Covering homotopy 3-spheres (Q1199244)

It is proved that, given a homotopy 3-sphere \(\Sigma\), any closed orientable 3-manifold \(M\) is a simple 3-fold branched covering of \(\Sigma\), thus generalizing the corresponding well-known result (as well as a proof of it) in case of the 3-sphere \(S^ 3\). In particular, the 3- sphere \(S^ 3\) itself is a 3-fold branched covering of every homotopy 3- sphere \(\Sigma\). In view of this it is then conjectured that for a primitive (i.e. surjective on fundamental groups) branched covering \(p: M\to N\) between closed orientable 3-manifolds, the Heegaard genus of \(M\) is larger than that of \(N\) (which is trivially true if the Heegaard genus is replaced by the rank of the fundamental groups). By the above, in case \(M=S^ 3\) this is equivalent to the Poincaré conjecture.