It is known that taken together, all collections of nonintersecting diagonals in a convex planar n-gon give rise to a (combinatorial type of a) convex (n − 3)-dimensional polytope As n called the Stasheff polytope, or associahedron. In the paper, we act in a similar way by taking a convex planar n-gon with k labeled punctures. All collections of mutually nonintersecting and mutually nonhomotopic topological diagonals yield a complex As n,k . We prove that it is a topological ball. We also show a natural cellular fibration Asn,k → As n,k−1 . A special example is delivered by the case k = 1. Here the vertices of the complex are labeled by all possible permutations together with all possible bracketings on n distinct entries. This hints to a relationship with M. Kapranov’s permutoassociahedron.
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References
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J. Stasheff, “Homotopy associativity of H-spaces. I, II,” Trans. Amer. Math. Soc., 108, 293–312 (1963).
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Published in Zapiski Nauchnykh Seminarov POMI, Vol. 448, 2016, pp. 246–251.
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Panina, G. Diagonal Complexes for Punctured Polygons. J Math Sci 224, 335–338 (2017). https://doi.org/10.1007/s10958-017-3418-0
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DOI: https://doi.org/10.1007/s10958-017-3418-0