Mapping class group ($\mathrm{Mod}(S)$)

The mapping class group $\mathrm{Mod}(S)$ of an oriented surface $S$ is the group of orientation-preserving self-homeomorphisms of $S$ modulo isotopy. It acts on isotopy classes of curves and arcs.

A mapping class is an isotopy class of self-homeomorphisms of a surface. Orientation-preserving classes form the mapping class group. Acting on markings changes a point of Teichmüller space while preserving the underlying unmarked Riemann surface.

If a finite group $G$ acts faithfully by orientation-preserving isometries on a closed hyperbolic surface $S$ of genus at least two, then $G\hookrightarrow \mathrm{Mod}(S)$. An isometry isotopic to the identity induces an inner automorphism of the surface group; its suitable lift commutes with every deck transformation and fixes their limit set, forcing the lift and the original isometry to be the identity.

Every finite group $G$ is the deck group of a connected finite regular covering of some closed orientable surface of genus at least two. Choose a surface group that surjects onto $G$ and take the covering corresponding to the kernel. The resulting free action embeds $G$ into the mapping class group of the covering surface.

A Dehn twist about a simple closed curve $\gamma$ in an oriented surface is a homeomorphism supported in an annulus around $\gamma$. In the coordinates and orientation convention of a right-handed Dehn twist, it has the formwhere $f$ is zero near one boundary circle and one near the other. It extends by the identity outside the annulus; subtracting $2\pi f(r)$ gives its inverse. Changing the interpolation without changing the end values gives the same mapping class.

For essential simple closed curves $\alpha ,\gamma$ and every integer $n$,Thus $i(\alpha ,\gamma )>0$ forces all iterates to have distinct isotopy classes. In minimal position of curves or arcs, each crossing strand winds $n$ times through the other strands; the bigon criterion proves the counted intersections cannot be removed.^[ref]

On a torus, take curves representing a basis $a,b$ of first homology, with $a\cdot b=1$. Under the stated shear convention, $T_a^n(b)=b+na$, and its geometric intersection number with $b$ is $|n|$. A twist about a curve bounding an unpunctured disc is instead isotopic to the identity by the Alexander trick. Boundary-parallel twists can remain nontrivial when isotopies must fix the boundary pointwise.

The mapping class group of a compact connected oriented topological surface, for orientation-preserving maps fixing its boundary pointwise and isotopies relative to that boundary, is generated by Dehn twists about simple closed curves. Boundary-parallel twists are allowed. For a closed surface of positive genus a finite generating collection exists. Permuting boundary components requires additional generators if that larger group is used.

If $\gamma \cdot u$ uses the ordered-tangent orientation convention for algebraic intersection number of curves on an oriented surface, a right-handed Dehn twist takes $u$ to $u+([\gamma ]\cdot u)[\gamma ]$ on first homology. In particular a separating-curve twist acts trivially there.

In orientation-preserving annular coordinates $(\theta ,r)$ with orientation $d\theta \wedge dr$, the twist is $(\theta ,r)\mapsto (\theta +2\pi f(r),r)$, where $f$ increases from zero to one. The map is the identity outside this annulus and near its boundary. The orientation of the ambient topological surface, rather than the orientation assigned to the curve, fixes the handedness.

A proper arc on a punctured surface approaches punctures at both ends. It is simple when its interior is embedded, and essential when it cannot be isotoped, relative to its ends, into a puncture.

Two transverse arcs form a bigon when subarcs between two consecutive intersection points together bound an embedded disc whose interior is disjoint from both arcs. Pushing one side across the disc removes the two corner intersections.

Two curves or proper arcs are in minimal position when their number of transverse intersection points is the least possible among representatives of their isotopy classes.

Two transverse essential simple curves or proper arcs on a surface are in minimal position exactly when they form no bigon. For proper arcs, the proof uses the compactification of the universal cover to control their ends at punctures.

The geometric intersection number $i(\alpha ,\beta )$ is the minimum number of transverse intersection points among representatives of the isotopy classes of $\alpha$ and $\beta$. Representatives in minimal position realize it.

This is the surface curve-pair version of an intersection number.

For transverse oriented curves on an oriented surface, the algebraic intersection number is the sum of the local signs of their crossings. It depends only on their oriented homology classes and is antisymmetric:Its absolute value is at most the geometric intersection number.

During a generic isotopy of one transverse curve, crossings with a fixed curve can only be created or removed in pairs of opposite local sign. Their signed sum, and hence the algebraic intersection number, is therefore invariant.

The vertices of the arc complex of a punctured surface $S$ are isotopy classes of unoriented essential simple proper arcs. A finite set spans a simplex when its classes have representatives with pairwise disjoint interiors. The mapping class group acts on the complex by simplicial automorphisms.

The three-punctured sphere has six arc classes: one joining each unordered pair of distinct punctures and one returning to each puncture. The three joining arcs span one triangle; for each puncture, its returning arc and the two joining arcs incident to it span another triangle. Its mapping class group is the symmetric group on the three punctures and has two vertex orbits, distinguished by whether the endpoints agree.

The mapping class group of the four-punctured sphere has two orbits on vertices of its arc complex: arcs whose endpoints are distinct and arcs whose endpoints coincide. Each orbit is infinite, and Dehn twists exhibit infinitely many classes of the second type based at any fixed puncture.

An Alexander system is a finite collection of pairwise nonisotopic essential simple curves and proper arcs in pairwise minimal position, arranged without triple intersections and with no three members intersecting pairwise. Pairwise isotopic Alexander systems can be carried to one another simultaneously by an ambient isotopy.

For a filling Alexander system $\{{\alpha }_i\}$, the structure graph is the embedded graphwith vertices at all curve intersections, arc endpoints, and punctures. Its edges are the curve, arc, and boundary segments between consecutive vertices. A homeomorphism preserving the system induces a graph automorphism.

Work on a compact oriented surface, possibly with finitely many marked points, and fix its boundary pointwise during isotopies. Suppose an Alexander system cuts the surface into discs, each containing at most one marked point. This is the filling hypothesis needed here.

A homeomorphism preserving every member up to isotopy can be adjusted by a simultaneous isotopy to preserve its structure graph of an Alexander system. If the induced graph action fixes every vertex and every oriented edge, the remaining maps on the complementary discs are isotopic to the identity by the Alexander trick. Thus the homeomorphism is isotopic to the identity. Since there are only finitely many automorphisms of the finite oriented graph, the pointwise curve-and-arc-class stabilizer is finite.^[ref]

The strong filling condition matters for a surface with boundary. A boundary-parallel Dehn twist can fix every interior curve class while remaining nontrivial relative to the boundary. A system cutting all the way into discs detects this missing information.

A homeomorphism of a closed disc that fixes its boundary pointwise is isotopic relative to the boundary to the identity. Radially interpolate the action toward the center after identifying the disc with the unit ball.

The center of $\mathrm{Mod}(T^2)$ has order two. A central class commutes with twists about two curves intersecting once, hence preserves both curve classes. The Alexander method reduces it to the identity or the elliptic involution $\iota :x\mapsto -x$, and both are central.

A self-homeomorphism of a connected surface induces an automorphism of its fundamental group after a path from the old basepoint to its image is chosen. Changing that path changes the automorphism by an inner automorphism, so isotopy classes define a homomorphism

The pure mapping class group consists of mapping classes that fix every puncture individually.

Pushing a distinguished puncture once around a loop $\gamma$ defines a mapping class. For a simple loop, the push is the product of opposite Dehn twists about the two boundary components of a thin annular neighborhood of $\gamma$.

For a finite-type surface $S$ of negative Euler characteristic and the surface $S^{\ast }$ obtained by adding a puncture, forgetting that puncture gives

The pure mapping class group of the three-punctured sphere is trivial. The three simple arcs joining distinct puncture pairs form an ideal triangulation; a pure homeomorphism can be isotoped to fix these arcs, and the Alexander trick on the two complementary discs finishes the isotopy to the identity.

The Birman exact sequence and $\mathrm{PMod}(S_{0,3})=1$ identify $\mathrm{PMod}(S_{0,4})$ with ${\pi }_1(S_{0,3})$, the free group of rank two.

Forgetting the fifth puncture gives a split Birman exact sequenceLifts of two free generators generate a copy of $F_2$ meeting the point-pushing kernel $F_3$ trivially.

The vertices of the curve complex are isotopy classes of essential simple closed curves. Distinct vertices span a simplex when they have pairwise disjoint representatives.

For every connected orientable surface of complexity above one, the one-skeleton of the curve complex is connected. Surgery replaces one curve by an essential curve disjoint from it while reducing intersection with a fixed target; induction on geometric intersection number gives a path.

A collection of essential simple closed curves fills a surface if every essential simple closed curve has positive geometric intersection number with at least one member. For such curves in minimal position of curves or arcs on a closed surface of positive genus, this is equivalent to every complementary component being a disc.

The Alexander method uses a stronger cutting condition when boundary or marked points are present: an Alexander system must cut the surface into discs with at most one marked point each. For example, a pair of pants has no essential nonperipheral simple closed curves, so the empty collection passes the curve-intersection test, but it does not cut the surface into discs. The boundary-relative version must therefore keep the stronger condition explicit.

A concrete filling set of curves consisting of two simple closed curves can be built from four unit squares. Label the squares $1,2,3,4$. Glue the right side of square $i$ to the left side of square $h(i)$ and its top side to the bottom side of square $v(i)$ by translations, whereThis gives a connected oriented square-tiled surface. The horizontal midlines form one simple closed curve $\alpha$, because $h$ has one cycle of a permutation of length four; the vertical midlines form one simple closed curve $\beta$, because $v$ has one cycle of a permutation of length four. They meet once at each square center.

The corner identifications have two classes, corresponding to the cycles of a permutation ofwith $1$ fixed. Thus the square CW complex has $V=2$, $E=8$, $F=4$, and Euler characteristic $2-8+4=-2$. The classification of compact surfaces gives genus two. The complement of $\alpha \cup \beta$ consists of the two open discs around these vertex classes, so the pair fills.

Orient both curves in their positive coordinate directions. All four crossings have the same local sign, giving algebraic intersection number of curves on an oriented surface $\alpha \cdot \beta =4$. Therefore $i(\alpha ,\beta )\ge 4$, and the displayed representatives attain equality. In particular both curves are nonseparating; a separating curve could not have this nonzero signed pairing.