Linear bounds for constants in Gromov’s systolic inequality and related results

Let $M^n$ be a closed Riemannian manifold. Larry Guth proved that there exists $c(n)$ with the following property: if for some $r>0$ the volume of each metric ball of radius $r$ is less than $({r\over c(n)})^n$, then there exists a continuous map from $M^n$ to a $(n-1)$-dimensional simplicial complex such that the inverse image of each point can be covered by a metric ball of radius $r$ in $M^n$. It was previously proven by Misha Gromov that this result would imply two famous Gromov's inequalities: $Fill Rad(M^n)\leq c(n)vol(M^n)^{1\over n}$ and, if $M^n$ is essential, then also $sys_1(M^n)\leq 6c(n)vol(M^n)^{1\over n}$ with the same constant $c(n)$. Here $sys_1(M^n)$ denotes the length of a shortest non-contractible curve in $M^n$. Here we prove that these results hold with $c(n)=n$. All previously known upper bounds for $c(n)$ were exponential in $n$. The proof uses ideas of Guth from [Gu 10] and of Panos Papasoglu from his recent work [P].