Classifying seven dimensional manifolds of fixed cohomology type (Q344187)

This paper considers a special class of seven dimensional manifolds. These manifolds have cohomology ring that is \(\mathbb Z\) in dimensions 0, 2, 5, and 7, \(0\) in dimensions 1, 3, and 6, and \({\mathbb Z}_{r}\) in dimension 4. If \(u\) is a generator of \(H^{2}\) then \(u^{2}\) is a generator of \(H^{4}\). Such manifolds are said to be of type \(r\). The authors give a classification of manifolds of type \(r\) that is an alternative to the one given by \textit{M. Kreck} and \textit{S. Stolz} [J. Differ. Geom. 33, No. 2, 465--486 (1991; Zbl 0733.53025)]. The classification is divided into the spin and non spin cases. The classification makes use of a self-linking number defined based on the element \(u^{2}\) and the \(s\)-invariants of Kreck and Stolz.