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In his work on singularities, expanders and topology of maps, Gromov showed,using isoperimetric inequalities in graded algebras, that every real valued mapon the $n$-torus admits a fibre whose homological size is bounded below by someuniversal constant depending on $n$. He obtained similar estimates for mapswith values in finite dimensional complexes, by a Lusternik--Schnirelmann typeargument.  We describe a new homological filling technique which enables us to derivesharp lower bounds in these theorems in certain situations. This partlyrealizes a programme envisaged by Gromov.  In contrast to previous approaches our methods imply similar lower bounds formaps defined on products of higher dimensional spheres.

Algebraic filling inequalities and cohomological width