Revista Matemática Iberoamericana


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Volume 29, Issue 2, 2013, pp. 611–634
DOI: 10.4171/RMI/732

Published online: 2013-04-22

Strongly isospectral manifolds with nonisomorphic cohomology rings

Emilio A. Lauret[1], Roberto J. Miatello[2] and Juan P. Rossetti[3]

(1) Universidad Nacional de Córdoba, Argentina
(2) Universidad Nacional de Córdoba, Argentina
(3) Universidad Nacional de Córdoba, Argentina

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.

Keywords: Isospectral, cohomology rings, primitive forms, flat manifolds

Lauret Emilio, Miatello Roberto, Rossetti Juan: Strongly isospectral manifolds with nonisomorphic cohomology rings. Rev. Mat. Iberoamericana 29 (2013), 611-634. doi: 10.4171/RMI/732