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.

arXiv comment25 pages, to appear in Revista Matemática Iberoamericana

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