Mathematical Research Letters
Volume 5 (1998)
Volumes, middle-dimensional systoles, and Whitehead products
Pages: 461 – 471
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.