Stable existence of incompressible 3-manifolds in 4-manifolds (Q2319621)

Let \(M\) be a closed \(4\) manifold such that \(\pi_1(M)\cong G_{-} \ast_{G_0} G_{+}\) with \(G_0 \subseteq G_{\pm}\). Is there a corresponding decomposition of \(M\)? In dimension \(3\), this problem is known as Kneser's conjecture and was proved by \textit{J. Stallings} [Group theory and three-dimensional manifolds. New Haven-London: Yale University Press (1971; Zbl 0241.57001)]. It fails in dimensions \(\geq 5\) by results of \textit{S. E. Cappell} [Invent. Math. 33, 171--179 (1976; Zbl 0335.57007)]. In dimension four, \textit{M. Kreck} et al. proved the conjecture is false [Comment. Math. Helv. 70, No. 3, 423--433 (1995; Zbl 0837.57016)], but is true if one allows stabilization i.e., if one allows additional connected sums with copies of \(S^2 \times S^2\) [\textit{M. Kreck} et al., Ann. Math. Stud. 138, 251--269 (1995; Zbl 0928.57019)]. The paper under review investigates the problem of stably realizing an injective amalgamated product decomposition of the fundamental group of a \(4\)-manifold via separating embedded codimension one submanifolds. The authors find an algebraic topological splitting criterion in terms of the orientation classes and universal covers. Also, they equivariantly generalize the Lickorish-Wallace theorem to regular covers.