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Register a new userAbelian covers of surfaces and the homology of the level L mapping class group
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Andrew Putman
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We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian $Z / L Z$-cover of the surface. If the surface has one marked point, then the answer is $Q^{tau(L)}$, where $tau(L)$ is the number of positive divisors of $L$. If the surface instead has one boundary component, then the answer is $Q$. We also perform the same calculation for the level $L$ subgroup of the mapping class group. Set $H_L = H_1(Sigma_g;Z/LZ)$. If the surface has one marked point, then the answer is $Q[H_L]$, the rational group ring of $H_L$. If the surface instead has one boundary component, then the answer is $Q$.
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We calculate the abelianizations of the level $L$ subgroup of the genus $g$ mapping class group and the level $L$ congruence subgroup of the $2g times 2g$ symplectic group for $L$ odd and $g geq 3$.
By the work of Harer, the reduced homology of the complex of curves is a fundamental cohomological object associated to all torsion free finite index subgroups of the mapping class group. We call this homology group the Steinberg module of the mapping class group. It was previously known that the curve complex has the homotopy type of a bouquet of spheres. Here, we give the first explicit homologically nontrivial sphere in the curve complex and show that under the action of the mapping class group, the orbit of this homology class generates the reduced homology of the curve complex.
In this paper, we study a series of $L^2$-torsion invariants from the viewpoint of the mapping class group of a surface. We establish some vanishing theorems for them. Moreover we explicitly calculate the first two invariants and compare them with hyperbolic volumes.
These are the lecture notes for my course at the 2011 Park City Mathematics Graduate Summer School. The first two lectures covered the basics of the Torelli group and the Johnson homomorphism, and the third and fourth lectures discussed the second cohomology group of the level p congruence subgroup of the mapping class group, following my papers The second rational homology group of the moduli space of curves with level structures and The Picard group of the moduli space of curves with level structures.
Let $text{Mod}(S_g)$ be the mapping class group of the closed orientable surface of genus $g geq 1$. For $k geq 2$, we consider the standard $k$-sheeted regular cover $p_k: S_{k(g-1)+1} to S_g$, and analyze the liftable mapping class group $text{LMod}_{p_k}(S_g)$ associated with the cover $p_k$. In particular, we show that $text{LMod}_{p_k}(S_g)$ is the stabilizer subgroup of $text{Mod}(S_g)$ with respect to a collection of vectors in $H_1(S_g,mathbb{Z}_k)$, and also derive a symplectic criterion for the liftability of a given mapping class under $p_k$. As an application of this criterion, we obtain a normal series of $text{LMod}_{p_k}(S_g)$, which generalizes of a well known normal series of congruence subgroups in $text{SL}(2,mathbb{Z})$. Among other applications, we describe a procedure for obtaining a finite generating set for $text{LMod}_{p_k}(S_g)$ and examine the liftability of certain finite-order and pseudo-Anosov mapping classes.
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