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Spaces of bracelets and the Stasheff complexes. (Les espaces de bracelets et les complexes de Stasheff.) (French) Zbl 0737.51016
Sémin. Théor. Spectrale Géom., Chambéry-Grenoble 9, Année 1990-1991, 111-118 (1991).
Let \(r_1,\ldots,r_n\in S^1\) be distinct points in counterclockwise order on the unit circle, thought of as the boundary of the closure of the hyperbolic disk. A bracelet tangent to \(\{r_1,\ldots,r_n\}\) is a set of horocircles \(\{h_1,\ldots,h_n\}\) so that \(h_i\cap S^1=\{r_i\}\) and \(\text{card}(h_i\cap h_{i+1})=\text{card}(h_n\cap h_1)=1\). The union of the segments of the horocircles \(h_i\) between the points of intersection with \(h_{i+1},h_{i-1}\) gives a closed curve. The bracelet is proper if the closed curve does not intersect itself.
The main result of the paper is that the closure of the space of proper bracelets is stratified and that the resulting cell complex is combinatorially isomorphic and homeomorphic to the Stasheff “associehedron” used to study homotopy associativity [J. D. Stasheff, Trans. Am. Math. Soc. 108, 275–292, 293–312 (1963; Zbl 0114.39402)].
MSC:
51M10 Hyperbolic and elliptic geometries (general) and generalizations
55P45 \(H\)-spaces and duals
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