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 Izv. RAN. Ser. Mat., 2018, Volume 82, Issue 5, Pages 3–22 (Mi izv8763)

Diagonal complexes

J. A. Gordonab, G. Yu. Paninacd

a National Research University Higher School of Economics, Moscow
b Chebyshev Laboratory, St. Petersburg State University, Department of Mathematics and Mechanics
c St. Petersburg Department of Steklov Mathematical Institute of Russian Academy of Sciences
d St. Petersburg State University, Mathematics and Mechanics Faculty

Abstract: It is known that the partially ordered set of all tuples of pairwise non-intersecting diagonals in an $n$-gon is isomorphic to the face lattice of a convex polytope called the associahedron. We replace the $n$-gon (viewed as a disc with $n$ marked points on the boundary) by an arbitrary oriented surface with a set of labelled marked points (‘vertices’). After appropriate definitions we arrive at a cell complex $\mathcal{D}$ (generalizing the associahedron) with the barycentric subdivision $\mathcal{BD}$. When the surface is closed, the complex $\mathcal{D}$ (as well as $\mathcal{BD}$) is homotopy equivalent to the space $RG_{g,n}^{\mathrm{met}}$ of metric ribbon graphs or, equivalently, to the decorated moduli space $\widetilde{\mathcal{M}}_{g,n}$. For bordered surfaces we prove the following. 1) Contraction of an edge does not change the homotopy type of the complex. 2) Contraction of a boundary component to a new marked point yields a forgetful map between two diagonal complexes which is homotopy equivalent to the Kontsevich tautological circle bundle. Thus we obtain a natural simplicial model for the tautological bundle. As an application, we compute the psi-class, that is, the first Chern class in combinatorial terms. This result is obtained by using a local combinatorial formula. 3) In the same way, contraction of several boundary components corresponds to the Whitney sum of tautological bundles.

Keywords: moduli space, ribbon graphs, curve complex, associahedron, Chern class.

 Funding Agency Grant Number Russian Science Foundation 16-11-10039 This work is supported by the Russian Science Foundation (grant no. 16-11-10039).

Author to whom correspondence should be addressed

DOI: https://doi.org/10.4213/im8763

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English version:
Izvestiya: Mathematics, 2018, 82:5, 861–879

Bibliographic databases:

UDC: 515.164.2
MSC: 52B70, 32G15
Revised: 14.03.2018

Citation: J. A. Gordon, G. Yu. Panina, “Diagonal complexes”, Izv. RAN. Ser. Mat., 82:5 (2018), 3–22; Izv. Math., 82:5 (2018), 861–879

Citation in format AMSBIB
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