Classification of closed oriented 4-manifolds modulo connected sum with simply connected manifolds
Two closed oriented 4-manifolds \(M\) and \(N\) are said to be stably equivalent if there are simply connected closed 4-manifolds \(M_0\) and \(N_0\) such that the connected sums \(M\# M_0\) and \(N\# N_0\) are orientation preserving homeomorphic. The authors classify the stable equivalence classes of connected oriented 4-manifolds \(M\) with finite presentable fundamental group \(\pi\) showing that they one-to-one correspond to the elements of the quotient \(H_4(B\pi;\mathbb{Z})/\Aut\pi\). The correspondence is given by \(M\to f_*[M]\), where \([M]\) denotes the fundamental class of \(M\), and \(f:M\to B\pi=K (\pi,1)\) is the classifying map. Applications are given for classes of closed 4-manifolds with special fundamental groups.
scientific article; zbMATH DE number 1216681