The Mapping Class Group Action

Introduction

The mapping class group of a surface acts on the homology of the surface, and the action preserves the intersection form. This gives the symplectic representation $$ \rho : \mathrm{Mod}(S_g)\longrightarrow \operatorname{Sp}(2g,\mathbb{Z}), $$ the bridge between the topology of the surface and the arithmetic of the symplectic group. It is surjective, and its kernel is the Torelli group: the representation loses exactly the mapping classes that move every curve but no homology class. This article treats the action as the operator-layer object of the group: the representation, its image, its kernel and what the finer actions recover.

The example to hold in mind is the torus, where the representation is an isomorphism $\mathrm{Mod}(T^2)\cong SL_2(\mathbb{Z})$, and the contrast with a surface of genus at least two, where the kernel is large and the action on the homology is far from faithful. The complementary actions — on the fundamental group (faithful, by Dehn–Nielsen–Baer) and on the curve complex (faithful, by Ivanov) — are the two repairs of the information that the homology action forgets.

The article assumes the mapping class group, the Dehn twist and its transvection action, and the intersection form and the symplectic group, that is, The Dehn Twist as an Operator, Bilinear Forms, Symplectic Forms and Poisson Brackets and the written Mapping Class Groups of this Part.

The boundaries of the article. The mapping class group itself — its definition, its presentations, the Nielsen–Thurston classification, the geometry of the curve complex and the detailed study of the Torelli group — is Mapping Class Groups, and is cited; this article owns the action on the homology, its kernel and its image. Teichmüller space and the moduli of surfaces belong to Teichmüller Theory; the hyperbolic structures to Hyperbolic Geometry; the mapping tori and the three-manifold applications to Low-Dimensional Topology. The congruence subgroup property and the arithmetic of $\operatorname{Sp}(2g,\mathbb{Z})$ are used only as stated, and the analytic proofs of the rigidity theorems are Part III.

The Action and the Symplectic Representation

Definition. The mapping class group $\mathrm{Mod}(S_g)$ acts on $H_1(S_g;\mathbb{Z})\cong\mathbb{Z}^{2g}$ by the induced maps $\phi\mapsto\phi_*$, and the action preserves the intersection form $i$. The resulting homomorphism $$ \rho : \mathrm{Mod}(S_g)\longrightarrow \operatorname{Sp}(2g,\mathbb{Z}) = \operatorname{Aut}\bigl(\mathbb{Z}^{2g},i\bigr), \qquad \rho\bigl([\phi]\bigr) = \phi_* , $$ is the symplectic representation, and the Torelli group is its kernel, $$ \mathcal{I}(S_g) = \ker\rho = \bigl\{[\phi] : \phi_* = \mathrm{id}\ \text{on}\ H_1(S_g;\mathbb{Z})\bigr\}. $$

Proposition (well-definedness and naturality). The representation is well defined: an isotopy of homeomorphisms induces the same map on the homology, and an orientation-preserving homeomorphism preserves the intersection form, so its class lands in the symplectic group. The representation is natural for the elementary operators: the image of a Dehn twist is the transvection of The Dehn Twist as an Operator, $$ \rho(T_a) = \bigl(x\mapsto x + i(A,x)A\bigr), \qquad A = [a]. $$

Proof. Isotopic maps induce equal maps on homology; a homeomorphism of a closed orientable surface preserves the intersection number of two curves, because it takes transverse intersections to transverse intersections with the same signs, and this is exactly the preservation of $i$. The image of the twist is the transvection formula.

Example (the torus and the sphere). For $g=1$ the intersection form is the standard alternating form on $\mathbb{Z}^2$ and $\operatorname{Sp}(2,\mathbb{Z}) = SL_2(\mathbb{Z})$, and the representation is an isomorphism $\mathrm{Mod}(T^2)\cong SL_2(\mathbb{Z})$, so the Torelli group is trivial. For $g=0$ the homology vanishes and the kernel is the whole of $\mathrm{Mod}(S^2)$, which is trivial as the group of orientation-preserving classes; in the full group $\mathrm{Mod}^{\pm}(S^2)$ the representation takes values in the trivial group and the kernel is $\mathbb{Z}/2$, generated by a reflection.

The Image and the Kernel

Theorem (surjectivity and the kernel). For $g\geq1$ the symplectic representation is surjective, $$ \rho\bigl(\mathrm{Mod}(S_g)\bigr) = \operatorname{Sp}(2g,\mathbb{Z}), $$ and the kernel is the Torelli group $\mathcal{I}(S_g)$. For $g=1$ the kernel is trivial and the representation is an isomorphism; for $g\geq2$ the Torelli group is infinite.

Proof sketch. The transvections generate $\operatorname{Sp}(2g,\mathbb{Z})$ as a group — the elementary generation of the symplectic group over $\mathbb{Z}$, a theorem of linear algebra — and the Dehn twists generate $\mathrm{Mod}(S_g)$ by Dehn–Lickorish, with images the transvections; hence the image contains all transvections and is the whole symplectic group. For $g\geq2$ a Dehn twist about a separating curve is a nontrivial mapping class whose image is the identity, so the kernel is infinite. The full generation statements are Mapping Class Groups.

Proposition (the Torelli group is the group of homology-trivial classes). An element of $\mathcal{I}(S_g)$ acts trivially on $H_1(S_g;\mathbb{Z})$ and hence on $H_1(S_g;\mathbb{Z}/m)$ for every $m$, and trivially on the intersection form. It need not act trivially on the fundamental group; the Torelli group is the smallest of the three kernels $$ \mathcal{I}(S_g)\subseteq \mathrm{Mod}(S_g)\longrightarrow \begin{cases} \operatorname{Sp}(2g,\mathbb{Z}), & \text{homology},\\ \operatorname{Out}(\pi_1(S_g)), & \text{fundamental group},\\ \operatorname{Aut}(\mathcal{C}(S_g)), & \text{curve complex}, \end{cases} $$ and the other two maps are injective: an orientation-preserving mapping class is determined by its outer action on the fundamental group (Dehn–Nielsen–Baer) and by its action on the curve complex (Ivanov).

Proof sketch. The containment is the definition of the kernel; if a class acts trivially on $\pi_1$ it acts trivially on the homology, and if it acts trivially on the curve complex then it is the identity by Ivanov's rigidity theorem, so the other two kernels vanish and the Torelli group is the only loss of the homology action. The theorems are those of Mapping Class Groups, quoted.

Theorem (the structure of the Torelli group). For $g\geq2$ the Torelli group $\mathcal{I}(S_g)$ is torsion-free, is generated by the Dehn twists about separating curves and by the bounding pair maps (Johnson), is finitely generated for $g\geq3$, and is not finitely generated for $g=2$ (McCullough–Miller). Its abelianisation is computed by the Johnson homomorphism $$ \tau : \mathcal{I}(S_{g,1})\longrightarrow \Lambda^3 H_1(S_{g,1};\mathbb{Z}), \qquad \tau(\text{separating twist}) = 0, $$ which is surjective onto the third exterior power of the homology for the surface with one boundary component; its kernel is the Johnson kernel, generated by the separating twists.

Proof sketch. A separating twist is the identity on the homology, and the bounding pair map (the product of the twists about two disjoint homologous non-separating curves) is also the identity on the homology; Johnson's theorem is that these two families generate. The Johnson homomorphism is defined by the action of a homology-trivial class on the second nilpotent quotient of the fundamental group, and the identification of its target with $\Lambda^3H_1$ is a computation in the free group. The details, tests and citations are in Mapping Class Groups.

Remark (the Johnson filtration). The kernel of the action of $\mathrm{Mod}(S_{g,1})$ on the $k$-th nilpotent quotient of $\pi_1$ is the $k$-th Johnson subgroup, and the filtration begins $$ \mathcal{I}(S_{g,1}) = \mathcal{J}_2 \supseteq \mathcal{J}_3 \supseteq \cdots , \qquad \mathcal{J}_2/\mathcal{J}_3 \cong \Lambda^3H_1 , $$ the quotient being the Johnson homomorphism. The filtration measures how far the mapping class group is from acting trivially on the successive nilpotent quotients, and its graded pieces are the natural home of the "Johnson classes" in the cohomology of the group. The filtrations and the cohomology of $\mathrm{Mod}$ are Mapping Class Groups and Stable Homotopy Theory respectively.

The Action on Other Modules

The representation is the first of a family: the mapping class group acts on every natural module built from the surface, and the family records more than the homology alone.

The dual action on the cohomology. The action on $H^1(S_g;\mathbb{Z})$ is the contragredient of the action on $H_1$ and is again symplectic, with the cup product corresponding to the intersection form; the fixed subrings and the invariants of the representation are the arithmetic invariants of the surface.

The action with twisted coefficients. For a module $V$ over $\mathbb{Z}[\pi_1(S_g)]$ the group acts on $H_*(S_g;V)$; the case of the Magnus representation, in which $V$ is the group ring of the free group of a once-punctured surface, gives a faithful representation of the Torelli group as units of a ring, and it is the tool by which the structure of $\mathcal{I}$ is studied. The construction is described in Mapping Class Groups.

The action on the homology of a covering. A finite covering $\tilde S\to S$ has a homology on which $\mathrm{Mod}(S)$ acts only through the subgroup preserving the covering; for the abelian cover one obtains the action on a symplectic lattice over the group ring of the abelianisation, whose invariants are the "higher" intersection forms and the Witt groups of The Witt Group and the Grothendieck–Witt Ring. The "quantum" and "HOMFLY" refinements are the algebra and the low-dimensional topology of this batch.

Proposition (the invariants of the action). A class of functions on $\mathrm{Mod}(S_g)$ that factors through the symplectic representation is an invariant of the symplectic group and hence of the intersection form alone; the refinements that see the Torelli group — the Johnson homomorphism, the Rochlin invariant of a homology sphere bounding the surface, the Casson invariant — are the invariants detecting the kernel, and they are Mapping Class Groups and Low-Dimensional Topology.

Examples and Applications

Example (the elementary matrices and the twists). On the torus the twists about the meridian and the longitude have images the two elementary matrices $\begin{pmatrix}1&1\\0&1\end{pmatrix}$ and $\begin{pmatrix}1&0\\1&1\end{pmatrix}$, which generate $SL_2(\mathbb{Z})$; the representation is an isomorphism, and the word problem and the conjugacy classification reduce to those of $SL_2(\mathbb{Z})$.

Example (the genus-two hyperelliptic case). On $S_2$ the hyperelliptic involution is the central element of $\mathrm{Mod}(S_2)$ modulo the centre generated by it, and the quotient is related to the mapping class group of the six-punctured sphere by the Birman–Hilden theorem; on the homology the hyperelliptic involution acts by $-I$, so it lies in the kernel only if $-I$ is the identity, which it is not; the example warns that the kernel is the Torelli group, not the centre. The statement and its exceptional small cases are Mapping Class Groups.

Example (the action and the Dehn twist formula). For every simple closed curve $a$ and every class $[\phi]\in\mathrm{Mod}(S_g)$ one has the conjugation formula $$ \rho(\phi T_a \phi^{-1}) = \rho(\phi)\,\rho(T_a)\,\rho(\phi)^{-1}, $$ so that the conjugates of a transvection are the transvections with direction the image class $\phi_*(A)$; the symplectic action permutes the transvections, as the symplectic group acts transitively on the primitive vectors. This is the operator-level statement that conjugating a twist about $a$ gives the twist about $\phi(a)$.

Example (the mapping torus and the Alexander polynomial). The mapping torus of a surface homeomorphism $\phi$ has a homology of rank one more than that of the fixed part of $\phi_*$, and the Alexander polynomial of the torus after abelianisation is the characteristic polynomial $\det(tI - \phi_*)$ of the symplectic matrix; the representation therefore computes the first homology of the mapping torus and the "Alexander invariant" of the $3$-manifold. The construction and the three-manifold examples are Low-Dimensional Topology.

Summary

The mapping class group acts on the homology of the surface by maps preserving the intersection form, and the resulting symplectic representation $\rho : \mathrm{Mod}(S_g)\to\operatorname{Sp}(2g,\mathbb{Z})$ is surjective with kernel the Torelli group $\mathcal{I}(S_g)$; the image of a Dehn twist is the transvection with its class as direction. The representation is an isomorphism for the torus and has infinite kernel for $g\geq2$, the kernel being torsion-free, generated by the separating twists and the bounding pairs, finitely generated for $g\geq3$ and not for $g=2$, with the Johnson homomorphism from the Torelli group of the once-punctured surface onto the third exterior power of the homology and Johnson kernel the separating twists. The action on the fundamental group and on the curve complex is faithful, so the Torelli group is exactly the information lost by the homology action, and the Johnson filtration measures the loss in the successive nilpotent quotients. The representation also governs the twisted actions, the Magnus representation of the Torelli group and the homology of a mapping torus, where the characteristic polynomial of the symplectic matrix computes the Alexander invariant.

Summary of Notation

Symbol	Meaning
$\mathrm{Mod}(S_g)$, $\mathrm{Mod}^{\pm}(S_g)$	mapping class group of orientation-preserving (all) isotopy classes of homeomorphisms
$\rho : \mathrm{Mod}(S_g)\to\operatorname{Sp}(2g,\mathbb{Z})$	the symplectic representation, the action on $H_1$
$\operatorname{Sp}(2g,\mathbb{Z})$	the automorphism group of $\mathbb{Z}^{2g}$ preserving the intersection form
$\mathcal{I}(S_g) = \ker\rho$	the Torelli group, the kernel of the homology action
$\rho(T_a) = x\mapsto x+i(A,x)A$	the image of a Dehn twist, a transvection of direction $A=[a]$
$\mathcal{I}(S_{g,1})$	the Torelli group of a surface with one boundary component
$\tau : \mathcal{I}(S_{g,1})\to\Lambda^3H_1$	the Johnson homomorphism; surjective onto $\Lambda^3H_1$
$\mathcal{J}_k$	the Johnson filtration, $\mathcal{J}_2 = \mathcal{I}$, $\mathcal{J}_2/\mathcal{J}_3\cong\Lambda^3H_1$
$\mathrm{Out}(\pi_1(S_g))$, $\mathrm{Aut}(\mathcal{C}(S_g))$	the faithful receptors of the mapping class group
$\det(tI-\phi_*)$	the characteristic polynomial of a symplectic matrix; the Alexander invariant of the mapping torus