Configuration spaces of squares in a rectangle

Plachta, Leonid

doi:10.2140/agt.2021.21.1445

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Configuration spaces of squares in a rectangle

Leonid Plachta

Algebraic & Geometric Topology 21 (2021) 1445–1478

DOI: 10.2140/agt.2021.21.1445

Abstract

The configuration space $F_{k} (Q, r)$ of $k$ squares of size $r$ in a rectangle $Q$ is studied with the help of the tautological function $𝜃$ defined on the affine polytope complex $Q^{k}$ . The critical points of the function $𝜃$ are described in geometric and combinatorial terms. We also show that under certain conditions, the space $F_{k} (Q, r)$ is connected.

Keywords

configuration space of squares, affine polytope complex, affine Morse-Bott function, tautological function, critical point, saturated graph, deformation retraction

Mathematical Subject Classification 2010

Primary: 57Q99, 57R25, 51M20

References

Publication

Received: 9 May 2019

Revised: 28 May 2020

Accepted: 6 July 2020

Published: 11 August 2021

Authors

Leonid Plachta
AGH University of Science and Technology Kraków Poland

Volume 21, issue 3 (2021)