#### Volume 21, issue 3 (2021)

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Configuration spaces of squares in a rectangle

### Leonid Plachta

Algebraic & Geometric Topology 21 (2021) 1445–1478
##### Abstract

The configuration space ${F}_{k}\left(Q,r\right)$ 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 ${Q}^{k}\phantom{\rule{-0.17em}{0ex}}$. The critical points of the function $𝜃$ are described in geometric and combinatorial terms. We also show that under certain conditions, the space ${F}_{k}\left(Q,r\right)$ 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