TY - JOUR

T1 - Diagonal complexes

AU - Гордон, Иосиф Александрович

AU - Панина, Гаянэ Юрьевна

N1 - Funding Information: This work is supported by the Russian Science Foundation (grant no. 16-11-10039). AMS 2010 Mathematics Subject Classification. 52B70, 32G15.

PY - 2018/10

Y1 - 2018/10

N2 - 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 D (generalizing the associahedron) with the barycentric subdivision BD. When the surface is closed, the complex D (as well as BD) is homotopy equivalent to the space RG g,n met of metric ribbon graphs or, equivalently, to the decorated moduli space fM 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 psiclass, 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.

AB - 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 D (generalizing the associahedron) with the barycentric subdivision BD. When the surface is closed, the complex D (as well as BD) is homotopy equivalent to the space RG g,n met of metric ribbon graphs or, equivalently, to the decorated moduli space fM 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 psiclass, 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.

KW - пространство модулей, расслоение Концевича, первый класс Черна

KW - Chern class

KW - associahedron

KW - curve complex

KW - moduli space

KW - ribbon graphs

KW - CURVES

UR - http://www.scopus.com/inward/record.url?scp=85056373024&partnerID=8YFLogxK

UR - http://arxiv.org/abs/1701.01603

UR - http://www.mendeley.com/research/diagonal-complexes

U2 - 10.1070/IM8763

DO - 10.1070/IM8763

M3 - Article

VL - 82

SP - 861

EP - 879

JO - Izvestiya Mathematics

JF - Izvestiya Mathematics

SN - 1064-5632

IS - 5

ER -