Strong intersystolic freedom of closed manifolds and of polyhedrons (Q1613979)

In the 70's M. Berger initiated the study of a new Riemannian invariant, called ``systole'', \(\text{sys}_{1}( g)\) defined as the minimum of the length of closed curves not homologous to zero. This definition has been generalized to many cases, for example to the \(k\)-systole, \(\text{sys}_{k}(g)\), defined via areas of nontrivial cycles represented by maps of \(k\)-dimensional manifolds into a Riemannian manifold \(( M,g) \). The important geometric meaning is played for a given \(n\)-dimensional Riemannian manifold by \[ \inf_{g}\frac{\text{vol}( g) }{\text{sys}_{k}( g) \cdot \text{sys}_{n-k}( g) }. \] If this limit is \(0\), the manifold is called systolically \(( k,n-k)\)-free. In his previous paper [Russ. Math. Surv. 55, 987-988 (2000; Zbl 1004.53031)] the author introduced the notions of homological systole \(\text{sys}_{k}\) and \(\text{stsys}_{k}\) via the volume of the \(k\)-dimensional class of homology with integer coefficients. The main goal is to prove that \[ \inf_{g}\frac{\text{vol}( g) }{\text{sys}_{k}( g) \cdot \text{sts}_{n-k}( g) }=0. \]