#### Volume 19, issue 6 (2019)

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Algebraic filling inequalities and cohomological width

### Meru Alagalingam

Algebraic & Geometric Topology 19 (2019) 2855–2898
##### 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.

##### Keywords
waist inequalities, space of cycles, filling inequalities, cohomological complexity, tori, essential manifolds, rational homotopy theory
##### Mathematical Subject Classification 2010
Primary: 55N05
Secondary: 55P62, 55S35