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
–torus admits
a fibre whose homological size is bounded below by some universal constant depending
on
.
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.
Keywords
waist inequalities, space of cycles, filling inequalities,
cohomological complexity, tori, essential manifolds,
rational homotopy theory
