Configuration spaces of squares in a rectangle
The author studies the configuration space \(F_k(Q;r)\) of \(k\) squares of size \(r\) in a rectangle \(Q\), by using the tautological function \(\theta\) defined on the affine polytope complex \(Q^k\). The critical points of the function \(\theta\) are described in geometric and combinatorial terms. It is also proved that under certain conditions, the space \(F_k(Q;r)\) is connected. The paper is organized into three sections dealing with the following aspects: Introduction, affine Morse-Bott functions on affine polytope complexes, configuration spaces of squares in a rectangle (first surgery of \(F_k(Q;r)\), connectivity of \(F_n(Q;r)\)).
scientific article; zbMATH DE number 7394052