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.

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 $T_{\gamma }$ cuts an oriented surface along a simple closed curve $\gamma$, rotates one side once, and glues it back. If a curve or arc $\alpha$ has nonzero geometric intersection with $\gamma$, the iterates $T_{\gamma }^n(\alpha )$ usually represent distinct isotopy classes, with intersection numbers growing linearly in $|n|$.

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.

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.

If an Alexander system fills a surface, a homeomorphism preserving every member up to isotopy is determined up to isotopy by its induced automorphism of the structure graph. A trivial graph action makes it isotopic to the identity; the finite graph automorphism group shows that the pointwise curve-class stabilizer is finite.

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 curves fills a surface when every essential simple closed curve intersects at least one member. For curves in minimal position on a closed surface, this is equivalent to every complementary component being a disc.

Split a closed genus-two surface along a separating curve $\alpha$ into two one-holed tori. In each torus choose two disjoint proper arcs that cut it into a disc, and match their four endpoints across $\alpha$ so that the four arcs join into one simple closed curve $\beta$. Then $i(\alpha ,\beta )=4$ and the complement of $\alpha \cup \beta$ is two discs, so the pair fills.