  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
realizes 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.
