Volumes, middle-dimensional systoles, and Whitehead products

Ivan K. Babenko, Mikhail G. Katz, Alexander I. Suciu
1998 Mathematical Research Letters  
Let X be a closed, orientable, smooth manifold of dimension 2m ≥ 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.
doi:10.4310/mrl.1998.v5.n4.a4 fatcat:etsbvdf55rcxhngnfy2nsdqp7q