# Mathematics Colloquia and Seminars

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### On isosystolic inequalities and Z/2Z homology

**Geometry/Topology**

Speaker: | Kei Nakamura, UC Davis |

Location: | 2112 MSB |

Start time: | Tue, Oct 8 2013, 4:10PM |

The systole Sys(M,g) of a complete Riemannian manifold (M,g) is the infimum of the length of non-contractible loops. An isosystolic inequality for a smooth n-manifold M states that the n-th power of Sys(M,g) is bounded above by the constant C times the volume of (M,g), where C is independent of metrics g on M. The celebrated result of Gromov says that, if M is an essential closed manifold, then an isosystolic inequality holds for M with some constant C. We show that, for any closed smooth n-manifold M that satisfy a certain (co)homological condition, an isosystolic inequality with constant C=n! holds. Our inequality can be regarded as a generalization of the inequality by Hebda and Burago, as well as a refinement of the inequality by Guth. The inequality applies to all compact euclidean space forms (e.g. n-tori), many spherical space forms (e.g. real projective n-spaces and Poincare dodecahedral space), and most closed essential 3-manifolds including all closed aspherical 3-manifolds.