Covering homotopy 3-spheres
type: Item
instance of: scholarly article
title: Covering homotopy 3-spheres
zbMATH Open document ID: 0759.57002
DOI: 10.1007/BF02566500
published in: Commentarii Mathematici Helvetici
publication date: 1993-01-16T00:00:00Z
full work available at URL: https://eudml.org/doc/140269
review text: 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.
Mathematics Subject Classification ID: 57M12
Mathematics Subject Classification ID: 57M40
Mathematics Subject Classification ID: 57N10
zbMATH DE Number: 93808
zbMATH Keywords: homotopy 3-sphere
zbMATH Keywords: simple 3-fold branched covering
zbMATH Keywords: Heegaard genus
zbMATH Keywords: Poincaré conjecture
author: Riccardo Piergallini
reviewed by: Bruno Zimmermann
MaRDI profile type: Publication
OpenAlex ID: W2011015880
description: scientific article; zbMATH DE number 93808