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The Salvetti complex and the little cubes

Dai Tamaki (2012)

Journal of the European Mathematical Society

For a real central arrangement 𝒜 , Salvetti introduced a construction of a finite complex Sal ( 𝒜 ) which is homotopy equivalent to the complement of the complexified arrangement in [Sal87]. For the braid arrangement 𝒜 k - 1 , the Salvetti complex Sal ( 𝒜 k - 1 ) serves as a good combinatorial model for the homotopy type of the configuration space F ( ℂ , k ) of k points in C , which is homotopy equivalent to the space C 2 ( k ) of k little 2 -cubes. Motivated by the importance of little cubes in homotopy theory, especially in the study of...

Tiling and spectral properties of near-cubic domains

Mihail N. Kolountzakis, Izabella Łaba (2004)

Studia Mathematica

We prove that if a measurable domain tiles ℝ or ℝ² by translations, and if it is "close enough" to a line segment or a square respectively, then it admits a lattice tiling. We also prove a similar result for spectral sets in dimension 1, and give an example showing that there is no analogue of the tiling result in dimensions 3 and higher.

Tilings associated with non-Pisot matrices

Maki Furukado, Shunji Ito, E. Arthur Robinson (2006)

Annales de l’institut Fourier

Suppose A ∈ G l d ( ℤ ) has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called “good star”, and has A * ≥ 0 , where A * is an oriented compound of A . A morphism θ of the free group on { 1 , 2 , ⋯ , d } is called a non-abelianization of A if it has structure matrix A . We show that there is a tiling substitution Θ whose “boundary substitution” θ = ∂ Θ is a non-abelianization of A . Such a tiling substitution Θ leads to a self-affine tiling of E u ∼ ℝ 2 with A u : = A | E u ∈ G L 2 ( ℝ ) as its expansion. In the last section we find conditions on A so...

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