Volumes, middle-dimensional systoles, and Whitehead products

Abstract

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

arXiv commentLaTeX2e, 11 pages. Revised version, to appear in Math. Research Letters