[Paper Review] Cohomology of mapping class groups and the abelian moduli space

Jørgen Ellegaard Andersen, Rasmus Villemoes|ArXiv.org|Mar 24, 2009

Algebraic Geometry and Number Theory12 references3 citations

TL;DR

This paper proves that the first cohomology groups $ H^1(\Gamma, L^2(M)) $ and $ H^1(\Gamma, C^\infty(M)) $ vanish for the mapping class group $ \Gamma $ of a genus $ g \geq 3 $ surface, where $ M $ is the abelian moduli space $ \operatorname{Hom}(H_1(\Sigma), \mathrm{U}(1)) $. The proof leverages the property (T) of $ \mathrm{Sp}(2g,\mathbb{Z}) $, the Hochschild-Serre spectral sequence, and a key vanishing condition on Dehn twist projections in unitary representations, establishing that no nontrivial 1-cocycles exist in these coefficient modules.

ABSTRACT

We consider a surface $Σ$ of genus $g \geq 3$, either closed or with exactly one puncture. The mapping class group $Γ$ of $Σ$ acts symplectically on the abelian moduli space $M = \Hom(π_1(Σ), U(1)) = \Hom(H_1(Σ),U(1))$, and hence both $L^2(M)$ and $C^\infty(M)$ are modules over $Γ$. In this paper, we prove that both the cohomology groups $H^1(Γ, L^2(M))$ and $H^1(Γ, C^\infty(M))$ vanish.

Motivation & Objective

• To establish the vanishing of the first cohomology group $ H^1(\Gamma, L^2(M)) $ for the mapping class group $ \Gamma $ acting on the $ L^2 $-functions of the abelian moduli space $ M $.
• To extend this result to the space of smooth functions, proving $ H^1(\Gamma, C^\infty(M)) = 0 $.
• To provide evidence against nontrivial 1-cocycles in infinite-dimensional unitary representations of $ \Gamma $, relevant to Kazhdan's property (T).
• To generalize a vanishing result for unitary representations by proving a necessary condition involving orthogonal projections onto fixed subspaces of Dehn twists.

Proposed method

• Use the Hochschild-Serre spectral sequence to relate $ H^1(\Gamma, \ell^2(H')) $ to $ H^1(\mathrm{Sp}(2g,\mathbb{Z}), \ell^2(H')) $ and $ H^1(\mathcal{T}, \ell^2(H'))^\Gamma $, where $ \mathcal{T} $ is the Torelli group.
• Prove that the restriction of any cocycle to the Torelli group is zero by exploiting the invariance of bounding pair maps under Dehn twists and the projection condition $ p_\gamma u(\tau_\gamma) = 0 $.
• Apply the fact that $ \mathrm{Sp}(2g,\mathbb{Z}) $ has property (T), which implies $ H^1(\mathrm{Sp}(2g,\mathbb{Z}), \ell^2(H')) = 0 $.
• Establish an isomorphism between $ \ell^2(H') $ and $ L^2_0(M) $ as $ \Gamma $-modules via an orthonormal basis indexed by $ H' $, the dual of $ H_1(\Sigma, \mathbb{Z}) $.
• For smooth functions, use the embedding $ C^\infty_0(M) \to \ell^2(H') $ and show that the lift of a cocycle to $ \ell^2(H') $ must be smooth by estimating decay of Fourier coefficients using telescoping sums along Dehn twist orbits.
• Use the strictly increasing norm growth of $ \tau_j^r m $ under Dehn twists to bound $ |f_m| $ via summation of differences $ g_{m,j}^\pm $, leading to $ |f_m| \leq G_{k+1}/(k|m|^k) $.

Experimental results

Research questions

• RQ1 Does the first cohomology group $ H^1(\Gamma, L^2(M)) $ vanish for the mapping class group $ \Gamma $ acting on the abelian moduli space $ M $?
• RQ2 Does the first cohomology group $ H^1(\Gamma, C^\infty(M)) $ vanish for the same action?
• RQ3 Can the vanishing of $ H^1(\Gamma, V) $ be established for unitary representations $ V $ via a necessary condition involving Dehn twist projections?
• RQ4 Is the Torelli group's action on $ \ell^2(H') $ such that any cocycle restricts to zero, enabling reduction to $ \mathrm{Sp}(2g,\mathbb{Z}) $?
• RQ5 Can the smoothness of a function in $ \ell^2(H') $ be deduced from the smoothness of its differences under Dehn twists?

Key findings

• The first cohomology group $ H^1(\Gamma, L^2(M)) $ vanishes, as shown by proving $ H^1(\mathrm{Sp}(2g,\mathbb{Z}), \ell^2(H')) = 0 $ and using the Hochschild-Serre sequence to reduce the problem to the Torelli group, whose cocycles vanish.
• The first cohomology group $ H^1(\Gamma, C^\infty(M)) $ vanishes, by lifting a smooth cocycle to $ \ell^2(H') $, showing the corresponding function $ f \in \ell^2(H') $ must be smooth via decay estimates on Fourier coefficients.
• The key technical tool is the identity $ p_\gamma u(\tau_\gamma) = 0 $ for any cocycle $ u $, where $ p_\gamma $ is the orthogonal projection onto the fixed subspace of a Dehn twist $ \tau_\gamma $, which holds for all unitary representations.
• The Torelli group is generated by genus-1 bounding pair maps, and any cocycle vanishes on these generators due to invariance under conjugation and the projection condition.
• The isomorphism between $ \ell^2(H') $ and $ L^2_0(M) $ as $ \Gamma $-modules allows transfer of cohomological results from $ \ell^2(H') $ to $ L^2(M) $.
• The decay estimate $ |f_m| \leq G_{k+1}/(k|m|^k) $ for $ f \in \ell^2(H') $, derived from telescoping sums along Dehn twist orbits, ensures that $ f $ is smooth if its differences under Dehn twists are smooth.