Abelian Covers of Surfaces and the Homology of the Level L Mapping Class Group

Putman, Andrew

doi:10.1142/s179352531100060x

Cited by 3 publications

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“…which is the subgroup of homomorphisms that induce the identity on H 1 (Σ; Z/LZ) [56,55]. Other examples involve mixtures of the lower central and derived subgroups of F .…”

Section: Introductionmentioning

confidence: 99%

“…Yet another important class of examples are the mod L versions of these subgroups. In particular, if L ∈ Z + and H = x∈F [F, F ]x L , then J(H) is the level L subgroup of M, sometimes denoted Mod(L), which is the subgroup of homomorphisms that induce the identity on H 1 (Σ; Z/LZ) [56,55]. Other examples involve mixtures of the lower central and derived subgroups of F .…”

Section: Introductionmentioning

confidence: 99%

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Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders

Cochran

Harvey

Horn

2011

International Mathematics Research Notices

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We define families of invariants for elements of the mapping class group of Σ, a compact-orientable surface. For any characteristic subgroup H π 1 (Σ), let J(H ) denote the subgroup of mapping classes that induce the identity on π 1 (Σ)/H . To any unitary representation ψ of π 1 (Σ)/H , we associate a higher-order ρ ψ -invariant and a signature 2-cocycle σ ψ . These signature cocycles are shown to be generalizations of the Meyer cocycle. In particular, each ρ ψ is a quasimorphism and each σ ψ is a bounded 2-cocycle on J(H ). In one of the simplest nontrivial cases, by varying ψ, we exhibit infinite families of linearly independent quasimorphisms and signature cocycles. We show that the ρ ψ restrict to homomorphisms on certain interesting subgroups. Many of these invariants extend naturally to the full mapping class group and some extend to the monoid of homology cylinders based on Σ.

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“…which is the subgroup of homomorphisms that induce the identity on H 1 (Σ; Z/LZ) [56,55]. Other examples involve mixtures of the lower central and derived subgroups of F .…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders

Cochran

Harvey

Horn

2011

International Mathematics Research Notices

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“…To solve this problem, we must perform a group cohomological computation to show that the stabilizers of any two simple closed nonseparating curves give the same "chunk" of homology, and thus that the map H 2 ((Mod g (L)) γ ; Q) → H 2 (Mod g (L); Q) is surjective. The key to this computation is a certain vanishing result of the author ( [35]; see Lemma 7.2 below) for the twisted first homology groups of Mod g (L) with coefficients in the homology groups of abelian covers of the surface.…”

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confidence: 99%

The second rational homology group of the moduli space of curves with level structures

Putman

2012

Advances in Mathematics

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Let Γ be a finite-index subgroup of the mapping class group of a closed genus g surface that contains the Torelli group. For instance, Γ can be the level L subgroup or the spin mapping class group. We show that H 2 (Γ; Q) ∼ = Q for g ≥ 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 Q. We also prove analogous results for surface with punctures and boundary components. IntroductionLet Σ g be a closed oriented genus g surface and let Mod g be its mapping class group, that is, the group of isotopy classes of orientation preserving homeomorphisms of Σ g (see [11,21] for surveys about Mod g ). Tremendous progress has been made over the last 40 years in understanding the homology of Mod g , culminating in the groundbreaking work of Madsen-Weiss [27], who identified H * (Mod g ; Q) in a stable range. However, little is known about the homology of finite-index subgroups of Mod g , or equivalently about the homology of finite covers of the moduli space of curves.Denote by I g the Torelli group, that is, the kernel of the representation Mod g → Sp 2g (Z) arising from the action of Mod g on H 1 (Σ g ; Z). Our main theorem is as follows. It answers in the affirmative a question of Hain [14] which has since appeared on problem lists of Farb [10, Conjecture 5.24] and Penner [30, Problem 11]. Theorem 1.1 (Rational H 2 of finite-index subgroups, closed case). For g ≥ 5, let Γ be a finite index subgroup of Mod g such that I g < Γ. Then H 2 (Γ; Q) ∼ = Q.We also have an analogous result for surfaces with punctures and boundary components; see Theorem 2.1 below.Examples. The subgroups of Mod g to which Theorem 1.1 applies are exactly the pullback to Mod g of finite-index subgroups of Sp 2g (Z). Two key examples are as follows.Example (Level L subgroup). For an integer L ≥ 2, the level L subgroup Mod g (L) of Mod g is the group of mapping classes that act trivially on H 1 (Σ g ; Z/L). The image of Mod g (L) in Sp 2g (Z) is the kernel of the natural map Sp 2g (Z) → Sp 2g (Z/L). This group of matrices, denoted Sp 2g (Z, L), is known as the level L subgroup of Sp 2g (Z).Example (Spin subgroup). Letting U Σ g be the unit tangent bundle of Σ g , a spin structure on Σ g is an element ω ∈ H 1 (U Σ g ; Z/2) such that ω(ℓ) = 1, where ℓ ∈ H 1 (U Σ g ; Z/2) is the loop around the fiber. If Σ g is given the structure of a Riemann surface, then spin structures on Σ g can be identified with theta characteristics, i.e. square roots of the canonical bundle [1, Proposition 3.2]. More topologically, Johnson [23] showed that spin structures on Σ g can be identified with Z/2-valued quadratic formsω on H 1 (Σ g ; Z/2), i.e. functionsω :Here i(·, ·) is the algebraic intersection pairing. Such quadratic forms are classified up to isomorphism by their Z/2-valued Arf invariant. The group Mod g acts on the set of spin structures on Σ g , and this action is transitive on the set of spin structures of a fixed Arf invariant. If ω is a spin structure on Σ g , then the stabilizer...

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Simple closed curves in stable covers of surfaces

Salter

2023

Trans. Amer. Math. Soc.

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Let f : X → Y f: X \to Y be a regular covering of a surface Y Y of finite type with nonempty boundary, with finitely-generated (possibly infinite) deck group G G . We give necessary and sufficient conditions for an integral homology class on X X to admit a representative as a connected component of the preimage of a nonseparating simple closed curve on Y Y , possibly after passing to a “stabilization”, i.e. a G G -equivariant embedding of covering spaces X ↪ X + X \hookrightarrow X^+ .

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Abelian Covers of Surfaces and the Homology of the Level L Mapping Class Group

Cited by 3 publications

References 21 publications

Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders

Higher-Order Signature Cocycles for Subgroups of Mapping Class Groups and Homology Cylinders

The second rational homology group of the moduli space of curves with level structures

Simple closed curves in stable covers of surfaces

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