The logarithms of Dehn twists on non-orientable surfaces

TL;DR

This paper introduces a Lie algebra for non-orientable surfaces and derives an explicit formula for Dehn twists along annulus curves, extending concepts from oriented surface theory.

Contribution

It develops a new Lie algebra framework for non-orientable surfaces and provides explicit formulas for Dehn twists, advancing the understanding of surface mapping class groups.

Findings

Defined a Lie algebra for non-orientable surfaces

Derived explicit Dehn twist formulas for annulus curves

Extended Goldman Lie algebra concepts to non-orientable cases

Abstract

We introduce a Lie algebra associated with a non-orientable surface, which is an analogue for the Goldman Lie algebra of an oriented surface. As an application, we deduce an explicit formula of the Dehn twist along an annulus simple closed curve on the surface as in Kawazumi-Kuno and Masseyeau-Turaev.