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\(\mathbb{Z}_2\)-systolic-freedom. (English) Zbl 1024.53029

Hass, Joel (ed.) et al., Proceedings of the Kirbyfest, Berkeley, CA, USA, June 22-26, 1998. Warwick: University of Warwick, Institute of Mathematics, Geom. Topol. Monogr. 2, 113-123 (1999).
By a result of M. Gromov, stable systolic rigidity holds for any Riemannian metric on products of spheres \(S^p\times S^q\). The author investigates \({\mathbb Z}_2-\)systoles in dimensions bigger than or equal to three which are of relevance in quantum information theory. He shows that the conjectured systolic rigidity over \({\mathbb Z}_2\) does not hold. In fact, he exhibits a family of Riemannian metrics on \(S^2\times S^1\) showing up \({\mathbb Z}_2\)-\((2,1)\)-systolic freedom and leading to more general \( {\mathbb Z}_2\)-freedoms.
For the entire collection see [Zbl 0939.00055].

MSC:

53C23 Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces
53C22 Geodesics in global differential geometry
81P68 Quantum computation
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