If a closed smooth n-manifold M admits a finite cover whose Z/2Z-cohomology
has the maximal cup-length, then for any riemannian metric g on M, we show that
the systole Sys(M,g) and the volume Vol(M,g) of the riemannian manifold (M,g)
are related by the following isosystolic inequality: Sys(M,g)^n \leq n!
Vol(M,g). The inequality can be regarded as a generalization of Burago and
Hebda's inequality for closed essential surfaces and as a refinement of Guth's
inequality for closed n-manifolds whose Z/2Z-cohomology has the maximal
cup-length. We also establish the same inequality in the context of possibly
non-compact manifolds under a similar cohomological condition. The inequality
applies to (i) T^n and all other compact euclidean space forms, (ii) RP^n and
many other spherical space forms including the Poincar\'e dodecahedral space,
and (iii) most closed essential 3-manifolds including all closed aspherical
3-manifolds.Comment: 34 pages, 0 figures. v2 contains expository revisions and some
additional reference
