Homotopy Spheres in Formal Language
Frank D. Farmer · Studies in Applied Mathematics · 1982
The set of nonempty proper subwords of a word is either contractible or a homotopy n‐sphere. There is a simple algorithm which computes n. The existence of spherical words is investigated, and the words which yield spheres are determined. A language which is closed under subwords has a finite number of components, and each component has a finitely generated fundamental group. For each n greater than 1, there is a language on two letters which has the homotopy type of an infinite cluster of n‐sphere. There is a language on two letters which has nontrivial homology in each dimension greater than 1. If a language is closed under subwords and has bounded period, then it has the homotopy type of a finite polyhedron.