Abstract
Given a separating embedded connected 3-manifold in a closed 4-manifold, the Seifert–van Kampen theorem implies that the fundamental group of the 4-manifold is an amalgamated product along the fundamental group of the 3-manifold. In the other direction, given a closed 4-manifold whose fundamental group admits an injective amalgamated product structure along the fundamental group of a 3-manifold, is there a corresponding geometric-topological decomposition of the 4-manifold in a stable sense? We find an algebraic-topological splitting criterion in terms of the orientation classes and universal covers. Also, we equivariantly generalize the Lickorish–Wallace theorem to regular covers.
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