Volume of Riemannian manifolds , geometric inequalities , and homotopy theory

We outline the current state of knowledge regarding geometric inequalities of systolic type, and prove new results, including systolic freedom in dimension 4. Namely, every compact, orientable, smooth 4-manifold X admits metrics of arbitrarily small volume such that every orientable, immersed surface of smaller than unit area is necessarily null-homologous in X. In other words, orientable 4-manifolds are 2-systolically free. More generally, let m ~ 2 be an even integer, and let n > m. Then all manifolds of dimension n are m-systolically free (modulo torsion) if all k-skeleta, m + 1 :::; k :::; n, of the loop space n(sm+l) are m-systolically free.