Classifying seven dimensional manifolds of fixed cohomology type

Abstract

We study seven dimensional manifolds of fixed cohomology type with integer coefficients: H0≅H2≅H5≅H7≅Z, H4≅Zr, H1=H3=H6=0, simply called manifolds of type r, where Zr, is a cyclic group of order r generated by the square of the generator of H2. Such manifolds include the Eschenburg spaces, the Witten manifolds and the generalized Witten manifolds. Most spaces from these three families admit a Riemannian metric of positive sectional curvature or an Einstein metric of positive Ricci curvature. In 1991 M. Kreck and S. Stolz introduced three invariants to classify manifolds M of type r up to homeomorphism and diffeomorphism. In this article, we show that for spin manifolds of type r we can replace two of the homeomorphism invariants by the first Pontrjagin class and the self-linking number of the manifolds. This replacement has been stated without proof and been used for calculations for the Eschenburg spaces, hence our result closes a gap in the literature.