Strong intersystolic softness of closed manifolds
Let \((M,g)\) be a Riemannian manifold of dimension \(m\). For an integral homology class \(a\in H_p(M,\mathbb{Z})\) we can define the volume \(\text{vol}_p(a,g)\). Now we define \(p\)-systole modulo torsion for \((M,g)\) by \(\text{sys}_p(M,g)= \inf\text{vol}_p(a,g)\), where \(a\) is \(\neq 0\) mod torsion and runs through \(H_p(M,\mathbb{Z})\).NEWLINENEWLINENEWLINEA closed \(m\)-dimensional manifold \(M\) is said to be \((p, m-p)\)-soft \((1\leq p\leq m/2)\) if NEWLINE\[NEWLINE\inf\{\text{vol}_m(M,g)/[\text{sys}_p (M,g) \text{sys}_{m-p} (M,g)]\}=0,NEWLINE\]NEWLINE where \(g\) is taken over all Riemannian metrics \(g\) on \(M\). The author proves that any closed \(m\)-dimensional manifold \(M\) is \((p,m-p)\)-soft for \(1\leq p\leq m/2\) by showing an equality which implies the condition of the \((p,m-p)\)-softness.
scientific article; zbMATH DE number 1729030