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Flat manifolds isospectral on \(p\)-forms. (English) Zbl 1040.58014

The main purpose of the present article is to exhibit examples of compact flat manifolds which are \(p\)-isospectral for some, but not all, values of \(p\). These examples comprise of (i) pairs of flat manifolds of dimension \(n=2p\), \(p\geq 2\), not homeomorphic to each other, isospectral on \(p\)-forms but not on \(q\)-forms for \(q\neq p\); (ii) pairs of manifolds isospectral on \(p\)-forms for every \(p\) odd, one of them orientable and the other not respectively one of them with holonomy group \(Z_4\) and the other with \(Z_2\); (iii) a pair of \(0\)-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure; (iv) pairs \(M,M'\) of manifolds of dimension \(n\geq 6\) isospectral on functions and such that the Betti numbers \(\beta_p (M) < \beta_p (M')\) for \(0<p<n\).

MSC:

58J53 Isospectrality
20H15 Other geometric groups, including crystallographic groups
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