Diagonal complexes for surfaces of finite type and surfaces with involution. (English) Zbl 1493.57018
St. Petersbg. Math. J. 33, No. 3, 465-481 (2022) and Algebra Anal. 33, No. 3, 51-72 (2021).
In this paper the authors present two constructions that are inspired from ideas that arose in their previous paper [Izv. Math. 82, No. 5, 861–879 (2018; Zbl 1406.52033); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 82, No. 5, 3–22 (2018)]. The constructions are:
(1) The diagonal complex \(\mathcal{D}\) and its barycentric subdivision \(\mathcal{BD}\) associated with an oriented surface \(F\) of finite type equipped with labeled marked points, and in which, unlike in the previous paper, boundary components without marked points are allowed. (In the paper mentioned, the fact that each boundary component contains at least one marked point was a requirement, and in the present paper, this condition is relaxed.)
(2) The symmetric diagonal complex \(\mathcal{D}^{inv}\) and its barycentric subdivision \(\mathcal{BD}^{inv}\) related to a symmetric oriented surface \(F\) (that is, a surface equipped with a distinguished involution) together with a certain number of (symmetrically placed) labeled marked points. The symmetric diagonal complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.
The paper is interesting and well written.
(1) The diagonal complex \(\mathcal{D}\) and its barycentric subdivision \(\mathcal{BD}\) associated with an oriented surface \(F\) of finite type equipped with labeled marked points, and in which, unlike in the previous paper, boundary components without marked points are allowed. (In the paper mentioned, the fact that each boundary component contains at least one marked point was a requirement, and in the present paper, this condition is relaxed.)
(2) The symmetric diagonal complex \(\mathcal{D}^{inv}\) and its barycentric subdivision \(\mathcal{BD}^{inv}\) related to a symmetric oriented surface \(F\) (that is, a surface equipped with a distinguished involution) together with a certain number of (symmetrically placed) labeled marked points. The symmetric diagonal complex is shown to be homotopy equivalent to the complex of a surface obtained by “taking a half” of the initial symmetric surface.
The paper is interesting and well written.
Reviewer: Athanase Papadopoulos (Strasbourg)
MSC:
| 57N60 | Cellularity in topological manifolds |
| 57Q70 | Discrete Morse theory and related ideas in manifold topology |
| 05E45 | Combinatorial aspects of simplicial complexes |
Keywords:
surface diffeomorphism; mapping class group; moduli space; ribbon graphs; curve complex; associahedronCitations:
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