On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients

Ivanov, N. V.

doi:10.1090/conm/150/01290

Cited by 52 publications

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“…Our inspiration for this theorem was the work of Ivanov [8] which gave the first stability results for the homology of mapping class groups with certain kinds of twisted coefficients. The following is a generalization of his results.…”

Section: Figure 2 T # Smentioning

confidence: 99%

“…The gluing procedure is completely analogous to the gluing in of the surface T described above. In section 1 we will define the notion of a coefficient system V = {V g,n }, generalizing the notion defined by Ivanov [8]. However for the purposes of the statement of the following theorem, its main property is the following.…”

Section: Figure 2 T # Smentioning

confidence: 99%

“…There is a morphism for each ambient isotopy class of embedding, e : We remark that this category is a slight variation of Ivanov's category of decorated surfaces in [8]. Notice that each C g,n ⊂ C is a subcategory.…”

Section: The Homology Of Mapping Class Groups With Twisted Coefficientsmentioning

confidence: 99%

“…In order to describe these operations more precisely, we use a graphical technique of Ivanov [8]. Namely, since these operations are all defined on surfaces with at least one boundary component, then we can identify ∂ 0 F with a rectangle using the parameterization, and we can picture F as a rectangular disk with handles attached, and disks removed from the interior.…”

Section: The Homology Of Mapping Class Groups With Twisted Coefficientsmentioning

confidence: 99%

“…Σ 1,0 (e) is defined similarly. See [8], section 2.5 for details of this construction. We now define the notion of a coefficient system.…”

Section: The Homology Of Mapping Class Groups With Twisted Coefficientsmentioning

confidence: 99%

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Surfaces in a background space and the homology of mapping class groups

Cohen¹,

Madsen²

2009

Algebraic Geometry

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Abstract. In this paper we study the topology of the space of Riemann surfaces in a simply connected space X, Sg,n(X, γ). This is the space consisting of triples, (Fg,n, φ, f ), where Fg,n is a Riemann surface of genus g and n-boundary components, φ is a parameterization of the boundary, ∂Fg,n, and f : Fg,n → X is a continuous map that satisfies a boundary condition γ. We prove three theorems about these spaces. Our main theorem is the identification of the stable homology type of the space S∞,n(X; γ), defined to be the limit as the genus g gets large, of the spaces Sg,n(X; γ). Our result about this stable topology is a parameterized version of the theorem of Madsen and Weiss proving a generalization of the Mumford conjecture on the stable cohomology of mapping class groups. Our second result describes a stable range in which the homology of Sg,n(X; γ) is isomorphic to the stable homology. Finally we prove a stability theorem about the homology of mapping class groups with certain families of twisted coefficients. The second and third theorems are generalizations of stability theorems of Harer and Ivanov.

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Section: Figure 2 T # Smentioning

confidence: 99%

Section: Figure 2 T # Smentioning

confidence: 99%

Section: The Homology Of Mapping Class Groups With Twisted Coefficientsmentioning

confidence: 99%

Section: The Homology Of Mapping Class Groups With Twisted Coefficientsmentioning

confidence: 99%

“…Σ 1,0 (e) is defined similarly. See [8], section 2.5 for details of this construction. We now define the notion of a coefficient system.…”

Section: The Homology Of Mapping Class Groups With Twisted Coefficientsmentioning

confidence: 99%

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Surfaces in a background space and the homology of mapping class groups

Cohen¹,

Madsen²

2009

Algebraic Geometry

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An orbifold partition of $$\bar M_g^n$$

Pikaart

1995

The Moduli Space of Curves

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Cohomology of automorphism groups of free groups with twisted coefficients

Randal-Williams

2017

Sel. Math. New Ser.

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We compute the groups H * (Aut(F n ); M) and H * (Out(F n ); M) in a stable range, where M is obtained by applying a Schur functor to H Q or H * Q , respectively the first rational homology and cohomology of F n . The answer may be described in terms of stable multiplicities of irreducibles in the plethysm Sym k • Sym l of symmetric powers. We also compute the stable integral cohomology groups of Aut(F n ) with coefficients in H or H * .

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On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients

Cited by 52 publications

References 0 publications

Surfaces in a background space and the homology of mapping class groups

Surfaces in a background space and the homology of mapping class groups

An orbifold partition of $$\bar M_g^n$$

Cohomology of automorphism groups of free groups with twisted coefficients

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