Volumes, middle-dimensional systoles, and Whitehead products

Let $X$ be a closed, orientable, smooth manifold of dimension $2m\ge 6$ with torsion-free middle-dimensional homology. We construct metrics on $X$ of arbitrarily small volume, such that every orientable, middle-dimensional submanifold of less than unit volume necessarily bounds. Thus, Loewner’s theorem has no higher-dimensional analogue.

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Published 1 January 1998