The systole Sys(M,g) of a Riemannian manifold (M,g) is the length of the shortest geodesic loop. An isosystolic inequality for a smooth n-manifold M states that the 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 a metric g on M. Gromov has shown that, if M is an essential closed manifold, then isosystolic inequality holds for M with some C.

We show that, for any closed smooth n-manifold M, satisfying certain homological/cohomological 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 readily applies to all euclidean space forms (e.g. n-tori) and many spherical space forms (e.g. real projective n-spaces). We will also show that all closed aspherical 3-manifolds satisfy our isosystolic inequality.