Diagonal complexes for surfaces of finite type and surfaces with involution

Panina, G.; Gordon, J.

doi:10.1090/spmj/1709

Diagonal complexes for surfaces of finite type and surfaces with involution
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by G. Panina and J. Gordon

St. Petersburg Math. J. 33 (2022), 465-481

DOI: https://doi.org/10.1090/spmj/1709

Published electronically: May 5, 2022

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Abstract:

Two constructions are studied that are inspired by the ideas of a recent paper by the authors.

— The diagonal complex $\mathcal {D}$ and its barycentric subdivision $\mathcal {BD}$ related to an oriented surface of finite type $F$ 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 $\mathcal {D}^{\operatorname {inv}}$ and its barycentric subdivision $\mathcal {BD}^{\operatorname {inv}}$ related to a symmetric (=with an involution) oriented surface $F$ 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.

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St. Petersburg Mathematical Journal

Diagonal complexes for surfaces of finite type and surfaces with involution
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Abstract: