Algebraic filling inequalities and cohomological width

Abstract

In his work on singularities, expanders and topology of maps, Gromov showed, using isoperimetric inequalities in graded algebras, that every real-valued map on the $n$–torus admits a fibre whose homological size is bounded below by some universal constant depending on $n$. He obtained similar estimates for maps with values in finite-dimensional complexes, by a Lusternik–Schnirelmann-type argument.

We describe a new homological filling technique which enables us to derive sharp lower bounds in these theorems in certain situations. This partly realises a programme envisaged by Gromov.

In contrast to previous approaches, our methods imply similar lower bounds for maps defined on products of higher-dimensional spheres.