6533b861fe1ef96bd12c5838

RESEARCH PRODUCT

Generalized Dehn twists in low-dimensional topology

Yusuke KunoGwénaël MassuyeauShunsuke Tsuji

subject

Mathematics - Geometric TopologyMathematics::Group Theory[MATH.MATH-GT]Mathematics [math]/Geometric Topology [math.GT]FOS: MathematicsGeometric Topology (math.GT)57M27 20F34 20F14Mathematics::Symplectic GeometryMathematics::Geometric Topology[MATH.MATH-GT] Mathematics [math]/Geometric Topology [math.GT]

description

The generalized Dehn twist along a closed curve in an oriented surface is an algebraic construction which involves intersections of loops in the surface. It is defined as an automorphism of the Malcev completion of the fundamental group of the surface. As the name suggests, for the case where the curve has no self-intersection, it is induced from the usual Dehn twist along the curve. In this expository article, after explaining their definition, we review several results about generalized Dehn twists such as their realizability as diffeomorphisms of the surface, their diagrammatic description in terms of decorated trees and the Hopf-algebraic framework underlying their construction. Going to the dimension three, we also overview the relation between generalized Dehn twists and $3$-dimensional homology cobordisms, and we survey the variants of generalized Dehn twists for skein algebras of the surface.

yearjournalcountryeditionlanguage
2021-07-01
https://dx.doi.org/10.48550/arxiv.1909.09496