VOLUMES, MIDDLE-DIMENSIONAL SYSTOLES, AND WHITEHEAD PRODUCTS

Let X be a closed, orientable, smooth manifold of dimension 2m greater than or equal to 6 with torsion-free middle-dimensional homology. We construct metrics on X of arbitrarily small volume, such that every or ientable, middle-dimensional submanifold of less than unit volume nece ssarily bounds. Thus, Loewner's theorem has no higher-dimensional anal ogue.