Algebraic filling inequalities and cohomological width

Alagalingam, Meru

doi:10.2140/agt.2019.19.2855

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Algebraic filling inequalities and cohomological width

Meru Alagalingam

Algebraic & Geometric Topology 19 (2019) 2855–2898

DOI: 10.2140/agt.2019.19.2855

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 $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ –torus admits a fibre whose homological size is bounded below by some universal constant depending on $<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><mi>n</mi></math>$ . 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

References

Publication

Received: 25 October 2017

Revised: 16 February 2019

Accepted: 24 February 2019

Published: 20 October 2019

Authors

Meru Alagalingam
Universität Augsburg Augsburg Germany

Volume 19, issue 6 (2019)