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. 
Mikhail G. Katz, Alexander I. Suciu