Open AccessSymmetry, isotopy, and irregular covers
Author(s) -
Rebecca R. Winarski
Publication year - 2014
Publication title -
geometriae dedicata
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.746
H-Index - 43
eISSN - 1572-9168
pISSN - 0046-5755
DOI - 10.1007/s10711-014-9986-y
Subject(s) - mathematics , isotopy , mapping class group , covering space , lift (data mining) , mathematical proof , group (periodic table) , pure mathematics , modulo , algebraic geometry , differential geometry , polyhedron , property (philosophy) , teichmüller space , class (philosophy) , space (punctuation) , simple (philosophy) , topology (electrical circuits) , combinatorics , surface (topology) , geometry , computer science , riemann surface , philosophy , chemistry , organic chemistry , epistemology , artificial intelligence , data mining , operating system
We say that a covering space of surfaces has the Birman–Hilden property if the subgroup of the mapping class group of consisting of mapping classes that have representatives that lift to embeds in the mapping class group of modulo the group of deck transformations. We identify one necessary condition and one sufficient condition for when a covering space has this property. We give new explicit examples of irregular branched covering spaces that do not satisfy the necessary condition as well as explicit covers that satisfy the sufficient condition. Our criteria are conditions on simple closed curves, and our proofs use the combinatorial topology of curves on surfaces.
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