A note on Hempel–McMillan coverings of 3-manifolds

Motivated by the concept of A -category of a manifold introduced by Clapp and Puppe, we give a different proof of a (slightly generalized) Theorem of Hempel and McMillan: If M is a closed 3-manifold that is a union of three open punctured balls then M is a connected sum of S 3 and S 2 -bundles over S 1 .