Decomposing Four-Manifolds up to Homotopy Type

Alberto Cavicchioli, Beatrice Ruini and Fulvia Spaggiari

Abstract: Let $M$ be a closed connected oriented topological $4$-manifold with fundamental group $\pi_1$. Let $\Lambda$ be the integral group ring of $\pi_1$. Suppose that $f : M\to P$ is a degree one map inducing an isomorphism on $\pi_1$. We give a homological condition on the intersection forms $\lambda^{\Bbb Z}_M$ and $\lambda^{\Lambda}_M$ under which $M$ is homotopy equivalent to a connected sum $P\# M'$ for some simply-connected closed (non-trivial) topological $4$-manifold $M'$. This gives a partial solution to a conjecture of Hillman [H] on the classification of closed $4$-manifolds with vanishing second homotopy group. Then some splitting results for closed $4$-manifolds with special homotopy complete the paper.

[H] Hillman, J. A.: Free products and $4$-dimensional connected sums. Bull. London Math. Soc. {\bf 27} (1995), 387--391.

Keywords: four-manifolds, homotopy type, connected sum, obstruction theory, intersection forms, homology with local coefficients, degree one map

Classification (MSC2000): 57N65, 57R67, 57Q10

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