Math 2.0

Manifolds and cell complexes Definition

Dehn twist

Builds on SurfaceOrientable surfaceHomeomorphismMapping class group

Definition

Let A=S1×[0,1]A = S^1 \times [0, 1] be the annulus, with S1=ℝ/2πℤS^1 = \R / 2\pi\Z and the orientation of the (θ,t)(\theta, t)-plane. The twist map T ⁣:A→AT \colon A \to A is T(θ,t)=(θ+2πt,t)T(\theta, t) = (\theta + 2\pi t, t); it is the identity on both boundary circles.

Let SS be an oriented surface and let cc be a simple closed curve in the interior of SS. Choose an orientation-preserving embedding φ ⁣:A→S\varphi \colon A \to S with φ(S1×{1/2})=c\varphi(S^1 \times \set{1/2}) = c. The Dehn twist about cc is the homeomorphism Tc ⁣:S→ST_c \colon S \to S given by Tc(x)={φ∘T∘φ−1(x)if x∈φ(A),xotherwise.T_c(x) = \begin{cases} \varphi \circ T \circ \varphi^{-1}(x) & \text{if } x \in \varphi(A), \\ x & \text{otherwise.} \end{cases}

Remarks

The isotopy class of TcT_c does not depend on the choice of φ\varphi. It depends only on the isotopy class of cc, and it is an element of the mapping class group of SS. Its definition does not depend on an orientation of cc, but it does depend on the orientation of SS: reversing it replaces TcT_c by its inverse.

Example: on the torus T2=ℝ2/ℤ2T^2 = \R^2 / \Z^2, let cc be the image of the horizontal line. The Dehn twist TcT_c acts on H1(T2;ℤ)=ℤ2H_1(T^2; \Z) = \Z^2 by the matrix (1101)\begin{pmatrix} 1 & 1 \\ 0 & 1 \end{pmatrix}, up to the sign convention. Since this matrix has infinite order, TcT_c has infinite order in the mapping class group.

Non-example: if cc bounds a disc in SS, then TcT_c is isotopic to the identity, so it is trivial in the mapping class group. And on a non-orientable surface, a curve whose neighbourhood is a Möbius strip has no annulus neighbourhood, so no Dehn twist about it is defined.

Reading path

To understand this page from the ground up, read these 7 pages in order. The foundations come first.

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Homeomorphism, concept, no evidence levelHomeomorphismconceptMapping class group, concept, no evidence levelMapping class groupconceptParametrised surface, concept, no evidence levelParametrised surfaceconceptSurface, concept, no evidence levelSurfaceconceptTangent plane and normal vector of a surface, concept, no evidence levelTangent plane and nor…conceptOrientable surface, concept, no evidence levelOrientable surfaceconceptDehn twist, definition, no evidence levelDehn twistdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Homeomorphism concept
  2. Mapping class group concept
  3. Parametrised surface concept
  4. Surface concept
  5. Tangent plane and normal vector of a surface concept
  6. Orientable surface concept
  7. Dehn twist definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Dehn_twist
About this page
TagXP21V9TVKK
URIm2://XP21V9TVKK
Kinddefinition
WritingW0 stub
MSC57K20, 57R50
DefinesDehn twist
UsesSurface, Orientable surface, Homeomorphism, Mapping class group
Changed2 November 2026

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