The generalized Dehn twist along a figure eight

2011/04/12 by Yusuke Kuno, Kuno, Yusuke · 1 citation
Mathematics · #Advanced Algebra and Geometry #Algebraic structures and combinatorial models #Geometric and Algebraic Topology #math.AT #math.GR #math.GT #msc:20F34 #msc:57N05

paper · pdf · doi:10.48550/arxiv.1104.2107

22 pages, 6 figures, 2 tables; minor corrections and modifications have been done in v2

arxiv created 2011/05/04 · arxiv updated 2011/05/05

Abstract

For any unoriented loop on a compact connected oriented surface with one boundary component, the generalized Dehn twist along the loop is defined as an automorphism of the completed group ring of the fundamental group of the surface. If the loop is simple, this is the usual right handed Dehn twist, in particular realized as a mapping class of the surface. We investigate the case when the loop has a single transverse double point, and show that in this case the generalized Dehn twist is not realized as a mapping class.

Cited by

• Intersections of curves on surfaces and their applications to mapping class groups

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• The logarithms of Dehn twists
• The Dehn twist coefficient for big and small mapping class groups
• A finite presentation of the mapping class group of an oriented surface
• A Presentation of the Mapping Class Groups
• An algebraic characterization of a Dehn twist for nonorientable surfaces