Welcome to MathOverflow

Question

9

2

It is known that every orientable 3-manfiold can be obtained as a ramified cover of S³ with a ramification (of some order) at a link in S³. I am curious if there is a reasonable characterization of 3-manifolds that cover 3-torus?

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 S² x S¹ does not admit a ramified cover of T³ (as far as I understand).

edited Nov 15 2009 at 23:27

Greg Kuperberg
36.3k●2●116●179

asked Nov 14 2009 at 18:33

Dmitri
16.6k●1●36●77

Any conjectures? Clearly there is an onto homomorphism $\pi_1M\to Z^3$; can you do better? – Anton Petrunin Nov 14 2009 at 23:01

I don't really know anything about this question, and can not do better than the homeomorphism to Z^3. For every M^3 its connected sum with several T^3 admits a cover to T^3 :) – Dmitri Nov 14 2009 at 23:58

Conjecture: M is a branched cover of T^3 iff the first homology group of M has rank at least three. – Sam Nead Nov 15 2009 at 0:59

(M assumed to be closed, orientable, connected.) – Sam Nead Nov 15 2009 at 1:00

1

No, in fact I did not read the proof for S^3. Also the existence of onto homeo $\pi_1M\to Z^3$ as well having rank of first homology at lest 3 is no sufficient because covers of T^3 don't admit a metric of positive scalar curvature, so the conncted sum of 3 S^2xS^1 will be a counterexample. – Dmitri Nov 15 2009 at 10:34

show 3 more comments

Allan Edmonds · Accepted Answer · 2009-11-15 22:01:02Z UTC

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.

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.

Ramified covers of 3-torus

Remember to vote up questions/answers you find interesting or helpful (requires 15 reputation points)

1 Answer

You can accept an answer to one of your own questions by clicking the check mark next to it. This awards 15 reputation points to the person who answered and 2 reputation points to you.