# Strongly isospectral manifolds with nonisomorphic cohomology rings

### Emilio A. Lauret

Universidad Nacional de Córdoba, Argentina### Roberto J. Miatello

Universidad Nacional de Córdoba, Argentina### Juan P. Rossetti

Universidad Nacional de Córdoba, Argentina

## Abstract

For any $n\ge 7$, $k\ge 3$, we give pairs of compact flat $n$-manifolds $M$, $M'$ 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 $M'$ 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.

## Cite this article

Emilio A. Lauret, Roberto J. Miatello, Juan P. Rossetti, Strongly isospectral manifolds with nonisomorphic cohomology rings. Rev. Mat. Iberoam. 29 (2013), no. 2, pp. 611–634

DOI 10.4171/RMI/732