Tile homotopy groups.
This beautiful and very interesting paper is firstly an exposition of the method of \textit{J. H. Conway} and \textit{J. C. Lagarias} [J. Comb. Theory, Ser. A 53, 183--208 (1990; Zbl 0741.05019)] to show the impossibility of certain tiling problems using groups of boundary words. The author shows how to use the method, which is in general not easy to apply, in many new cases. He is mainly concerned with tiling rectangles by a given set of polyomino prototiles. The author proves a theorem that enables one to show in certain cases that the tile homotopy group of a given set of prototiles is finitely generated, which helps in calculating these groups. He then presents some examples of such calculations, and uses them to show the impossibility of certain tilings. He shows that the tile homology group is not sufficient to obtain these results. The software package GAP is used to make the group theoretical calculations. The author also gives a sufficient condition for the tile homotopy group to be abelian, and uses this to calculate some examples of abelian homotopy groups (which are then isomorphic to the corresponding homology groups). In particular, he shows that tile homotopy cannot reproduce the theorem of \textit{D. W. Walkup} [Am. Math. Mon. 72, 986--988 (1965; Zbl 0132.00603)] that an \(m\times n\) rectangle can be tiled by T-tetrominoes only if \(m\) and \(n\) are both multiples of \(4\). In an appendix the author presents some more examples (with calculational details to appear elsewhere) of impossibility results given by the homotopy group which are not obtainable from the homology group.
scientific article; zbMATH DE number 2028613