## 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.

### MSC:

 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)

### Keywords:

four-manifolds; homotopy type; decomposition; intersection form
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