Isospectral Hantzsche-Wendt manifolds

Miatello, Roberto J.; Rossetti, Juan Pablo

doi:10.1515/crll.1999.077

Cited by 28 publications

(54 citation statements)

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“…Indeed, one computes that the Betti numbers are: 1, 3, 18, 46, 60, 60, 46, 18, 3, 1 for M and 1, 6, 18, 38, 60, 66, 46, 18, 3, , which are rational homology spheres. We will recall some basic facts from [MR1]. Let n be odd.…”

Section: If and Only Ifmentioning

confidence: 99%

Flat manifolds isospectral on p-forms

Miatello

Rossetti

2001

J Geom Anal

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Abstract. We study isospectrality on p-forms of compact flat manifolds by using the equivariant spectrum of the Hodge-Laplacian on the torus. We give an explicit formula for the multiplicity of eigenvalues and a criterion for isospectrality. We construct a variety of new isospectral pairs, some of which are the first such examples in the context of compact Riemannian manifolds. For instance, we give pairs of flat manifolds of dimension n = 2p, p ≥ 2, not homeomorphic to each other, which are isospectral on p-forms but not on q-forms for q = p, 0 ≤ q ≤ n. Also, we give manifolds isospectral on p-forms if and only if p is odd, one of them orientable and the other not, and a pair of 0-isospectral flat manifolds, one of them Kähler, and the other not admitting any Kähler structure. We also construct pairs, M, M ′ of dimension n ≥ 6, which are isospectral on functions and such that β p (M ) < β p (M ′ ), for 0 < p < n and pairs isospectral on p-forms for every p odd, and having different holonomy groups, Z 4 and Z 2 2 respectively.

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Section: If and Only Ifmentioning

confidence: 99%

Flat manifolds isospectral on p-forms

Miatello

Rossetti

2001

J Geom Anal

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“…In the oriented case, when G is contained in SOðnÞ y R n , which is only possible for n odd, they are rational homology spheres (cf. [11]), and there are pairs of such manifolds that are isospectral and but are not homeomorphic (see [7]). It is also known that for all oriented GHW manifolds, the first homology group is exactly the holonomy group (cf.…”

Section: Introductionmentioning

confidence: 99%

“…[2]). The number of GHW manifolds grows exponentially with the dimension (see [7]). Recently, Dekimpe and Petrosyan [3] determined the homology of some oriented GHW groups in low dimension.…”

Section: Introductionmentioning

confidence: 99%

Properties of generalized Hantzsche–Wendt groups

Szczepański¹

2009

Journal of Group Theory

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“…Though here they are considered for first time together, these manifolds have been studied partially in [12], [13], [14], [15], [17], mainly in the oriented case, where we shall call them Hantzsche-Wendt manifolds (HW manifolds for short; respectively HW (Bieberbach) groups, if we consider their fundamental groups, which are Bieberbach groups). They have many interesting properties, for instance, there are pairs of isospectral HW manifolds non homeomorphic to each other (cf.…”

Section: Introductionmentioning

confidence: 99%

“…They have many interesting properties, for instance, there are pairs of isospectral HW manifolds non homeomorphic to each other (cf. [14]). Moreover, HW manifolds have the Q-homology of spheres (cf.…”

Section: Introductionmentioning

confidence: 99%