--- title: Mapping Class Groups author: D. Zack Garza slideNumber: true width: 1200 height: 900 transition: fade notoc: true date: 2021-04-28 18:09:50 --- [mapping class group](mapping%20class%20group.md) Tags: #geomtop/MCG #projects/my-talks #projects/review # Setup - All manifolds: - Connected - Oriented - 2nd countable (countable basis) - Hausdorff (separate with disjoint neighborhoods, uniqueness of limits) - With boundary (possibly empty) - Weakly Hausdorff: every continuous image of a compact Hausdorff space into it is closed. - Compactly generated: sets are closed iff their intersection with every compact subspace is closed. - Curves: simple, closed, oriented - Surfaces: these guys - For $X, Y$ topological spaces, consider \[ Y^X = C(X, Y) = \hom_\Top(X, Y) \da \ts{f: X\to Y \st f\,\,\text{is continuous}} .\] \newpage ## The Compact-Open Topology - General idea: *cartesian closed* categories, require *exponential objects* or *internal homs*: i.e. for every hom set, there is some object in the category that represents it - Slogan: we'd like homs to be spaces. - Can make this work if we assume WHCG: weakly Hausdorff and compactly generated. - Topologize with the *compact-open* topology $\OO_{\text{CO}}$: \[ U \in \OO_{\text{CO}} \iff \forall f\in U, \quad f(K) \subset Y \text{ is open for every compact } K\subseteq X .\] ### Mapping Spaces - So define \[ \Map(X, Y) \da (\hom_\Top(X, Y), \OO_{\text{CO}}) \qquad\text{where }\OO_{\text{CO}}\text{ is the compact-open topology} .\] - Can immediately define interesting derived spaces: - $\Homeo(X, Y)$ the subspace of homeomorphisms - $\mathrm{Imm}(X, Y)$, the subspace of immersions (injective map on tangent spaces) - $\mathrm{Emb}(X, Y)$, the subspace of embeddings (immersion + diffeomorphic onto image) - $C^k(X, Y)$, the subspace of $k\times$ differentiable maps - $C^\infty(X, Y)$ the subspace of smooth maps - $\mathrm{Diffeo}(X ,Y)$ the subspace of diffeomorphisms - $C^\omega(X, Y)$ the subspace of analytic maps - $\mathrm{Isom}(X, Y)$ the subspace of isometric maps (for Riemannian metrics) - $[X, Y]$ homotopy classes of maps \newpage ## Aside on Analysis - If $Y = (Y, d)$ is a metric space, this is the topology of "uniform convergence on compact sets": for $f_n \to f$ in this topology iff \[ \norm{f_n - f}_{\infty, K} \da \sup \ts{d(f_n(x), f(x)) \st x\in K}\converges{n\to\infty}\to 0 \quad \forall K\subseteq X \,\,\text{compact} .\] - In words: $f_n\to f$ uniformly on every compact set. - If $X$ itself is compact and $Y$ is a metric space, $C(X, Y)$ can be promoted to a metric space with \[ d(f, g) = \sup_{x\in X}(f(x), g(x)) .\] \newpage --- # Path Spaces - Can immediately consider some interesting spaces via the functor $\Map(\wait, Y)$: \[ X = \pt &\leadsto \quad \Map(\pt, Y) \cong Y \\ X= I &\leadsto\quad \mathcal{P}Y \da \ts{f: I\to Y} = Y^I \\ X= S^1 &\leadsto\quad \mathcal{L}Y \da \ts{f: S^1\to Y } = Y^{S^1} .\] - Adjoint property: there is a homeomorphism \[ \Map(X\cross Z, Y) &\leftrightarrow_{\cong} \Map(Z, Y^X) \\ H:X\cross Z \to Y &\iff \tilde H: Z\to \Map(X, Y)\\ (x, z) \mapsto H(x,z) &\iff z \mapsto H(\wait, z) .\] - Categorically, $\hom(X, \wait) \leftrightarrow (X\cross \wait)$ form an adjoint pair in $\Top$. - A form of this adjunction holds in any cartesian closed category (terminal objects, products, and exponentials) \newpage ## Homotopy and Isotopy in Terms of Path Spaces - Can take basepoints to obtain the base path space $PY$, the based loop space $\Omega Y$. - Importance in homotopy theory: the path space fibration \[ \Omega Y \injects PY \mapsvia{\gamma \mapsto \gamma(1)} Y \] - Plays a role in "homotopy replacement", allows you to assume everything is a fibration and use homotopy long exact sequences. - Fun fact: with some mild point-set conditions (Locally compact and Hausdorff), \[ \pi_0 \Map(X, Y) = \ts{[f],\, \text{homotopy classes of maps }f: X\to Y} ,\] i.e. two maps $f, g$ are homotopic $\iff$ they are connected by a path in $\Map(X, Y)$. > Picture! \newpage ### Proof \[ \mathcal{P}\Map(X, Y) = \Map(I, Y^X) \cong \Map(X\cross I, Y) ,\] and just check that $\gamma(0) = f \iff H(x, 0) = f$ and $\gamma(1) = g \iff H(x, 1) = g$. - Interpretation: the RHS contains homotopies for maps $X\to Y$, the LHS are paths in the space of maps. --- \newpage # Defining the Mapping Class Group ## Isotopy - Define a homotopy between $f, g: X\to Y$ as a map $F:X\cross I \to Y$ restricting to $f, g$ on the ends. - Equivalently: a *path*, an element of $\Map(I, C(X, Y))$. - Isotopy: require the partially-applied function $F_t:X\to Y$ to be homeomorphisms for every $t$. - Equivalently: a path in the subspace of homeomorphisms, an element of $\Map(I, \Homeo(X, Y))$ > Picture: picture of homotopy, paths in $\Map(X, Y)$, subspace of homeomorphisms. \newpage ## Self-Homeomorphisms - In any category, the automorphisms form a group. - In a general category $\mathcal{C}$, we can always define the group $\Aut_{\mathcal{C}}(X)$. - If the group has a topology, we can consider $\pi_0 \Aut_{\mathcal{C}}(X)$, the set of path components. - Since groups have identities, we can consider $\Aut^0_{\mathcal{C}}(X)$, the path component containing the identity. - So we make a general definition, the *extended mapping class group*: \[ \MCG^\pm_{\mathcal{C}}(X) \da \Aut_{\mathcal{C}}(X) / \Aut_{\mathcal{C}}^0(X) .\] - Here the $\pm$ indicates that we take both orientation preserving and non-preserving automorphisms. - Has an index 2 subgroup of orientation-preserving automorphisms, $\MCG^+(X)$. - Can define $\MCG_\bd(X)$ as those that restrict to the identity on $\bd X$. > Picture: quotienting out by identity component \newpage ## Definitions in Several Categories - Now restrict attention to \[ \Homeo(X) \da \Aut_{\Top}(X) = \ts{f\in \Map(X, X) \st f \text{ is an isomorphism}} \\ \qquad\text{equipped with }\OO_{\text{CO}} .\] - Taking $\MCG^\pm_\Top(X)$ yields *homeomorphism up to homotopy* - Similarly, we can do all of this in the smooth category: \[ \Diffeo(X) \da \Aut_{C^\infty}(X) .\] - Taking $\MCG_{C^\infty}(X)$ yields *diffeomorphism up to isotopy* - Similarly, we can do this for the homotopy category of spaces: \[ \text{ho}(X) \da \ts{[f] \in [X, Y]} .\] - Taking $\MCG(X)$ here yields *homotopy classes of self-homotopy equivalences*. \newpage \newpage ## Relation to Moduli Spaces - For topological manifolds: Isotopy classes of homeomorphisms - In the compact-open topology, two maps are isotopic iff they are in the same component of $\pi_0 \Aut(X)$. - For surfaces: For $\Sigma$ a genus $g$ surface, $\MCG(S)$ acts on the Teichmuller space $T(S)$, yielding a SES \[ 0 \to \MCG(\Sigma) \to T(\Sigma) \to {\mathcal{M}}_g \to 0 \] where the last term is the moduli space of Riemann surfaces homeomorphic to $X$. - $T(S)$ is the moduli space of complex structures on $S$, up to the action of homeomorphisms that are isotopic to the identity: - Points are isomorphism classes of marked Riemann surfaces - Equivalently the space of hyperbolic metrics - Used in the Neilsen-Thurston Classification: for a compact orientable surface, a self-homeomorphism is isotopic to one which is any of: - Periodic, - Reducible (preserves some simple closed curves), or - Pseudo-Anosov (has directions of expansion/contraction) > Picture: $\mathcal{M}_g$. \newpage # Examples of MCG ## The Plane: Straight Lines - $\MCG_\Top(\RR^2) = 1$: for any $f:\RR^2\to \RR^2$, take the straight-line homotopy: \[ F: \RR^2 \cross I &\to \RR^2 \\ F(x, t) &= tf(x) + (1-t)x .\] > Picture: parameterize line between $x$ and $f(x)$ and flow along it over time. \newpage ## The Closed Disc: The Alexander Trick - $\MCG_\Top(\bar \DD^2) = 1$: for any $f: \bar\DD^2\to\bar\DD^2$ such that $\ro{f}{\bd \bar\DD^2} = \id$, take \[ F: \bar\DD^2 \cross I &\to \bar\DD^2 \\ F(x, t) &\da \begin{cases} t f \qty{x\over t} & \norm{x} \in [0, t) \\ x & \norm{x} \in [1-t, 1] \end{cases} .\] - This is an isotopy from $f$ to the identity. - Interpretation: "cone off" your homeomorphism over time: - Note that this won't work in the smooth category: singularity at origin \newpage ## Overview of Big Results - The word problem in $\MCG(\Sigma_g)$ is solvable - Any finite group is $\MCG(X)$ for some compact hyperbolic 3-manifold $X$. - For $g\geq 3$, the center of $\MCG(\Sigma_g)$ is trivial and $H_1(\MCG(\Sigma_g); \ZZ) = 1$ - Why care: same as abelianization of the group. :::{.theorem title="Dehn-Neilsen-Baer"} Let $\Sigma_g$ be compact and oriented with $\chi(\Sigma_g) < 0$. Then \[ \MCG^+_\del(\Sigma_g) \cong \Out_\del(\pi_1(\Sigma_g)) \cong_{\Grp} \pi_0 \mathrm{ho}_\del(\Sigma_g) .\] ::: - For $g\geq 4$, $H_2(\MCG(\Sigma_g); \ZZ) = \ZZ$ - Why care: used to understand surface bundles \begin{center} \begin{tikzcd} \Sigma_g \ar[r] & E \ar[d] \\ & B \\ \end{tikzcd} \end{center} - Find the classifying space $B\Diffeo$ - Understand its homotopy type, since the homotopy LES yields \[ [S^n, B\Diffeo(\Sigma_g)] \cong [S^{n-1}, \Diffeo(\Sigma_g)] \] - Theorem (Earle-Ells): For $g\geq 2$, $\Diffeo_0(\Sigma_g)$ is contractible. As a consequence, $\Diffeo(\Sigma_g) \surjects \Mod(\Sigma_g)$ is a homotopy equivalence, and there is a correspondence: \[ \correspond{\text{Oriented $\Sigma_g$ bundles} \\ \text{over } B }/\text{\tiny Bundle isomorphism} \iff \correspond{\text{Monodromy Representations} \\ \rho: \pi_1(B) \to \MCG(\Sigma_g)}/\text{\tiny Conjugacy} .\] --- \newpage # Dehn Twists - $\MCG(\Sigma_g)$ is generated by finitely many **Dehn twists**, and always has a finite presentation :::{.claim} Let $A \da \ts{z\in \CC \st 1\leq \abs z \leq 2}$, then $\MCG(A) \cong \ZZ$, generated by the map \[ \tau_0: \CC &\to \CC \\ z & \mapsto \exp{2\pi i \abs z}\, z .\] > See complex function plotter ::: \newpage # MCG of the Torus ## Setup :::{.definition title="Special Linear Group"} \[ \SL(n, \kk) = \ts{M\in \GL(n, \kk) \mid \det M = 1} = \ker \det_{\GG_m} .\] ::: :::{.definition title="Symplectic Group"} \[ \Sp(2n, \kk) = \ts{M\in \GL(2n, \kk) \mid M^t\Omega M = \Omega} \leq \SL(2n, \kk) \] where $\Omega$ is a nondegenerate skew-symmetric bilinear form on $\kk$. Example: \[ \Omega = \begin{bmatrix} 0 & I_n \\ -I_n & 0 \end{bmatrix} .\] ::: :::{.definition title="Algebraic Intersection"} A bilinear antisymmetric form on middle homology: \[ \hat{\iota}: H_1(\Sigma_g; \ZZ) \tensor H_1(\Sigma_g; \ZZ) \to \ZZ .\] > Note that this is a symplectic pairing. ::: - There is a natural action of $\MCG(\Sigma)$ on $H_1(\Sigma; \ZZ)$, i.e. a *homology representation* of $\MCG(\Sigma)$: \[ \rho: \MCG(\Sigma) &\to \Aut_{\Grp}(H_1(\Sigma; \ZZ)) \\ f &\mapsto f_* .\] \newpage - For a surface of finite genus $g\geq 1$, elements in $\im \rho$ preserve the *algebraic intersection form* - Thus there is an interesting surjective representation: \[ 0 \to \mathrm{Tor}(\Sigma_g) \injects \MCG(\Sigma_g) \surjects \Sp(2g; \ZZ) \to 0 .\] - Kernel is the *Torelli group*, interesting because the symplectic group is well understood, so questions about $\MCG$ reduce to questions about $\Tor$. :::{.theorem title="Mapping Class Group of the Torus"} The homology representation of the torus induces an isomorphism \[ \sigma: \MCG(\Sigma_2) \mapsvia{\cong} \SL(2, \ZZ) \] ::: \newpage ## Proof - For $f$ any automorphism, the induced map $f_*: \ZZ^2 \to \ZZ^2$ is a group automorphism, so we can consider the group morphism \[ \tilde \sigma: (\Homeo(X,X), \circ) &\to (\GL(2, \ZZ), \circ) \\ f &\mapsto f_* .\] - This will descend to the quotient $\MCG(X)$ iff \[\Homeo^0(X, X) \subseteq \ker \tilde \sigma = \tilde\sigma\inv(\id)\] - This is true here, since any map in the identity component is homotopic to the identity, and homotopic maps induce the equal maps on homology. - So we have a (now injective) map \[ \tilde \sigma:\MCG(X) &\to \GL(2, \ZZ) \\ f &\mapsto f_* .\] \newpage :::{.claim} $\im(\tilde\sigma )\subseteq \SL(2, \ZZ)$. ::: :::{.proof} \hfill - Algebraic intersection numbers in $\Sigma_2$ correspond to determinants - $f\in \Homeo^+(X)$ preserve algebraic intersection numbers. - See section 1.2 of Farb and Margalit ::: - We can thus freely restrict the codomain to define the map \[ \sigma:\MCG(X) &\to \SL(2, \ZZ) \\ f &\mapsto f_* .\] \newpage ### Surjectivity :::{.claim} $\sigma$ is surjective. ::: - $\RR^2$ is the universal cover of $\Sigma_2$, with deck transformation group $\ZZ^2$. - Any $A\in \SL(2, \ZZ)$ extends to $\tilde A \in \GL(2, \RR)$, a linear self-homeomorphism of the plane that is orientation-preserving. :::{.claim} $\tilde A$ is equivariant wrt $\ZZ^2$ ::: :::{.proof} \[ \SL(2, \ZZ) = \gens{ S = \begin{bmatrix} 0 & -1 \\ 1 & 0 \end{bmatrix} , T = \begin{bmatrix} 1 & 1 \\ 0 & 1 \end{bmatrix} } .\] Note that \[ S^2 = 1, \qquad T^n = \begin{bmatrix} 1 & n \\ 0 & 1 \end{bmatrix} \] So if $\vector x = \thevector{x_1, x_2} \in \ZZ \oplus \ZZ$ and $\tilde A\in \SL(2, \ZZ)$, we have $\tilde A\vector x \in \ZZ\oplus \ZZ$, i.e. $A$ preserves any integer lattice \[ \Lambda = \ts{p \vector v_1 + q\vector v_2 \st p, q\in \ZZ} .\] ::: - So $\tilde A$ descends to a well-defined map \[ \psi_{\tilde A}: \Sigma_2 \selfmap = \RR^2 / \ZZ^2 \selfmap \] which is still a linear self-homeomorphism. \newpage - There is a correspondence \[ \correspond{\text{Primitive curves in } \\ \pi_1(\Sigma_2) \cong \ZZ^2} \iff \correspond{\text{Primitive vectors in }\ZZ^2} \iff \correspond{\text{Oriented simple closed} \\\text{curves in } \Sigma_2}/\text{\tiny homotopy} ,\] where an element $x$ is *primitive* iff it is not a multiple of another element. - By changing basis, you can associate a unique primitive vector to $M$ (all components coprime) - By the correspondence, changing a map by a homotopy corresponds to the same primitive vector - Thus $\sigma([\psi_{\tilde A}]) = \tilde A$, and we have surjectivity. \newpage ### Injectivity :::{.claim} $\sigma$ is injective. ::: - Useful fact: $\Sigma_2 \simeq K(\ZZ^2, 1)$. :::{.proposition title="Hatcher 1B.9"} Let $X$ be a connected CW complex and $Y$ a $K(G, 1)$. Then there is a map \[ \hom_\Grp(\pi_1(X; x_0), \pi_1(Y; y_0)) \to \hom_\Top((X; x_0), (Y; y_0)) ,\] i.e. every homomorphism of fundamental groups is induced by a continuous pointed map. Moreover, the map is unique up to homotopies fixing $x_0$. ::: - Thus there is a correspondence \[ \correspond{\text{Homotopy classes of } \\ \text{maps }\Sigma_2 \selfmap} \iff \correspond{\text{Group morphisms } \ZZ^2\selfmap} .\] - Claim: any element $f\in \MCG(\Sigma_2)$ has a representative $\phi$ which fixes any given basepoint - So if $f\in \ker \sigma$, then $f\simeq \phi \simeq \id$ are homotopic, so $\ker \sigma = 1$.