Diagonal complexes for surfaces of finite type and surfaces with involution

Abstract

Two constructions are studied that are inspired by the ideas of a recent paper by the authors. — The diagonal complex D \mathcal {D} and its barycentric subdivision B D \mathcal {BD} related to an oriented surface of finite type F F equipped with a number of labeled marked points. This time, unlike the paper mentioned above, boundary components without marked points are allowed, called holes. — The symmetric diagonal complex D inv \mathcal {D}^{\operatorname {inv}} and its barycentric subdivision B D inv \mathcal {BD}^{\operatorname {inv}} related to a symmetric (=with an involution) oriented surface F F equipped with a number of (symmetrically placed) labeled marked points. The symmetric complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.