Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Volume 44 (2003), No. 1, 189-201.

Decomposing Four-Manifolds up to Homotopy Type Alberto Cavicchioli  Beatrice Ruini  Fulvia Spaggiari ^{∗} Dipartimento di Matematica, Università di Modena e Reggio Emilia Via Campi 213/B, 41100 Modena, Italy -mail: cavicchioli.alberto@unimo.it Abstract. Let M be a closed connected oriented topological 4-manifold with undamental group π_{1}. Let Λ be the integral group ring of π_{1}. Suppose that f : M → P is a degree one map inducing an isomorphism on π_{1}. We give a omological condition on the intersection forms λ_{M}^{ℤ} and λ_{M}^{Λ} under which M is omotopy 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 f Hillman [16] on the classification of closed 4-manifolds with vanishing second omotopy group. Then some splitting results for closed 4-manifolds with special omotopy complete the paper. SC 2000: 57N65, 57R67, 57Q10 eywords: four-manifolds, homotopy type, connected sum, obstruction theory, ntersection forms, homology with local coefficients, degree one map 1. Introduction Let M^{4} be a closed connected orientable topological 4-manifold with fundamental group

π_{1} = π_{1}(M). We assume that all manifolds have a CW-structure with one 4-cell and one 0-cell, hence the 3-skeleton M^{(3)} of M is obtained from M by removing an open 4-cell.

∗ Work performed under the auspices of the G.N.S.A.G.A. of the C.N.R. (National Research Council) of Italy, partially supported by the Ministero per la Ricerca Scientifica e Tecnologica of Italy within the project “Proprietà Geometriche delle Varietà Reali e Complesse”, and by a research grant of the University of Modena and Reggio Emilia.

0138-4821/93 $ 2.50 \begin{matrix}c \\ ⊗ \end{matrix} 2003 Heldermann Verlag