Mathematical Research Letters

Volume 5 (1998)

Number 4

Volumes, middle-dimensional systoles, and Whitehead products

Pages: 461 – 471



Ivan K. Babenko (Université de Montpellier 2)

Mikhail G. Katz (Université de Nancy 1)

Alexander I. Suciu (Northeastern University)


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.

Published 1 January 1998