Four-manifolds with \(\pi_1\)-free second homotopy
The author studies closed connected topological 4-manifolds \(M\) for which \(\pi_2(M)\) is a \(\Lambda\)-free module, where \(\Lambda\) is the integral group ring \({\mathbb Z}[\pi_1(M)]\). The \textit{quadratic 2-type} of \(M\) is defined as the quadruple of data \([\pi_1(M),\pi_2(M),\lambda_{M},k(M)]\), where \(\pi_2\cong H_2(M;\Lambda)\) is regarded as a \(\Lambda\)-module, \(\lambda_{M}\) is the intersection form of \(M\) (over \(\Lambda\)), and \(k(M)\in H^3(\pi_1;\pi_2)\) is the first \(k\)-invariant of \(M\). The main theorem states that if \(M\) satisfies (i.) \(\pi_1(M)\) is infinite and \(\pi_2\) is \(\Lambda\)-free (with both finitely generated), and (ii.)\(H_4(B\pi_1;{\mathbb Z}) \cong H_5(B\pi_1;{\mathbb Z})\cong 0\), then the homotopy type of \(M\) is determined by the isometry class of its quadratic 2-type. Examples of 4-manifolds for which the conditions of the theorem hold are those where \(\pi_1\) is free, or isomorphic to the fundamental group of a closed aspherical manifold of dimension less than or equal to four. One consequence of this paper's main theorem therefore is to give a different, more algebraic proof of the homotopy classification of closed orientable topological 4-manifolds with free fundamental group than that given originally in [\textit{A. Cavicchioli} and \textit{F. Hegenbarth}, Forum Math. 6, 415-429 (1994; Zbl 0822.57015)].
scientific article; zbMATH DE number 1965507