Aspectos topológicos de las  simetrías en superficies: Topological aspects of symmetries on surfaces

Juan Armando Parra; Israel Morales

doi:10.56754/0719-0646.2702.411

DOI:

https://doi.org/10.56754/0719-0646.2702.411

Abstract

The homeomorphism group of a topological surface \(\Sigma\), Homeo(\(\Sigma\)), admits a topology known as the compact-open topology, with which it becomes a topological group. In this work, we provide a self-contained proof of this fact. Moreover, we use elementary tools to prove that Homeo(\(\Sigma\)) is a Polish group (i.e., it is separable and completely metrizable). We translate the isotopy relation in Homeo(\(\Sigma\)) as path-connectedness in Homeo(\(\Sigma\)) and, denoting by Homeo₀(\(\Sigma\)) the identity path component, we use classic results in Descriptive Set Theory to prove that the Extended Mapping Class Group of \(\Sigma\), Mod^±(\(\Sigma\)) := Homeo(\(\Sigma\))/ Homeo₀(\(\Sigma\)), is a Polish group with the quotient topology. At the end of this survey, we discuss an alternative proof of this result based on realizing the Extended Mapping Class Group as the automorphism group of the complex of curves; this connection arises as one of the most important and beautiful in the theory of Mapping Class Groups.

Resumen

El grupo de homeomorfismos de una superficie topológica \(\Sigma\), Homeo(\(\Sigma\)), admite una topología conocida como la topología compacto-abierta, con la cual es un grupo topológico. En este escrito damos una demostración autocontenida de este hecho. Del mismo modo, utilizamos herramientas elementales para demostrar que Homeo(\(\Sigma\)) es un grupo polaco (es decir, es separable y completamente metrizable). Traducimos la relación de isotopía en Homeo(\(\Sigma\)) como arcoconexidad en Homeo(\(\Sigma\)) y, denotando por Homeo₀(\(\Sigma\)) a la componente arcoconexa de la identidad, usamos resultados clásicos de la Teoría Descriptiva de Conjuntos para probar que el Grupo Modular Extendido de \(\Sigma\) (o mapping class group extendido), Mod^±(\(\Sigma\)) := Homeo(\(\Sigma\))/ Homeo₀(\(\Sigma\)), es un grupo polaco con la topología cociente. Al final de este compendio, discutimos una demostración alternativa de este resultado que se basa en ver al Grupo Modular Extendido como el grupo de automorfismo del grafo de curvas; esta conexión figura como una de las más importantes y bellas en toda la teoría de Grupos Modulares.

Keywords

Topological surfaces , homeomorphism group , mapping class group of surfaces , compact-open topology , isotopy , complex of curves

Mathematics Subject Classification:

57K20 , 57S05 , 54H11 , 20F65 , 03E15

Juan Armando Parra Centro de Investigación en Matemáticas, A.C. CIMAT. Guanajuato, México. https://orcid.org/0009-0003-6217-3193
Israel Morales Departamento de Matemática y Estadística, Universidad de La Frontera, Temuco, Chile. https://orcid.org/0000-0002-9645-9435

Pages: 411–459
Date Published: 2025-10-21
Vol. 27 No. 2 (2025): Spanish Edition (40th Anniversary)

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