Small separators, upper bounds for $l^\infty$-widths, and systolic geometry

Abstract

We investigate the dependence on the dimension in the inequalities that relate the Euclidean volume of a closed submanifold $M^n\subset \mathbb{R}^N$ with its $l^\infty$-width $W^{l^\infty}_{n-1}(M^n)$ defined as the infimum over all continuous maps $ϕ:M^n\longrightarrow K^{n-1}\subset\mathbb{R}^N$ of $sup_{x\in M^n}\Vert ϕ(x)-x\Vert_{l^\infty}$. We prove that $W^{l^\infty}_{n-1}(M^n)\leq const\ \sqrt{n}\ vol(M^n)^{\frac{1}{n}}$, and if the codimension $N-n$ is equal to $1$, then $W^{l^\infty}_{n-1}(M^n)\leq \sqrt{3}\ vol(M^n)^{\frac{1}{n}}$. As a corollary, we prove that if $M^n\subset \mathbb{R}^N$ is {\it essential}, then there exists a non-contractible closed curve on $M^n$ contained in a cube in $\mathbb{R}^N$ with side length $const\ \sqrt{n}\ vol^{\frac{1}{n}}(M^n)$ with sides parallel to the coordinate axes. If the codimension is $1$, then the side length of the cube is $4\ vol^{\frac{1}{n}}(M^n)$. To prove these results we introduce a new approach to systolic geometry that can be described as a non-linear version of the classical Federer-Fleming argument, where we push out from a specially constructed non-linear $(N-n)$-dimensional complex in $\mathbb{R}^N$ that does not intersect $M^n$. To construct these complexes we first prove a version of kinematic formula where one averages over isometries of $l^N_\infty$ (Theorem 3.5), and introduce high-codimension analogs of optimal foams recently discovered in [KORW] and [AK].