Abstract: We prove that in a stable range, the rational cohomology of the moduli space of curves with level structures is the same as that of the ordinary moduli space of curves: a polynomial algebra in the Miller-Morita-Mumford classes.

Abstract: For surface groups and right-angled Artin groups, we prove lower bounds on the shortest word in the generators representing a nontrivial element of the k^th term of the lower central series.

Abstract: We give two proofs that appropriately defined congruence subgroups of the mapping class group of a surface with punctures/boundary have enormous amounts of rational cohomology in their virtual cohomological dimension. In particular we give bounds that are super-exponential in each of three variables: number of punctures, number of boundary components, and genus, generalizing work of Fullarton-Putman. Along the way, we give a simplified account of a theorem of Harer explaining how to relate the homotopy type of the curve complex of a multiply-punctured surface to the curve complex of a once-punctured surface through a process that can be viewed as an analogue of a Birman exact sequence for curve complexes. As an application, we prove upper and lower bounds on the coherent cohomological dimension of the moduli space of curves with marked points. For g $\leq$ 5, we compute this coherent cohomological dimension for any number of marked points. In contrast to our bounds on cohomology, when the surface has n$\geq$ 1 marked points, these bounds turn out to be independent of n, and depend only on the genus.

Abstract: Putman and Wieland conjectured that if $\widetilde{\Sigma} \rightarrow \Sigma$ is a finite branched cover between closed oriented surfaces of sufficiently high genus, then the orbits of all nonzero elements of H₁($\widetilde{\Sigma}$;$\mathbb{Q}$) under the action of lifts to $\widetilde{\Sigma}$ of mapping classes on $\Sigma$ are infinite. We prove that this holds if H₁($\widetilde{\Sigma}$;$\mathbb{Q}$) is generated the homology classes of lifts of simple closed curves on $\Sigma$. We also prove that the subspace of H₁($\widetilde{\Sigma}$;$\mathbb{Q}$) spanned by such lifts is a symplectic subspace. Finally, simple closed curves lie on subsurfaces homeomorphic to 2-holed spheres, and we prove that H₁($\widetilde{\Sigma}$;$\mathbb{Q}$) is generated by the homology classes of lifts of loops on $\Sigma$ lying on subsurfaces homeomorphic to 3-holed spheres.

Abstract: We introduce a new method for proving twisted homological stability, and use it to prove such results for symmetric groups and general linear groups. In addition to sometimes slightly improving the stable range given by the traditional method (due to Dwyer), it is easier to adapt to nonstandard situations. As an illustration of this, we generalize to GL_n of many rings R a theorem of Borel which says that passing from GL_n of a number ring to a finite-index subgroup does not change the rational cohomology. Charney proved this generalization for trivial coefficients, and we extend it to twisted coefficients.

Abstract: Answering a question of Hatcher-Vogtmann, we prove that the top homology group of the free factor complex is not the dualizing module for Aut(F_n), at least for n=5.

Abstract: We prove a homological stability theorem for the subgroup of the mapping class group acting as the identity on some fixed portion of the first homology group of the surface. We also prove a similar theorem for the subgroup of the mapping class group preserving a fixed map from the fundamental group to a finite group, which can be viewed as a mapping class group version of a theorem of Ellenberg-Venkatesh-Westerland about braid groups. These results require studying various simplicial complexes formed by subsurfaces of the surface, generalizing work of Hatcher-Vogtmann.

Abstract: We prove that the Steinberg representation of a connected reductive group over an infinite field is irreducible. For finite fields, this is a classical theorem of Steinberg and Curtis.

Abstract: Let M_n be the connect sum of n copies of S² $\times$ S¹. A classical theorem of Laudenbach says that the mapping class group Mod(M_n) is an extension of Out(F_n) by a group ($\mathbb{Z}$/2)ⁿ generated by sphere twists. We prove that this extension splits, so Mod(M_n) is the semidirect product of Out(F_n) by ($\mathbb{Z}$/2)ⁿ, which Out(F_n) acts on via the dual of the natural surjection Out(F_n) $\rightarrow$ GL_n($\mathbb{Z}$/2). Our splitting takes Out(F_n) to the subgroup of Mod(M_n) consisting of mapping classes that fix the homotopy class of a trivialization of the tangent bundle of M_n. Our techniques also simplify various aspects of Laudenbach's original proof, including the identification of the twist subgroup with ($\mathbb{Z}$/2)ⁿ.

Abstract: For a finite ring R, not necessarily commutative, we prove that the category of VIC(R)-modules over a left Noetherian ring k is locally Noetherian, generalizing a theorem of the authors that dealt with commutative R. As an application, we prove a very general twisted homology stability for GL_n(R) with R a finite noncommutative ring.

Abstract: A beautifully simple free generating set for the commutator subgroup of a free group was constructed by Tomaszewski. We give a new geometric proof of his theorem, and show how to give a similar free generating set for the commutator subgroup of a surface group. We also give a simple representation-theoretic description of the structure of the abelianizations of these commutator subgroups and calculate their homology.

Abstract: We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group is finitely generated in a linear stable range. This was originally proved for the second terms by Ershov and He.

Abstract: For a number ring $\mathcal{O}$, Borel and Serre proved that SL_n($\mathcal{O}$) is a virtual duality group whose dualizing module is the Steinberg module. They also proved that GL_n($\mathcal{O}$) is a virtual duality group. In contrast to SL_n($\mathcal{O}$), we prove that the dualizing module of GL_n($\mathcal{O}$) is sometimes the Steinberg module, but sometimes instead is a variant that takes into account a sort of orientation. Using this, we obtain vanishing and nonvanishing theorems for the cohomology of GL_n($\mathcal{O}$) in its virtual cohomological dimension.

Abstract: Let $\Gamma$_n(p) be the level-p congruence subgroup of SL_n($\mathbb{Z}$). Borel-Serre proved that the cohomology of $\Gamma$_n(p) vanishes above degree $\binom{n}{2}$. We study the cohomology in this top degree $\binom{n}{2}$. Let $\mathcal{T}$_n($\mathbb{Q}$) denote the Tits building of SL_n($\mathbb{Q}$). Lee-Szczarba conjectured that $H$^{$\binom{n}{2}$}($\Gamma$_n(p)) is isomorphic to $\widetilde{H}$_n-2($\mathcal{T}$_n($\mathbb{Q}$)/$\Gamma$_n(p)) and proved that this holds for p=3. We partially prove and partially disprove this conjecture by showing that a natural map $H$^{$\binom{n}{2}$}($\Gamma$_n(p)) $\rightarrow$ $\widetilde{H}$_n-2($\mathcal{T}$_n($\mathbb{Q}$)/$\Gamma$_n(p)) is always surjective, but is only injective for p$\leq$5. In particular, we completely calculate $H$^{$\binom{n}{2}$}($\Gamma$_n(5)) and improve known lower bounds for the ranks of $H$^{$\binom{n}{2}$}($\Gamma$_n(p)) for p$\geq$5.

Abstract: We give a new proof of a theorem of D. Calegari that says that the Cayley graph of a surface group with respect to any generating set lying in finitely many mapping class group orbits has infinite diameter. This applies, for instance, to the generating set consisting of all simple closed curves.

Abstract: Let R be a (not necessarily commutative) ring whose additive group is finitely generated and let U_n(R) $\subset$ GL_n(R) be the group of upper-triangular unipotent matrices over R. We study how the homology groups of U_n(R) vary with n from the point of view of representation stability. Our main theorem asserts that if for each n we have representations M_n of U_n(R) over a ring k that are appropriately compatible and satisfy suitable finiteness hypotheses, then the rule [n] $\mapsto$ $\widetilde{H}$_i(U_n(R),M_n) defines a finitely generated OI-module. As a consequence, if k is a field then dim $\widetilde{H}$_i(U_n(R),k) is eventually equal to a polynomial in n. We also prove similar results for the Iwahori subgroups of GL_n$\mathcal{O}$ for number rings $\mathcal{O}$.

Abstract: We prove that the moduli space of curves with level structures has an enormous amount of rational cohomology in its cohomological dimension. As an application, we prove that the coherent cohomological dimension of the moduli space of curves is at least g-2. Well known conjectures of Looijenga would imply that this is sharp.

Abstract: We develop a theory of equivariant group presentations and relate them to the second homology group of a group. Our main application says that the second homology group of the Torelli subgroup of the mapping class group is finitely generated as an Sp_2g($\mathbb{Z}$)-module.

Abstract: We construct examples of finite covers of punctured surfaces where the first rational homology is not spanned by lifts of simple closed curves. More generally, for any set $\mathcal{O} \subset$ F_n consisting of the union of finitely many Aut(F_n)-orbits, we construct finite-index normal subgroups of F_n whose first rational homology is not spanned by powers of elements of $\mathcal{O}$. These examples answer questions of Farb-Hensel, Looijenga, and Marché. We also show that the normal subgroup of Out(F_n) generated by k^th powers of transvections is often infinite index. Finally, for any set $\mathcal{O} \subset$ F_n consisting of the union of finitely many Aut(F_n)-orbits, we construct integral linear representations of free groups with infinite image which maps all elements of $\mathcal{O}$ to torsion elements.

Abstract: We prove a new structural result for the spherical Tits building attached to SL_n K for many number fields K, and more generally for the fraction fields of many Dedekind domains $\mathcal{O}$: the Steinberg module St_n(K) is generated by integral apartments if and only if the ideal class group cl($\mathcal{O}$) is trivial. We deduce this integrality by proving that the complex of partial bases of $\mathcal{O}$ⁿ is Cohen-Macaulay. We apply this to prove new vanishing and nonvanishing results for H^$\nu$(SL_n$\mathcal{O}$_K;$\mathbb{Q}$), where $\mathcal{O}$_K is the ring of integers in a number field and $\nu$ is the virtual cohomological dimension of SL_n$\mathcal{O}$_K. The (non)vanishing depends on the (non)triviality of the class group of $\mathcal{O}$_K. We also obtain a vanishing theorem for the cohomology H^$\nu$(SL_n$\mathcal{O}$_K;V) with twisted coefficients V.

Abstract: For a field k, we prove that the i^th homology of the groups GL_n(k), SL_n(k), Sp_2n(k), SO_n,n(k), and SO_n,n+1(k) with coefficients in their Steinberg representations vanish for n $\geq$ 2i+2.

Abstract: We give a new proof of a celebrated theorem of Dennis Johnson that asserts that the kernel of the Johnson homomorphism on the Torelli subgroup of the mapping class group is generated by separating twists. In fact, we prove a more general result that also applies to "subsurface Torelli groups". Using this, we extend Johnson's calculation of the rational abelianization of the Torelli group not only to the subsurface Torelli groups, but also to finite-index subgroups of the Torelli group that contain the kernel of the Johnson homomorphism.

Abstract: Let IA_n be the Torelli subgroup of Aut(F_n). We give an explicit finite set of generators for H₂(IA_n) as a GL_n($\mathbb{Z}$)-module. Corollaries include a version of surjective representation stability for H₂(IA_n), the vanishing of the GL_n($\mathbb{Z}$)-coinvariants of H₂(IA_n), and the vanishing of the second rational homology group of the level l congruence subgroup of Aut(F_n). Our generating set is derived from a new group presentation for IA_n which is infinite but which has a simple recursive form.

Abstract: We prove that H^{$\binom{n}{2}$-1}(SL_n $\mathbb{Z}$;$\mathbb{Q}$) = 0, where $\binom{n}{2}$ is the cohomological dimension of SL_n$\mathbb{Z}$, and similarly for GL_n$\mathbb{Z}$. We also prove analogous vanishing theorems for cohomology with coefficients in a rational representation of the algebraic group GL_n. These theorems are derived from a presentation of the Steinberg module for SL_n$\mathbb{Z}$ whose generators are integral apartment classes, generalizing Manin's presentation for the Steinberg module of SL₂$\mathbb{Z}$. This presentation was originally constructed by Bykovskii. We give a new topological proof of it.

Abstract: We construct analogues of FI-modules where the role of the symmetric group is played by the general linear groups and the symplectic groups over finite rings and prove basic structural properties such as Noetherianity. Applications include a proof of Schwartz's Artinian conjecture in the generic representation theory of finite fields and representation-theoretic versions of homological stability for congruence subgroups of the general linear group, the automorphism group of a free group, the symplectic group, and the mapping class group.

Abstract: We develop an analogue of the Birman exact sequence for the Torelli subgroup of Aut(F_n). This builds on earlier work of the authors who studied an analogue of the Birman exact sequence for the entire group Aut(F_n). These results play an important role in the authors' recent work on the second homology group of the Torelli group.

Abstract: Answering a question of Farb-Leininger-Margalit, we give explicit lower bounds for the dilatations of pseudo-Anosov mapping classes lying in the d^th term of the Johnson filtration of the mapping class group.

Abstract: The homology groups of many natural sequences of groups $\{G_n\}_{n=1}^{\infty}$ (e.g. general linear groups, mapping class groups, etc.) stabilize as $n \rightarrow \infty$. Indeed, there is a well-known machine for proving such results that goes back to early work of Quillen. Church and Farb discovered that many sequences of groups whose homology groups do not stabilize in the classical sense actually stabilize in some sense as representations. They called this phenomena {\em representation stability}. We prove that the homology groups of congruence subgroups of GL_n(R) (for almost any reasonable ring R) satisfy a strong version of representation stability that we call central stability. The definition of central stability is very different from Church-Farb's definition of representation stability (it is defined via a universal property), but we prove that it implies representation stability. Our main tool is a new machine for proving central stability that is analogous to the classical homological stability machine.

Abstract: For k$\geq$1, let $\mathcal{I}_g^1$(k) be the k^th term in the Johnson filtration of the mapping class group of a genus g surface with one boundary component. We prove that for all k$\geq$1, there exists some G_k such that $\mathcal{I}_g^1$(k) is generated by elements which are supported on subsurfaces whose genus is at most G_k. We also prove similar theorems for the Johnson filtration of Aut(F_n) and for certain mod-p analogues of the Johnson filtrations of both the mapping class group and of Aut(F_n). The main tools used in the proofs are the related theories of FI-modules (due to the first author together with Ellenberg and Farb) and central stability (due to the second author), both of which concern the representation theory of the symmetric groups over $\mathbb{Z}$.

Abstract: We prove that the hyperelliptic Torelli group is generated by Dehn twists about separating curves that are preserved by the hyperelliptic involution. This verifies a conjecture of Hain. The hyperelliptic Torelli group can be identified with the kernel of the Burau representation evaluated at t=-1 and also the fundamental group of the branch locus of the period mapping. One application is that each component in Torelli space of the locus of hyperelliptic curves becomes simply-connected when curves of compact type are added.

Abstract: In this paper we give a conjectural picture of a large piece of the unstable rational cohomology of SL_n($\mathbb{Z}$), of mapping class groups, and of Aut(F_n).

Abstract: A well-known conjecture asserts that the mapping class group of a surface (possibly with punctures/boundary) does not virtually surject onto $\mathbb{Z}$ if the genus of the surface is large. We prove that if this conjecture holds for some genus, then it also holds for all larger genera. We also prove that if there is a counterexample to this conjecture, then there must be a counterexample of a particularly simple form. We prove these results by relating the conjecture to a family of linear representations of the mapping class group that we call the higher Prym representations. They generalize the classical symplectic representation.

Abstract: We study the complex of partial bases of a free group, which is an analogue for Aut(F_n) of the curve complex for the mapping class group. We prove that it is connected and simply connected, and we also prove that its quotient by the Torelli subgroup of Aut(F_n) is highly connected. Using these results, we give a new, topological proof of a theorem of Magnus that asserts that the Torelli subgroup of Aut(F_n) is finitely generated.

Abstract: 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".

Abstract: Let Mod_g be the mapping class group of a genus g$\geq$2 surface. The group Mod_g has virtual cohomological dimension 4g-5. In this note we use a theorem of Broaddus and the combinatorics of chord diagrams to prove that H^4g-5(Mod_g;$\mathbb{Q}$)=0.

Abstract: Proving a conjecture of Dennis Johnson, we show that the Torelli subgroup of the mapping class group has a finite generating set whose size grows cubically with respect to the genus of the surface. Our main tool is a new space (the handle graph of a surface) on which the Torelli group acts cocompactly.

Abstract: The Birman exact sequence describes the effect on the mapping class group of a surface with boundary of gluing discs to the boundary components. We construct an analogous exact sequence for the automorphism group of a free group. For the mapping class group, the kernel of the Birman exact sequence is a surface braid group. We prove that in the context of the automorphism group of a free group, the natural kernel is finitely generated. However, it is not finitely presentable; indeed, we prove that its second rational homology group has infinite rank by constructing an explicit infinite collection of linearly independent abelian cycles. We also determine the abelianization of our kernel and build a simple infinite presentation for it. The key to many of our proofs are several new generalizations of the Johnson homomorphisms.

Abstract: For 4$\nmid$L and g large, we calculate the integral Picard groups of the moduli spaces of curves and principally polarized abelian varieties with level L structures. In particular, we determine the divisibility properties of the standard line bundles over these moduli spaces and we calculate the second integral cohomology group of the level L subgroup of the mapping class group (in a previous paper, the author determined this rationally). This entails calculating the abelianization of the level L subgroup of the mapping class group, generalizing previous results of Perron, Sato, and the author. Finally, along the way we calculate the first homology group of the mod L symplectic group with coefficients in the adjoint representation.

Abstract: Let $\Gamma$ be a finite-index subgroup of the mapping class group of a closed genus g surface that contains the Torelli group. For instance, $\Gamma$ can be the level L subgroup or the spin mapping class group. We show that H₂($\Gamma$;$\mathbb{Q}$) $\cong$ $\mathbb{Q}$ for g$\geq$5. A corollary of this is that the rational Picard groups of the associated finite covers of the moduli space of curves are equal to $\mathbb{Q}$. We also prove analogous results for surface with punctures and boundary components.

Abstract: We calculate the first homology group of the mapping class group with coefficients in the first rational homology group of the universal abelian $\mathbb{Z}$/L$\mathbb{Z}$-cover of the surface. If the surface has one marked point, then the answer is $\mathbb{Q}$^$\tau$(L), where $\tau$(L) is the number of positive divisors of L. If the surface instead has one boundary component, then the answer is $\mathbb{Q}$. We also perform the same calculation for the level L subgroup of the mapping class group. Set H_L=H₁($\Sigma$_g;$\mathbb{Z}$/L$\mathbb{Z}$). If the surface has one marked point, then the answer is $\mathbb{Q}$[H_L], the rational group ring of H_L. If the surface instead has one boundary component, then the answer is $\mathbb{Q}$.

Abstract: Consider a group G acting nicely on a simply-connected simplicial complex X. Numerous classical methods exist for using this group action to produce a presentation for G. For the case that X/G is 2-connected, we give a new method that has the novelty that one does not have to identify a fundamental domain for the action. Indeed, the resulting presentation is canonical in the sense that no arbitrary choices need to be made. It can be viewed as a nonabelian analogue of a simple result in the study of equivariant homology.

Abstract: We prove that various subgroups of the mapping class group Mod($\Sigma$) of a surface $\Sigma$ are at least exponentially distorted. Examples include the Torelli group (answering a question of Hamenstädt), the "point-pushing" and surface braid subgroups, and the Lagrangian subgroup. Our techniques include a method to compute lower bounds on distortion via representation theory and an extension of Johnson theory to arbitrary subgroups of H₁($\Sigma$;$\mathbb{Z}$).

Abstract: We give various estimates of the minimal number of self-intersections of a nontrivial element of the k^th term of the lower central series and derived series of the fundamental group of a surface. As an application, we obtain a new topological proof of the fact that free groups and fundamental groups of closed surfaces are residually nilpotent. Along the way, we prove that a nontrivial element of the k^th term of the lower central series of a nonabelian free group has to have word length at least k in a free generating set.

Abstract: For some g$\geq$3, let $\Gamma$ be a finite index subgroup of the mapping class group of a genus g surface (possibly with boundary components and punctures). An old conjecture of Ivanov says that the abelianization of $\Gamma$ should be finite. In this note, we prove two theorems supporting this conjecture. For the first, let $T_x$ denote the Dehn twist about a simple closed curve x. For some n$\geq$1, we have $T_x^n \in \Gamma$. We prove that $T_x^n$ is torsion in the abelianization of $\Gamma$. Our second result shows that the abelianization of $\Gamma$ is finite if $\Gamma$ contains a "large chunk" (in a certain technical sense) of the Johnson kernel, that is, the subgroup of the mapping class group generated by twists about separating curves.

Abstract: In this paper, we construct a presentation of the Torelli subgroup of the mapping class group of a surface whose generators consist of the set of all "separating twists", all "bounding pair maps", and all "commutators of simply intersecting pairs" and whose relations all come from a short list of topological configurations of these generators on the surface. Aside from a few obvious ones, all of these relations come from a set of embeddings of groups derived from surface groups into the Torelli group. In the process of analyzing these embeddings, we derive a novel presentation for the fundamental group of a closed surface whose generating set is the set of all simple closed curves. Our main tool for analyzing the Torelli group is a new theorem which allows us to obtain presentations for groups acting on simplicial complexes without identifying a fundamental domain. We apply this to the action of the Torelli group on a variant of the complex of curves, yielding an inductive description of the Torelli group in terms of the the subgroups stabilizing simple closed curves on the surface.

Abstract: Let f be the gluing map of a Heegaard splitting of a 3-manifold W. The goal of this paper is to determine the information about W contained in the image of f under the symplectic representation of the mapping class group. We prove three main results. First, we show that the first homology group of the three manifold together with Seifert's linking form provides a complete set of stable invariants. Second, we give a complete, computable set of invariants for these linking forms. Third, we show that a slight augmentation of Birman's determinantal invariant for a Heegaard splitting gives a complete set of unstable invariants.

Abstract: This note is devoted to a trick which yields almost trivial proofs that certain complexes associated to topological surfaces are connected or simply connected. Applications include new proofs that the complexes of curves, separating curves, nonseparating curves, pants, and cut systems are all connected for genus g$\gg$0. We also prove that two new complexes are connected : one involves curves which split a genus 2g surface into two genus g pieces, and the other involves curves which are homologous to a fixed curve. The connectivity of the latter complex can be interpreted as saying the "homology" relation on the surface is (for g$\geq$3) generated by "embedded/disjoint homologies". We finally prove that the complex of separating curves is simply connected for g$\geq$4.

Abstract: We bound the value of the Casson invariant of any integral homology 3-sphere M by a constant times the distance-squared to the identity, measured in any word metric on the Torelli group $\mathcal{I}$, of the element of $\mathcal{I}$ associated to any Heegaard splitting of M. We construct examples which show this bound is asymptotically sharp.

Abstract: We introduce machinery to allow "cut-and-paste"-style inductive arguments in the Torelli subgroups of the mapping class groups. In the past these arguments have been problematic because the restriction of the Torelli groups to subsurfaces gives different groups depending on how the subsurfaces are embedded. We define a category TSur whose objects are surfaces together with a decoration restricting how they can be embedded into larger surfaces and whose morphisms are embeddings which respect the decoration. There is a natural "Torelli functor" on this category which extends the usual definition of the Torelli group on a closed surface. Additionally, we prove an analogue of the Birman exact sequence for the Torelli groups of surfaces with boundary and use the action of the Torelli groups on the complex of curves to find generators for the Torelli groups. For genus g$\geq$1 only twists about (certain) separating curves and bounding pairs are needed, while for genus g=0 a new type of generator (a "commutator of a simply intersecting pair") is needed. As a special case, our methods provide a new, more conceptual proof of the classical result of Birman-Powell which says that the Torelli groups on closed surfaces are generated by twists about separating curves and bounding pairs.

Abstract: Let $\Gamma$ be the fundamental group of a manifold modeled on three dimensional Sol geometry. We prove that $\Gamma$ has a finite index subgroup G which has a rational growth series with respect to a natural generating set. We do this by enumerating G by a regular language. However, in contrast to most earlier proofs of this sort our regular language is not a language of words in the generating set, but rather reflects a different geometric structure in G.

Abstract: We prove that every term of the lower central series and Johnson filtrations of the Torelli subgroups of the mapping class group and the automorphism group of a free group are finitely generated in a stable range. This was originally proved for the commutator subgroup by Ershov-He.

Historical note. After we distributed a preliminary version of this paper, we learned that Ershov had independently proved similar results. The three of us decided to write a joint paper, and in the course of writing this joint paper we managed to significantly improve the ranges of our theorems. The combined paper is here. The paper here is a permanent preprint and will not be submitted for publication.

Abstract: 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.

Historical note. I originally wrote this paper in March of 2008. Towards the end of that month, I gave a master class on the Torelli group at the University of Aarhus. That master class ended in a conference, and I had intended to speak about this paper at that conference. However, I learned that both Bernard Perron and Masatoshi Sato had proven similar theorems and intended to speak about them at the same conference! Sato was a graduate student and had actually proved somewhat better results (in particular, he could deal with L=2), so I decided not to publish this paper. Sato's work appeared here, and Perron's work was sketched in here. See my later paper here for results for $L$ not divisible by $4$. Dealing with the case where $L$ is divisible by $4$ is still open (as of August 2023).