Decomposing four-manifolds up to homotopy type

Let M be a closed connected oriented topological 4-manifold with fundamental group π1. Let Λ be the integral group ring of π1. Suppose that f : M → P is a degree one map inducing an isomorphism on π1. We give a homological condition on the intersection forms λM and λ Λ M under which M is homotopy equivalent to a connected sum P#M ′ for some simply-connected closed (non-trivial) topological 4-manifoldM . This gives a partial solution to a conjecture of Hillman [16] 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. MSC 2000: 57N65, 57R67, 57Q10

[1]  A. Cavicchioli,et al.  On the homotopy classification of 4-manifolds having the fundamental group of an aspherical 4-manifold , 2000 .

[2]  A. Kawauchi.  TORSION LINKING FORMS ON SURFACE-KNOTS AND EXACT 4-MANIFOLDS , 2000 .

[3]  P. Teichner,et al.  A non-extended hermitian form over ℤ[ℤ] , 1997 .

[4]  J. Hillman.  Free Products and 4‐Dimensional Connected Sums , 1995 .

[5]  A. Kawauchi.  Splitting a 4-manifold with infinite cyclic fundamental group , 1994 .

[6]  J. Hillman.  The Algebraic Characterization of Geometric 4-Manifolds , 1994 .

[7]  R. Stong.  Uniqueness of connected sum decompositions in dimension 4 , 1994 .

[8]  R. Stong.  Uniqueness of π1-negligible embeddings in 4-manifolds: A correction to theorem 10.5 of Freedman and Quinn , 1993 .

[9]  J. Hillman.  On 4-manifolds homotopy equivalent to surface bundles over surfaces , 1991 .

[10]  Michael H. Freedman,et al.  Topology of 4-Manifolds (PMS-39), Volume 39 , 1990 .

[11]  M. Kreck,et al.  On the classification of topological 4-manifolds with finite fundamental group , 1988 .

[12]  J. Whitehead.  A Certain Exact Sequence , 1950 .

[13]  P. Teichner,et al.  A NON-EXTENDED HERMITIAN FORM OVER Z[Z] , 2005 .

[14]  Dušan D. Repovš,et al.  Four-Manifolds With Surface Fundamental Groups , 1997 .

[15]  A. Cavicchioli,et al.  A splitting theorem for homotopy equivalent smooth 4-manifolds , 1997 .

[16]  T. Matumoto,et al.  On 4-dimensional closed manifolds with free fundamental groups , 1995 .

[17]  R. Stong.  A structure theorem and a splitting theorem for simply-connected smooth 4-manifolds , 1995 .

[18]  A. Cavicchioli,et al.  On 4-manifolds with free fundamental group , 1994 .

[19]  A. Cavicchioli,et al.  ON THE INTERSECTION FORMS OF CLOSED 4-MANIFOLDS , 1992 .

[20]  N. Habegger,et al.  On the homotopy theory of simply connected four manifolds , 1990 .

[21]  M. Freedman.  Poincaré transversality and four-dimensional surgery , 1988 .

[22]  S. Bauer.  The homotopy type of a 4-manifold with finite fundamental group , 1988 .

[23]  Michael H. Freedman,et al.  The topology of four-dimensional manifolds , 1982 .

[24]  M. Freedman,et al.  Λ-splitting 4-manifolds , 1977 .

[25]  Charles Terence Clegg Wall,et al.  Surgery on compact manifolds , 1970 .