Systolic freedom

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A metric phenomenon in differential geometry, systolic freedom was first detected by Mikhail Gromov in an I.H.E.S. preprint in 1992 (which eventually appeared as [6]), and was further developed by Michael Freedman and others. Gromov's observation was elaborated on by Marcel Berger in [1]. One of the first publications to study systolic freedom in detail is [7].

Systolic freedom refers to the fact that systolic invariants, or products thereof, in general provide no universal (i.e. curvature-free) lower bounds for the total volume of a closed Riemannian manifold. Systolic freedom has applications in quantum error correction. A survey of the main results on systolic freedom appears in [2].

[edit] Example

The complex projective plane admits Riemannian metrics of arbitrarily small volume, such that every essential surface is of area at least 1. Here a surface is called "essential" if it cannot be contracted to a point in the ambient 4-manifold.

[edit] References

• [1] Berger, Marcel: Systoles et applications selon Gromov. (French. French summary) [Systoles and their applications according to Gromov] Séminaire Bourbaki, Vol. 1992/93. Astérisque No. 216 (1993), Exp. No. 771, 5, 279–310.

• [2] Croke, Christopher B.; Katz, Mikhail: Universal volume bounds in Riemannian manifolds. Surveys in differential geometry, Vol. VIII (Boston, MA, 2002), 109–137, Surv. Differ. Geom., VIII, Int. Press, Somerville, MA, 2003.

• [3] Freedman, Michael H.: Z₂-systolic-freedom. Proceedings of the Kirbyfest (Berkeley, CA, 1998), 113–123 (electronic), Geom. Topol. Monogr., 2, Geom. Topol. Publ., Coventry, 1999.

• [4] Freedman, Michael H.; Meyer, David A.; Luo, Feng: Z₂-systolic freedom and quantum codes. Mathematics of quantum computation, 287–320, Comput. Math. Ser., Chapman & Hall/CRC, Boca Raton, FL, 2002.

• [5] Freedman, Michael H.; Meyer, David A.: Projective plane and planar quantum codes. Found. Comput. Math. 1 (2001), no. 3, 325–332.

• [6] Gromov, Mikhael: Systoles and intersystolic inequalities. (English, French summary) Actes de la Table Ronde de Géométrie Différentielle (Luminy, 1992), 291–362, Sémin. Congr., 1, Soc. Math. France, Paris, 1996.

• [7] Katz, Mikhail: Counterexamples to isosystolic inequalities. Geom. Dedicata 57 (1995), no. 2, 195–206.