The authors prove that if \(M\) is a closed connected orientable topological \(2n\)-manifold, \(n\geq 2\), with poly-surface fundamental group, and if \(M\) is simple homotopy equivalent to the total space \(E\) of an \(Y\)-bundle over a closed connected orientable manifold with negative Euler characteristic, where \(Y\) is a closed connected aspherical \((2n-2)\)-manifold such that \(\widetilde K_0(\mathbb{Z} [\pi_1(Y)])=0\), then \(M\) is topologically \(s\)-cobordant to \(E\). This extends to higher dimensions the result of J. A. Hillman [Topology Appl. 40, No. 3, 275–286 (1991; Zbl 0754.57014)] obtained for 4-manifolds. The proof given by the authors is different from the one provided by Hillman: the authors use Mayer-Vietoris techniques and the properties of the \(\mathbb{L}\)-theory assembly maps for such bundles.