Strongly isospectral manifolds with nonisomorphic cohomology rings

Abstract

For any n≥7n≥7, k≥3k≥3, we give pairs of compact flat nn-manifolds MM, M′M′ with holonomy groups Zk2Z2k, that are strongly isospectral, hence isospectral on pp-forms for all values of pp, having nonisomorphic cohomology rings. Moreover, if nn is even, MM is Kähler while M′M′ is not. Furthermore, with the help of a computer program we show the existence of large Sunada isospectral families; for instance, for n=24n=24 and k=3k=3 there is a family of eight compact flat manifolds (four of them Kähler) having very different cohomology rings. In particular, the cardinalities of the sets of primitive forms are different for all manifolds