Strongly isospectral manifolds with nonisomorphic cohomology rings

Abstract

For any $n\ge 7$, $k\ge 3$, we give pairs of compact flat $n$-manifolds $M$, $M^{\prime }$ with holonomy groups ${\mathbb{Z}}_2^k$, that are strongly isospectral, hence isospectral on $p$-forms for all values of $p$, having nonisomorphic cohomology rings. Moreover, if $n$ is even, $M$ is Kähler while $M^{\prime }$ is not. Furthermore, with the help of a computer program we show the existence of large Sunada isospectral families; for instance, for $n=24$ and $k=3$ 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.