# Algebraic filling inequalities and cohomological width

## TL;DR

This paper introduces a new homological filling technique that provides sharp lower bounds for the homological size of fibers in maps from tori and products of spheres, advancing Gromov's program.

## Contribution

It develops a novel algebraic filling method that extends Gromov's inequalities to higher-dimensional sphere products and improves lower bound estimates.

## Key findings

- Derived sharp lower bounds for homological sizes of fibers.
- Extended Gromov's inequalities to products of higher-dimensional spheres.
- Introduced a new homological filling technique for topological maps.

## 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 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.