Decomposing four-manifolds up to homotopy type. (English) Zbl 1024.57022

Let \(M\) be a closed connected orientable topological \(4\)-manifold with fundamental group \(\pi_1=\pi_1(M)\). This paper investigates the question of when \(M\) is homotopy equivalent to a connected sum of the form \(P\#M'\), with \(M'\) a simply-connected closed \(4\)-manifold not homeomorphic to the \(4\)-sphere. The main Theorem 1 provides a characterization, given a degree one map \(f:M\to P\) which induces an isomorphism on \(\pi_1\). In this case, \(M\) is homotopy equivalent to \(P\#M'\) if and only if \[ \lambda_M^\Lambda |_{K_2(f,\Lambda)}\cong \lambda_M^{\mathbb Z} |_{K_2(f,{\mathbb Z})}\otimes_{\mathbb Z} \Lambda, \] where \(\Lambda={\mathbb Z}[\pi_1]\), and \(\lambda_M^\Lambda |_{K_2(f,\Lambda)}\) and \(\lambda_M^{\mathbb Z} |_{K_2(f,{\mathbb Z})}\) are the intersection forms of \(M\) with coefficients \(\Lambda\) and \({\mathbb Z}\), respectively, restricted to the kernels of the maps induced by \(f\) on second homology.
After giving two different proofs of Theorem 1, the authors list some consequences. They give various splitting theorems, providing algebraic conditions for \(M\) to be identified, up to homotopy equivalence and in certain cases up to homeomorphism, as a connected sum of \(M'\) with common manifolds such as \(S^1\times S^3\), and \(F\times S^2\) with \(F\) a closed aspherical surface. A further consequence concerns \(4\)-manifolds \(M\) with \(\pi_1\) torsion free and infinite, and \(\pi_2(M)\) trivial. Hillman has conjectured that any such \(M\) is homeomorphic to a connected sum of aspherical closed \(4\)-manifolds and \(S^1\times S^3\) factors. Theorem 1 implies that \(M\) is homotopy equivalent to a connected sum of that form, verifying Hillman’s conjecture up to homotopy type.


57N65 Algebraic topology of manifolds
55P10 Homotopy equivalences in algebraic topology
57Q10 Simple homotopy type, Whitehead torsion, Reidemeister-Franz torsion, etc.
57R67 Surgery obstructions, Wall groups
57N13 Topology of the Euclidean \(4\)-space, \(4\)-manifolds (MSC2010)
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