A non-extended Hermitian form over \(\mathbb{Z}[\mathbb{Z}]\) (Q1365302)

The authors describe a nonsingular Hermitian form of rank 4 over the group ring \(\mathbb{Z}[\mathbb{Z}]\) which is not extended from the integers. They also show that under certain indefiniteness assumptions, every nonsingular Hermitian form on a free \(\mathbb{Z}[\mathbb{Z}]\)-module is extended from the integers. Therefore there exists a closed oriented 4-dimensional manifold with fundamental group \(\mathbb{Z}\) which is not the connected sum of a simply connected 4-manifold and \(S^1\times S^3\).