^3$ if necessary we obtain such a map that is also surjective on $\pi_1$. Assuming the resulting map is of degree $\ge 3$, the main result of [Edmonds, Allan L.Deformation of maps to branched coverings in dimension three. Math. Ann. 245 (1979), no. 3, 273--279.] implies that this map is homotopic to a branched covering.</p>