mapping class group

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 \begin{align*} Y^X = C(X, Y) = \hom_{\mathsf{Top}}(X, Y) \coloneqq\left\{{f: X\to Y {~\mathrel{\Big\vert}~}f\,\,\text{is continuous}}\right\} .\end{align*}

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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 \({\mathcal{O}}_{\text{CO}}\): \begin{align*} U \in {\mathcal{O}}_{\text{CO}} \iff \forall f\in U, \quad f(K) \subset Y \text{ is open for every compact } K\subseteq X .\end{align*}

Mapping Spaces

• So define \begin{align*} \mathop{\mathrm{Maps}}(X, Y) \coloneqq(\hom_{\mathsf{Top}}(X, Y), {\mathcal{O}}_{\text{CO}}) \qquad\text{where }{\mathcal{O}}_{\text{CO}}\text{ is the compact-open topology} .\end{align*}

• Can immediately define interesting derived spaces:

  • \({\operatorname{Homeo}}(X, Y)\) 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

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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 \begin{align*} {\left\lVert {f_n - f} \right\rVert}_{\infty, K} \coloneqq\sup \left\{{d(f_n(x), f(x)) {~\mathrel{\Big\vert}~}x\in K}\right\}\overset{n\to\infty}\to 0 \quad \forall K\subseteq X \,\,\text{compact} .\end{align*}

  • 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

\begin{align*} d(f, g) = \sup_{x\in X}(f(x), g(x)) .\end{align*}

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Path Spaces

• Can immediately consider some interesting spaces via the functor \(\mathop{\mathrm{Maps}}({-}, Y)\): \begin{align*} X = {\operatorname{pt}}&\leadsto \quad \mathop{\mathrm{Maps}}({\operatorname{pt}}, Y) \cong Y \\ X= I &\leadsto\quad \mathcal{P}Y \coloneqq\left\{{f: I\to Y}\right\} = Y^I \\ X= S^1 &\leadsto\quad \mathcal{L}Y \coloneqq\left\{{f: S^1\to Y }\right\} = Y^{S^1} .\end{align*}

• Adjoint property: there is a homeomorphism \begin{align*} \mathop{\mathrm{Maps}}(X\times Z, Y) &\leftrightarrow_{\cong} \mathop{\mathrm{Maps}}(Z, Y^X) \\ H:X\times Z \to Y &\iff \tilde H: Z\to \mathop{\mathrm{Maps}}(X, Y)\\ (x, z) \mapsto H(x,z) &\iff z \mapsto H({-}, z) .\end{align*}

  • Categorically, \(\hom(X, {-}) \leftrightarrow (X\times{-})\) form an adjoint pair in \({\mathsf{Top}}\).

  • A form of this adjunction holds in any cartesian closed category (terminal objects, products, and exponentials)

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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 \begin{align*} \Omega Y \hookrightarrow PY \xrightarrow{\gamma \mapsto \gamma(1)} Y \end{align*}

  • 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), \begin{align*} \pi_0 \mathop{\mathrm{Maps}}(X, Y) = \left\{{[f],\, \text{homotopy classes of maps }f: X\to Y}\right\} ,\end{align*} i.e. two maps \(f, g\) are homotopic \(\iff\) they are connected by a path in \(\mathop{\mathrm{Maps}}(X, Y)\).

  Picture!

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Proof

\begin{align*} \mathcal{P}\mathop{\mathrm{Maps}}(X, Y) = \mathop{\mathrm{Maps}}(I, Y^X) \cong \mathop{\mathrm{Maps}}(X\times I, Y) ,\end{align*} 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.

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Defining the Mapping Class Group

Isotopy

• Define a homotopy between \(f, g: X\to Y\) as a map \(F:X\times I \to Y\) restricting to \(f, g\) on the ends.
  • Equivalently: a path, an element of \(\mathop{\mathrm{Maps}}(I, C(X, Y))\).
• 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 \(\mathop{\mathrm{Maps}}(I, {\operatorname{Homeo}}(X, Y))\)

Picture: picture of homotopy, paths in \(\mathop{\mathrm{Maps}}(X, Y)\), subspace of homeomorphisms.

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Self-Homeomorphisms

• In any category, the automorphisms form a group.

  • In a general category \(\mathcal{C}\), we can always define the group \(\mathop{\mathrm{Aut}}_{\mathcal{C}}(X)\).

    • If the group has a topology, we can consider \(\pi_0 \mathop{\mathrm{Aut}}_{\mathcal{C}}(X)\), the set of path components.
    • Since groups have identities, we can consider \(\mathop{\mathrm{Aut}}^0_{\mathcal{C}}(X)\), the path component containing the identity.

  • So we make a general definition, the extended mapping class group: \begin{align*} {\operatorname{MCG}}^\pm_{\mathcal{C}}(X) \coloneqq\mathop{\mathrm{Aut}}_{\mathcal{C}}(X) / \mathop{\mathrm{Aut}}_{\mathcal{C}}^0(X) .\end{align*}

  • Here the \(\pm\) indicates that we take both orientation preserving and non-preserving automorphisms.

  • Has an index 2 subgroup of orientation-preserving automorphisms, \({\operatorname{MCG}}^+(X)\).

  • Can define \({\operatorname{MCG}}_{{\partial}}(X)\) as those that restrict to the identity on \({{\partial}}X\).

Picture: quotienting out by identity component

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Definitions in Several Categories

• Now restrict attention to \begin{align*} {\operatorname{Homeo}}(X) \coloneqq\mathop{\mathrm{Aut}}_{{\mathsf{Top}}}(X) = \left\{{f\in \mathop{\mathrm{Maps}}(X, X) {~\mathrel{\Big\vert}~}f \text{ is an isomorphism}}\right\} \\ \qquad\text{equipped with }{\mathcal{O}}_{\text{CO}} .\end{align*}
  • Taking \({\operatorname{MCG}}^\pm_{\mathsf{Top}}(X)\) yields homeomorphism up to homotopy
• Similarly, we can do all of this in the smooth category: \begin{align*} {\operatorname{Diffeo}}(X) \coloneqq\mathop{\mathrm{Aut}}_{C^\infty}(X) .\end{align*}
  • Taking \({\operatorname{MCG}}_{C^\infty}(X)\) yields diffeomorphism up to isotopy
• Similarly, we can do this for the homotopy category of spaces: \begin{align*} \text{ho}(X) \coloneqq\left\{{[f] \in [X, Y]}\right\} .\end{align*}
  • Taking \({\operatorname{MCG}}(X)\) here yields homotopy classes of self-homotopy equivalences.

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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 \(\pi_0 \mathop{\mathrm{Aut}}(X)\).

• For surfaces: For \(\Sigma\) a genus \(g\) surface, \({\operatorname{MCG}}(S)\) acts on the Teichmuller space \(T(S)\), yielding a SES \begin{align*} 0 \to {\operatorname{MCG}}(\Sigma) \to T(\Sigma) \to {\mathcal{M}}_g \to 0 \end{align*} 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: \(\mathcal{M}_g\).

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Examples of MCG

The Plane: Straight Lines

• \({\operatorname{MCG}}_{\mathsf{Top}}({\mathbf{R}}^2) = 1\): for any \(f:{\mathbf{R}}^2\to {\mathbf{R}}^2\), take the straight-line homotopy: \begin{align*} F: {\mathbf{R}}^2 \times I &\to {\mathbf{R}}^2 \\ F(x, t) &= tf(x) + (1-t)x .\end{align*}

Picture: parameterize line between \(x\) and \(f(x)\) and flow along it over time.

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The Closed Disc: The Alexander Trick

• \({\operatorname{MCG}}_{\mathsf{Top}}(\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu^2) = 1\): for any \(f: \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu^2\to\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu^2\) such that \({ \left.{{f}} \right|_{{{{\partial}}\mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu^2}} } = \operatorname{id}\), take \begin{align*} F: \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu^2 \times I &\to \mkern 1.5mu\overline{\mkern-1.5mu{\mathbb{D}}\mkern-1.5mu}\mkern 1.5mu^2 \\ F(x, t) &\coloneqq \begin{cases} t f \qty{x\over t} & {\left\lVert {x} \right\rVert} \in [0, t) \\ x & {\left\lVert {x} \right\rVert} \in [1-t, 1] \end{cases} .\end{align*}

  • 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

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Overview of Big Results

• The word problem in \({\operatorname{MCG}}(\Sigma_g)\) is solvable

• Any finite group is \({\operatorname{MCG}}(X)\) for some compact hyperbolic 3-manifold \(X\).

• For \(g\geq 3\), the center of \({\operatorname{MCG}}(\Sigma_g)\) is trivial and \(H_1({\operatorname{MCG}}(\Sigma_g); {\mathbf{Z}}) = 1\)

  • Why care: same as abelianization of the group.

• For \(g\geq 4\), \(H_2({\operatorname{MCG}}(\Sigma_g); {\mathbf{Z}}) = {\mathbf{Z}}\)

  • 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 \(B{\operatorname{Diffeo}}\)

  • Understand its homotopy type, since the homotopy LES yields \begin{align*} [S^n, B{\operatorname{Diffeo}}(\Sigma_g)] \cong [S^{n-1}, {\operatorname{Diffeo}}(\Sigma_g)] \end{align*}

  • Theorem (Earle-Ells): For \(g\geq 2\), \({\operatorname{Diffeo}}_0(\Sigma_g)\) is contractible. As a consequence, \({\operatorname{Diffeo}}(\Sigma_g) \twoheadrightarrow{\mathsf{Mod}}(\Sigma_g)\) is a homotopy equivalence, and there is a correspondence: \begin{align*} \left\{{\substack{\text{Oriented $\Sigma_g$ bundles} \\ \text{over } B }}\right\}/\text{\tiny Bundle isomorphism} \iff \left\{{\substack{\text{Monodromy Representations} \\ \rho: \pi_1(B) \to {\operatorname{MCG}}(\Sigma_g)}}\right\}/\text{\tiny Conjugacy} .\end{align*}

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Dehn Twists

• \({\operatorname{MCG}}(\Sigma_g)\) is generated by finitely many Dehn twists, and always has a finite presentation

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MCG of the Torus

Setup

• There is a natural action of \({\operatorname{MCG}}(\Sigma)\) on \(H_1(\Sigma; {\mathbf{Z}})\), i.e. a homology representation of \({\operatorname{MCG}}(\Sigma)\): \begin{align*} \rho: {\operatorname{MCG}}(\Sigma) &\to \mathop{\mathrm{Aut}}_{{\mathsf{Grp}}}(H_1(\Sigma; {\mathbf{Z}})) \\ f &\mapsto f_* .\end{align*}

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• For a surface of finite genus \(g\geq 1\), elements in \(\operatorname{im}\rho\) preserve the algebraic intersection form

• Thus there is an interesting surjective representation: \begin{align*} 0 \to \mathrm{Tor}(\Sigma_g) \hookrightarrow{\operatorname{MCG}}(\Sigma_g) \twoheadrightarrow{\mathsf{Sp}}(2g; {\mathbf{Z}}) \to 0 .\end{align*}

• Kernel is the Torelli group, interesting because the symplectic group is well understood, so questions about \({\operatorname{MCG}}\) reduce to questions about \(\operatorname{Tor}\).

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Proof

• For \(f\) any automorphism, the induced map \(f_*: {\mathbf{Z}}^2 \to {\mathbf{Z}}^2\) is a group automorphism, so we can consider the group morphism \begin{align*} \tilde \sigma: ({\operatorname{Homeo}}(X,X), \circ) &\to (\operatorname{GL}(2, {\mathbf{Z}}), \circ) \\ f &\mapsto f_* .\end{align*}

• This will descend to the quotient \({\operatorname{MCG}}(X)\) iff \begin{align*}{\operatorname{Homeo}}^0(X, X) \subseteq \ker \tilde \sigma = \tilde\sigma^{-1}(\operatorname{id})\end{align*}

  • 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 \begin{align*} \tilde \sigma:{\operatorname{MCG}}(X) &\to \operatorname{GL}(2, {\mathbf{Z}}) \\ f &\mapsto f_* .\end{align*}

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• We can thus freely restrict the codomain to define the map \begin{align*} \sigma:{\operatorname{MCG}}(X) &\to {\operatorname{SL}}(2, {\mathbf{Z}}) \\ f &\mapsto f_* .\end{align*}

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Surjectivity

• \({\mathbf{R}}^2\) is the universal cover of \(\Sigma_2\), with deck transformation group \({\mathbf{Z}}^2\).

• Any \(A\in {\operatorname{SL}}(2, {\mathbf{Z}})\) extends to \(\tilde A \in \operatorname{GL}(2, {\mathbf{R}})\), a linear self-homeomorphism of the plane that is orientation-preserving.

• So \(\tilde A\) descends to a well-defined map \begin{align*} \psi_{\tilde A}: \Sigma_2 {\circlearrowleft}= {\mathbf{R}}^2 / {\mathbf{Z}}^2 {\circlearrowleft} \end{align*} which is still a linear self-homeomorphism.

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• There is a correspondence \begin{align*} \left\{{\substack{\text{Primitive curves in } \\ \pi_1(\Sigma_2) \cong {\mathbf{Z}}^2}}\right\} \iff \left\{{\substack{\text{Primitive vectors in }{\mathbf{Z}}^2}}\right\} \iff \left\{{\substack{\text{Oriented simple closed} \\\text{curves in } \Sigma_2}}\right\}/\text{\tiny homotopy} ,\end{align*} 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.

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Injectivity

• Useful fact: \(\Sigma_2 \simeq K({\mathbf{Z}}^2, 1)\).

• Thus there is a correspondence \begin{align*} \left\{{\substack{\text{Homotopy classes of } \\ \text{maps }\Sigma_2 {\circlearrowleft}}}\right\} \iff \left\{{\substack{\text{Group morphisms } {\mathbf{Z}}^2{\circlearrowleft}}}\right\} .\end{align*}

• Claim: any element \(f\in {\operatorname{MCG}}(\Sigma_2)\) has a representative \(\phi\) which fixes any given basepoint

• So if \(f\in \ker \sigma\), then \(f\simeq \phi \simeq \operatorname{id}\) are homotopic, so \(\ker \sigma = 1\).