## The primary approximation to the cohomology of the moduli space of curves and cocycles for the Mumford-Morita-Miller classes (2001)

Citations: | 19 - 11 self |

### BibTeX

@MISC{Kawazumi01theprimary,

author = {Nariya Kawazumi and Shigeyuki Morita},

title = {The primary approximation to the cohomology of the moduli space of curves and cocycles for the Mumford-Morita-Miller classes},

year = {2001}

}

### OpenURL

### Abstract

### Citations

282 |
On the Vassiliev Knot Invariants, Topology 34
- Bar-Natan
- 1995
(Show Context)
Citation Context ...ll such a graph a linear chord diagram because if we close the straight line to obtain a circle, then we obtain the usual chord diagram which appears in the theory of Vassiliev’s knot invariants (see =-=[2]-=-). For any linear chord diagram C, let (7) aC ∈ (H ⊗2k Q )Sp be the invariant tensor defined by permuting the tensor product (ω0) ⊗k in such a way that the s-th part (ω0)s goes to (HQ)is ⊗ (HQ)js, whe... |

202 | Towards an enumerative geometry of the moduli space of curves
- Mumford
- 1983
(Show Context)
Citation Context ...H∗ (Mg,∗; Q). Let e ∈ H2 (Mg,∗; Q) be the universal Euler class of the tangent bundle along the fiber of the universal Σg-bundle and let ei ∈ H2i (Mg; Q) be the i-th MumfordMorita-Miller classes (see =-=[58]-=-[44][43]). Actually Mumford defined these classes at the level of the rational Chow algebras of the moduli spaces. Namely he defined classes KCg/Mg ∈A1 (Cg), κi ∈Ai (Mg) and called them tautological c... |

153 |
Feynman diagrams and low-dimensional topology. First European
- Kontsevich
- 1992
(Show Context)
Citation Context ... Sp-modules explicitly. To do so, we use a fundamental result of Weyl in the classical representation theory. It turned out that this is a specific case of Kontsevich’s general framework given in [37]=-=[38]-=-.6 NARIYA KAWAZUMI AND SHIGEYUKI MORITA We consider Λ3H as a natural Sp-submodule of H⊗3 by the injection Λ3H ⊂ H⊗3 defined by Λ 3 H ∋ a1 ∧ a2 ∧ a3 ↦−→ ∑ sgn σaσ(1) ⊗ aσ(2) ⊗ aσ(3) ∈ H ⊗3 σ∈S3 where ... |

141 |
Formal (non-)commutative symplectic geometry
- Kontsevich
- 1993
(Show Context)
Citation Context ...bove Sp-modules explicitly. To do so, we use a fundamental result of Weyl in the classical representation theory. It turned out that this is a specific case of Kontsevich’s general framework given in =-=[37]-=-[38].6 NARIYA KAWAZUMI AND SHIGEYUKI MORITA We consider Λ3H as a natural Sp-submodule of H⊗3 by the injection Λ3H ⊂ H⊗3 defined by Λ 3 H ∋ a1 ∧ a2 ∧ a3 ↦−→ ∑ sgn σaσ(1) ⊗ aσ(2) ⊗ aσ(3) ∈ H ⊗3 σ∈S3 wh... |

135 |
Stable real cohomology of arithmetic groups
- Borel
- 1974
(Show Context)
Citation Context ...ficiently large g. Then if we consider the spectral sequence of the rational cohomology of the morphism (37) and use the spectral sequence comparison theorem together with the Borel vanishing theorem =-=[3]-=-[4] concerning the stable cohomology of Sp(2g, Z) with non-trivial rational representations as coefficients, we can deduce for any k that H∗ (Mg,1/Mg,1(k); Q) stabilizes. More precisely, the E2 term o... |

123 |
Representation Theory, Graduate Texts
- Fulton, Harris
- 1991
(Show Context)
Citation Context ...reducible representation of the algebraic group Sp(HQ) corresponding to the Young diagram [13] (see [16][55][56] for details of symplectic representation theory related to the mapping class group and =-=[8]-=- for generalities). Extending earlier results in [52][53], the second author constructed in [56] a morphism (1) π1Σg −−−−→ [12]˜×HQ ⏐ ⏐ ↓ ↓ Mg,∗ ⏐ ↓ ρ2 −−−−→ (([1 ) 2 2 ] ⊕ [2 ] ˜×torelliΛ3 ) HQ ⋊ Sp(... |

115 |
Stability of the homology of the mapping class group of an orientable surface
- Harer
- 1985
(Show Context)
Citation Context ...ents in H does not admit the stability [46]. On the basis of Looijenga’s noteworthy idea [42] that the stable cohomology with symplectic coefficients is computed only from the Harer stability theorem =-=[21]-=- with14 NARIYA KAWAZUMI AND SHIGEYUKI MORITA trivial coefficients, the first author has deduced the following result: the stable integral cohomology of with coefficients in H⊗n is a free module over ... |

84 |
The second homology group of the mapping class group of an orientable surface
- Harer
(Show Context)
Citation Context ...rational cohomology of the mapping class group is isomorphic to the polynomial algebra generated by the classes ei. We mention that this conjecture has been proved to be true for degrees ≤ 4 by Harer =-=[20]-=-[22][24] and for degree 5 by Arbarello and Cornalba [1]. See also [31][32] for another evidence for the conjecture. In the above discussion, we can replace {Mg,1(k)}k by the other filtration {M′ g,1(k... |

77 |
Finite type invariants of integral homology 3-spheres, Jour. of Knot Theory and its Ramifications 5(1
- Ohtsuki
- 1996
(Show Context)
Citation Context ...le cohomology class for genus 3 moduli space given in [40] can be detected by the above homomorphism or not. Remark 14.2. Garoufalidis and Levine [9] proved that the filtration, introduced by Ohtsuki =-=[59]-=- (see also [60]), on the vector space generated by oriented homology 3spheres can be described by the lower central series of the Torelli group Ig. Moreover they showed in [10] a relation between the ... |

61 |
A conjectural description of the tautological ring of the moduli space of curves
- Faber
- 1999
(Show Context)
Citation Context ... (up to signs). The tautological algebras R∗ (Cg) and R∗ (Mg) of the moduli spaces are defined to be the subalgebras of the rational Chow algebras generated by the above tautological classes (see [41]=-=[7]-=-[18][25]). Similarly we define the tautological algebras R∗ (Mg,∗) and R∗ (Mg) of the mapping class groups to be the subalgebras of H∗ (Mg,∗; Q) and H∗ (Mg; Q) generated by the classes e, e1,e2, ···. ... |

61 | Infinitesimal presentations of the Torelli groups
- Hain
- 1997
(Show Context)
Citation Context ...HQ where ω0 ∈ Λ2HQ is the symplectic class. We denote the quotient Λ3HQ/HQ simply by UQ. UQ is an irreducible representation of the algebraic group Sp(HQ) corresponding to the Young diagram [13] (see =-=[16]-=-[55][56] for details of symplectic representation theory related to the mapping class group and [8] for generalities). Extending earlier results in [52][53], the second author constructed in [56] a mo... |

57 |
On a universal perturbative invariant of 3-manifolds, Topology 37
- Le, Murakami, et al.
- 1998
(Show Context)
Citation Context ...from J. Murakami that the restriction to Ig of the projective representation of the mapping class group associated to the universal perturbative 3-manifolds invariants due to Le, Murakami and Ohtsuki =-=[39]-=- is a unipotent representation after taking canonical truncations. Hence it should be described in terms of the Malcev completion of the Torelli group, though explicit description is far from being un... |

57 |
Characteristic classes of surface bundles
- Morita
- 1987
(Show Context)
Citation Context ...Mg,∗; Q). Let e ∈ H2 (Mg,∗; Q) be the universal Euler class of the tangent bundle along the fiber of the universal Σg-bundle and let ei ∈ H2i (Mg; Q) be the i-th MumfordMorita-Miller classes (see [58]=-=[44]-=-[43]). Actually Mumford defined these classes at the level of the rational Chow algebras of the moduli spaces. Namely he defined classes KCg/Mg ∈A1 (Cg), κi ∈Ai (Mg) and called them tautological class... |

55 |
Calculating cohomology groups of moduli spaces of curves via algebraic geometry
- Arbarello, Cornalba
- 1998
(Show Context)
Citation Context ...phic to the polynomial algebra generated by the classes ei. We mention that this conjecture has been proved to be true for degrees ≤ 4 by Harer [20][22][24] and for degree 5 by Arbarello and Cornalba =-=[1]-=-. See also [31][32] for another evidence for the conjecture. In the above discussion, we can replace {Mg,1(k)}k by the other filtration {M′ g,1(k)}k and we have the following result concerning it.COH... |

46 |
Abelian quotients of subgroups of the mapping class group of surfaces
- Morita
- 1993
(Show Context)
Citation Context ...tient Γk−1(π1Σ 0 g)/Γk(π1Σ 0 g)of π1Σ 0 g trivially. In particular, Mg,1(1) is the Torelli group Ig,1 ⊂Mg,1. We also have similar filtrations {Mg,∗(k)}k and {Mg(k)}k for Mg,∗ and Mg respectively (see =-=[51]-=-[53][55] for details). There is another filtration {M ′ g,1 (k)}k of Mg,1 where M ′ g,1 (1) = Ig,1 and for k ≥ 2, M ′ g,1(k) is defined to be the (k − 1)-th term Γk−1(Ig,1) in the lower central series... |

46 |
A polynomial invariant of rational homology 3-spheres, Invent
- Ohtsuki
- 1996
(Show Context)
Citation Context ...lass for genus 3 moduli space given in [40] can be detected by the above homomorphism or not. Remark 14.2. Garoufalidis and Levine [9] proved that the filtration, introduced by Ohtsuki [59] (see also =-=[60]-=-), on the vector space generated by oriented homology 3spheres can be described by the lower central series of the Torelli group Ig. Moreover they showed in [10] a relation between the finite type inv... |

45 |
An abelian quotient of the mapping class group Ig
- Johnson
- 1980
(Show Context)
Citation Context ...ively. Namely they are subgroups consisting of elements which act on H trivially. Then it is one of the fundamental results of Johnson that H1(Ig,∗; Q) ∼ = Λ 3 HQ, H1(Ig; Q) ∼ = UQ for any g ≥ 3 (see =-=[28]-=-[30]). In this section, we describe the Sp(2g, Q)-invariant part of the rational cohomology algebras H∗ (Λ3H; Q) =H ∗ (Λ3HQ) ∼ = Λ∗ (Λ3H ∗ Q ) and H∗ (U; Q) =H∗(UQ) ∼ = Λ∗U ∗ Q of the above Sp-modules... |

44 |
Casson’s Invariant for Homology 3–spheres and characteristic Classes of Surface Bundles I, Topology 28
- Morita
- 1989
(Show Context)
Citation Context ...−1(Ig,1) in the lower central series of the Torelli group Ig,1. Johnson [29] showed that M ′ g,1 (k) ⊂Mg,1(k) for all k and asked whether they coincide after tensoring with Q or not. It was proved in =-=[48]-=- that M ′ g,1(3) has an infinite index in Mg,1(3) and Hain [14] proved that the same is true for all k ≥ 3. There are similar filtrations {M ′ g,∗ (k)}k, {M ′ g (k)}k for Mg,∗, Mg and similar results ... |

43 |
The structure of the Torelli group. III. The abelianization of T , Topology 24
- Johnson
- 1985
(Show Context)
Citation Context ...y. Namely they are subgroups consisting of elements which act on H trivially. Then it is one of the fundamental results of Johnson that H1(Ig,∗; Q) ∼ = Λ 3 HQ, H1(Ig; Q) ∼ = UQ for any g ≥ 3 (see [28]=-=[30]-=-). In this section, we describe the Sp(2g, Q)-invariant part of the rational cohomology algebras H∗ (Λ3H; Q) =H ∗ (Λ3HQ) ∼ = Λ∗ (Λ3H ∗ Q ) and H∗ (U; Q) =H∗(UQ) ∼ = Λ∗U ∗ Q of the above Sp-modules exp... |

41 |
The homology of the mapping class group
- Miller
- 1986
(Show Context)
Citation Context ...; Q). Let e ∈ H2 (Mg,∗; Q) be the universal Euler class of the tangent bundle along the fiber of the universal Σg-bundle and let ei ∈ H2i (Mg; Q) be the i-th MumfordMorita-Miller classes (see [58][44]=-=[43]-=-). Actually Mumford defined these classes at the level of the rational Chow algebras of the moduli spaces. Namely he defined classes KCg/Mg ∈A1 (Cg), κi ∈Ai (Mg) and called them tautological classes. ... |

39 |
Cohomology of group extensions
- Hochschild, Serre
- 1953
(Show Context)
Citation Context ...ducts. For a group G and a (left) G-module M, we denote by C∗ (G; M) the standard normalized cochain complex of G with values in M and by Z∗ (G; M) the set of cocycles in C∗ (G; M). See, for example, =-=[26]-=-. Suppose a group Q acts on a group N, namely there is given a homomorphism of groups Q → Aut(N). Then a group law on the product set N × Q is defined by (n1,q1)(n2,q2) =(n1q1(n2),q1q2) , (n1,n2 ∈ N, ... |

38 |
A survey of the Torelli group
- Johnson
- 1983
(Show Context)
Citation Context ...her filtration {M ′ g,1 (k)}k of Mg,1 where M ′ g,1 (1) = Ig,1 and for k ≥ 2, M ′ g,1(k) is defined to be the (k − 1)-th term Γk−1(Ig,1) in the lower central series of the Torelli group Ig,1. Johnson =-=[29]-=- showed that M ′ g,1 (k) ⊂Mg,1(k) for all k and asked whether they coincide after tensoring with Q or not. It was proved in [48] that M ′ g,1(3) has an infinite index in Mg,1(3) and Hain [14] proved t... |

38 | Structure of the mapping class groups of surfaces: a survey and a prospect
- Morita
- 1998
(Show Context)
Citation Context ...here ω0 ∈ Λ2HQ is the symplectic class. We denote the quotient Λ3HQ/HQ simply by UQ. UQ is an irreducible representation of the algebraic group Sp(HQ) corresponding to the Young diagram [13] (see [16]=-=[55]-=-[56] for details of symplectic representation theory related to the mapping class group and [8] for generalities). Extending earlier results in [52][53], the second author constructed in [56] a morphi... |

37 | Torelli groups and geometry of moduli spaces of curves, from: “Current topics in complex algebraic geometry
- Hain
- 1992
(Show Context)
Citation Context ... be the homomorphism defined by forgetting the point p1 (resp. p0). Consider a diffeomorphism ψℓ :(Σg,p1)→(Σg,p0) given by sliding the point p1 along the curve ℓ. We introduce a homomorphism αℓ : M 2 =-=(15)-=- g−→Mg,∗ by the correspondence ϕ ↦→ (π(ϕ),ψℓ¯π(ϕ)ψℓ −1 ). Then we have a commutative diagram (16) M1 g,1 −−−−→ M2g ⏐ ⏐ π↓ π↓ αℓ −−−−→ Mg,∗ π ⏐ ↓ Mg,1 −−−−→ Mg,∗ Mg,∗. Consequently the following lemma ... |

32 | Mapping class groups and moduli spaces of curves
- Hain, Looijenga
- 1997
(Show Context)
Citation Context ...p to signs). The tautological algebras R∗ (Cg) and R∗ (Mg) of the moduli spaces are defined to be the subalgebras of the rational Chow algebras generated by the above tautological classes (see [41][7]=-=[18]-=-[25]). Similarly we define the tautological algebras R∗ (Mg,∗) and R∗ (Mg) of the mapping class groups to be the subalgebras of H∗ (Mg,∗; Q) and H∗ (Mg; Q) generated by the classes e, e1,e2, ···. Thes... |

32 |
On the homology stability for Teichmüller modular groups: closed surfaces and twisted coefficients, in Mapping class groups and moduli spaces of Riemann surfaces
- Ivanov
- 1991
(Show Context)
Citation Context ... − π!(θχ) = 0. This proves (ii). Next we consider (iii). Since π!(χ) =π!(θχ) = 0, we have ɛ(H∗−1 (Π; HA⊗M)) ⊂ Ker π! ∩ Ker ϕ0. Moreover we have π♯(χ) =ι ∗ (χ) =(µ ′ ) −1 (1H) =−ω0 ∈ H 0 (Π; H ⊗2 A ). =-=(27)-=- Hence π♯ɛ(v) =π♯(µ ⊗ 1M)∗(χ ⊗ π ∗ v)=(1H⊗ µ ⊗ 1M )∗(π♯(χ) ⊗ v) =v for any v ∈ H∗−1 (Π; HA ⊗ M). This proves (iii). Finally we prove (iv). We observe π!(χ ⊗2 )=ω0 ∈ H 0 (Π; H ⊗2 A ). (28) In fact, we ... |

30 |
On the Structure of the Torelli Group and the Casson Invariant, Topology 30
- Morita
- 1991
(Show Context)
Citation Context ...g,1(k) proves the first half of the theorem. For the latter half, we again use Hain’s theory of relative Malcev completion of the mapping class group in [14]. It is a consequence of the result of [48]=-=[49]-=- that the two classes b1 and −12s1 coincide in H ∗ c ′(M∞,1; Q)1. It remains to prove that b2i−1 and s2i−1 are linearly independent in it for any i>1. As mentioned before (see (38)(39)), Hain construc... |

26 |
The extension of Johnson’s homomorphism from the Torelli group to the mapping class group
- Morita
- 1993
(Show Context)
Citation Context ...Q) corresponding to the Young diagram [13] (see [16][55][56] for details of symplectic representation theory related to the mapping class group and [8] for generalities). Extending earlier results in =-=[52]-=-[53], the second author constructed in [56] a morphism (1) π1Σg −−−−→ [12]˜×HQ ⏐ ⏐ ↓ ↓ Mg,∗ ⏐ ↓ ρ2 −−−−→ (([1 ) 2 2 ] ⊕ [2 ] ˜×torelliΛ3 ) HQ ⋊ Sp(HQ) ⏐ ↓ Mg −−−−→ ρ2 ( ) 2 [2 ]˜×UQ ⋊ Sp(HQ)COHOMOLOG... |

26 |
A linear representation of the mapping class group of orientable surfaces and characteristic classes of surface bundles, from: “Topology and Teichmüller spaces (Katinkulta
- Morita
- 1995
(Show Context)
Citation Context ...orresponding to the Young diagram [13] (see [16][55][56] for details of symplectic representation theory related to the mapping class group and [8] for generalities). Extending earlier results in [52]=-=[53]-=-, the second author constructed in [56] a morphism (1) π1Σg −−−−→ [12]˜×HQ ⏐ ⏐ ↓ ↓ Mg,∗ ⏐ ↓ ρ2 −−−−→ (([1 ) 2 2 ] ⊕ [2 ] ˜×torelliΛ3 ) HQ ⋊ Sp(HQ) ⏐ ↓ Mg −−−−→ ρ2 ( ) 2 [2 ]˜×UQ ⋊ Sp(HQ)COHOMOLOGY OF... |

25 | Finite Type 3–Manifold Invariants, the Mapping Class Group and Blinks, Journal of differential Geometry 47
- Garoufalidis, Levine
- 1997
(Show Context)
Citation Context ...Q). For example, we may ask whether Looijenga’s unstable cohomology class for genus 3 moduli space given in [40] can be detected by the above homomorphism or not. Remark 14.2. Garoufalidis and Levine =-=[9]-=- proved that the filtration, introduced by Ohtsuki [59] (see also [60]), on the vector space generated by oriented homology 3spheres can be described by the lower central series of the Torelli group I... |

25 |
Completions of mapping class groups and the cycles C
- Hain
- 1993
(Show Context)
Citation Context ... ρ1 −−−−→ 1 −−−−→ ρ1 2Λ3H ⋊ Sp(2g, Z) ⏐ ↓ 1 2 Λ3 H/H ⋊ Sp(2g, Z) of group extensions which was constructed in [53] and is a projection of the morphism (1) in §1 (see also Hain’s closely related works =-=[14]-=-[15] in the context of algebraic geometry). Here the top horizontal homomorphism is the abelianization and the other two homomorphisms ρ1 are described as follows. Namely certain crossed homomorphisms... |

25 |
On the tautological ring of
- Looijenga
- 1995
(Show Context)
Citation Context ...logy (up to signs). The tautological algebras R∗ (Cg) and R∗ (Mg) of the moduli spaces are defined to be the subalgebras of the rational Chow algebras generated by the above tautological classes (see =-=[41]-=-[7][18][25]). Similarly we define the tautological algebras R∗ (Mg,∗) and R∗ (Mg) of the mapping class groups to be the subalgebras of H∗ (Mg,∗; Q) and H∗ (Mg; Q) generated by the classes e, e1,e2, ··... |

21 |
Chow Rings of Moduli Spaces of Curves I: The Chow Ring of ¯ M3
- Faber
(Show Context)
Citation Context ...) Sp = 4 so that we have 4 relations. Similar computations as above yield the following relations e 2 1 =0, ee1 +4e 2 =0. We can also show that e3 = 0 by our method, but we omit the details here. See =-=[5]-=- for the structure of the Chow algebra. (IV) The case of g = 2. In this case, we can obtain the well-known facts e1 =0, e 2 =0. The details are also omitted. See [57] for further unstable relations in... |

17 | The Hodge De Rham theory of relative Malčev completion,Ann
- Hain
- 1998
(Show Context)
Citation Context ...he inclusion homomorphism Rg = Ker(π 0 1→π1). The kernel of π : M2 g→Mg,∗ is naturally identified with π0 1 . Hence we obtain a morphism of group extensions 1 −−−−→ Rg −−−−→ π0 1 −−−−→ π1 −−−−→ 1 ⏐ ⏐ =-=(17)-=- ∥ ↓ ψℓ↓ 1 −−−−→ Rg −−−−→M 2 g αℓ −−−−→ Mg,∗ −−−−→ 1. Let M be an Mg,∗-module. We regard it as an Mg,∗-module by the homomorphism π : Mg,∗→Mg,∗. Since the group Rg is a subgroup of a free group π0 1 ,... |

17 |
The third homology group of the moduli space of curves
- Harer
- 1992
(Show Context)
Citation Context ...alcev completion of π1Σg and the targets of the other two homomorphisms ρ2 are semi-direct products of Sp(HQ) with two step nilpotent groups whose extension classes are given by certain Sp-submodules =-=[22]-=- ⊂ H2 (UQ) =Λ2U∗ Q and [12] torelli ⊕ [22] ⊂ H2 (Λ3HQ) = Λ2 (Λ3H ∗ Q ) (see the above cited papers for details). By a general property of cohomology of semi-direct products (see §3), the diagram (1) i... |

15 |
Families of Jacobian manifolds and characteristic classes of surface bundles
- Morita
- 1989
(Show Context)
Citation Context ...heory. As for Mg, this result cannot be generalized to integral symplectic coefficients. In fact, for example, the second integral cohomology of Mg with coefficients in H does not admit the stability =-=[46]-=-. On the basis of Looijenga’s noteworthy idea [42] that the stable cohomology with symplectic coefficients is computed only from the Harer stability theorem [21] with14 NARIYA KAWAZUMI AND SHIGEYUKI ... |

14 | Stable cohomology of the mapping class group with symplectic coefficients and of the universal Abel-Jacobi map
- Looijenga
- 1996
(Show Context)
Citation Context ... =−ωℓ ∈ Z 1 (M 2 g; H). Proof. For any ϕ ∈M2 g , we have α ∗ ( ℓ (k0) =k0 π(ϕ),ψℓ¯π(ϕ)ψℓ −1) = [ ψℓ¯π(ϕ)ψℓ −1 π(ϕ) −1] = [ −1 ψℓψϕ(ℓ) ] = [ ℓϕ(ℓ) −1] = [ ℓ − ϕ(ℓ) ] = − ωℓ(ϕ) completing the proof. In =-=[42]-=- Looijenga obtained a remarkable result that the rational stable cohomology of Mg with coefficients in any finite dimensional irreducible representation of the algebraic group Sp(2g, Q) is isomorphic ... |

12 |
Chow rings of moduli spaces of curves II: some results on the Chow ringg of Atj. Annals of Mathematics
- Faber
- 1990
(Show Context)
Citation Context ... − 100 3 ee1 − 200e 2 =0. From these, we can conclude that the following two relations 32e2 +3e 2 1 =0 7e2 − 9ee1 − 54e 2 =0 hold for g = 4. The first relation coincides with κ 2 32 1 = 3 κ2 given in =-=[6]-=- (see also [7]). Here it is amusing to observe that the fiber integral of the second equality above yields a trivial identity 54e1 − 54e1 = 0, while that of e(7e2 − 9ee1 − 54e 2 )=0 yields −42e2 − 9e ... |

11 |
Casson invariant, signature defect of framed manifolds and the secondary characteristic classes of surface bundles
- Morita
- 1997
(Show Context)
Citation Context ...l generated by all odd classes b3,b5, ··· except for the first one b1. Remark 13.6. The above result suggests (but not prove) that the secondary characterstic classes of surface bundles introduced in =-=[54]-=- are non-trivial. Remark 13.7. If we compare Hain’s presentation of the Lie algebras associated to the Malcev completions of Ig,1 and Ig, it is not difficult to show that the natural homomorphism H ∗ ... |

10 |
On the complex analytic Gel’fand-Fuks cohomology of open Riemann surfaces, Ann. Inst. Fourier 43
- Kawazumi
- 1993
(Show Context)
Citation Context ...lynomial algebra generated by the classes ei. We mention that this conjecture has been proved to be true for degrees ≤ 4 by Harer [20][22][24] and for degree 5 by Arbarello and Cornalba [1]. See also =-=[31]-=-[32] for another evidence for the conjecture. In the above discussion, we can replace {Mg,1(k)}k by the other filtration {M′ g,1(k)}k and we have the following result concerning it.COHOMOLOGY OF THE ... |

10 |
A generalization of the Morita-Mumford classes to extended mapping class groups for surfaces, Invent
- Kawazumi
- 1998
(Show Context)
Citation Context ...≥ 1, we have mi+1,0 = ei.COHOMOLOGY OF THE MODULI SPACE OF CURVES 13 On the mapping class group Mg,1 these classes are nothing but the cohomology classes (−1) jmi,j introduced by the first author in =-=[34]-=-, where they are called the generalized Morita-Mumford classes. In order to verify this, we introduce the mapping class group M2 g of Σg fixing two distinct points p0,p1 ∈ Σg pointwise. Choose a simpl... |

10 | Generators for the tautological algebra of the moduli space of curves
- Morita
(Show Context)
Citation Context ... two linear chord diagrams with 2k vertices. Then αC(aC ′)=(−1)k−r (2g) r where r = r(C, C ′ ) is the number of connected components of the union C ∪ C ′ . Proof. To prove this, we recall a result of =-=[57]-=-, Lemma 3.3. Let i, j be two indices with 1 ≤ i<j≤ 2k and let pij : H ⊗2k Q →H⊗2k Q be the map defined by first taking the contraction of the i-th and the j-th entries by the intersection pairing and ... |

9 |
Improved stability for the homology of the mapping class groups of surfaces
- Harer
- 1993
(Show Context)
Citation Context ...now that the tautological algebras have no relations in the stable range (see [43][45]). The claim now follows. The precise value of the stable range g is due to Harer’s improved stability theorem in =-=[23]-=-. ≤ 2 3 Remark 11.2. Garoufalidis and Nakamura claim in [11] that there exsits a canonical isomorphism Q[Γ ; Γ ∈G 0 ]/(IH0) ∼ = Λ ∗ UQ/([2 2 ]) in the stable range where IH0 denotes the ideal generate... |

8 |
Johnson’s homomorphisms of subgroups of the mapping class group, the Magnus expansion and Massey higher products of mapping tori
- Kitano
- 1996
(Show Context)
Citation Context ...the free differentials and [ ] : Z[π1(Σ0 g )]→H denotes the homomorphism induced by the abelianization π1(Σ0 g )→H. The crossed homomorphism ˜ k can be represented in terms of the cochain θ (see also =-=[36]-=-). Here we regard ˜ k as a crossed homomorphism of Mg,1 into Hom(H, 1 2Λ2H). We define the twisted product 1 2Λ2H ˜×H to be the product set 1 2Λ2H × H equipped with the group law (ξ,u)(η, v) =(ξ+ η + ... |

8 |
Structure of the mapping class group and symplectic representation theory
- Morita
(Show Context)
Citation Context ... ω0 ∈ Λ2HQ is the symplectic class. We denote the quotient Λ3HQ/HQ simply by UQ. UQ is an irreducible representation of the algebraic group Sp(HQ) corresponding to the Young diagram [13] (see [16][55]=-=[56]-=- for details of symplectic representation theory related to the mapping class group and [8] for generalities). Extending earlier results in [52][53], the second author constructed in [56] a morphism (... |

7 |
Geometric proofs of some results
- Hain, Reed
- 2001
(Show Context)
Citation Context ...vertices which has two loops and let Γ2 be a trivalent graph with two vertices without loop, namely a theta graph. Then it was proved in [47][50] that αΓ1 = −e1 − 4g(g − 1)e, αΓ2 = −e1 +6ge (see also =-=[19]-=- for a proof in the context of algebraic geometry). Now there is an embedding I ⊂ Γ1 such that Γ2 is obtained from Γ1 by replacing I by H. Then Γ2 \ τ2 is a circle while Γ1 \ τ1 is the disjoint union ... |

7 |
The structure of the mapping class group and characteristic classes of surface bundles, inMapping
- Morita
- 1993
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Citation Context ...example. Example 1.4. Let Γ1 be a trivalent graph with two vertices which has two loops and let Γ2 be a trivalent graph with two vertices without loop, namely a theta graph. Then it was proved in [47]=-=[50]-=- that αΓ1 = −e1 − 4g(g − 1)e, αΓ2 = −e1 +6ge (see also [19] for a proof in the context of algebraic geometry). Now there is an embedding I ⊂ Γ1 such that Γ2 is obtained from Γ1 by replacing I by H. Th... |

6 |
The fourth homology group of the moduli space of curves, Duke University preprint
- Harer
(Show Context)
Citation Context ... cohomology of the mapping class group is isomorphic to the polynomial algebra generated by the classes ei. We mention that this conjecture has been proved to be true for degrees ≤ 4 by Harer [20][22]=-=[24]-=- and for degree 5 by Arbarello and Cornalba [1]. See also [31][32] for another evidence for the conjecture. In the above discussion, we can replace {Mg,1(k)}k by the other filtration {M′ g,1(k)}k and ... |

5 | Finite type 3-manifold invariants and the structure of the Torelli group
- Garoufalidis, Levine
- 1998
(Show Context)
Citation Context ..., introduced by Ohtsuki [59] (see also [60]), on the vector space generated by oriented homology 3spheres can be described by the lower central series of the Torelli group Ig. Moreover they showed in =-=[10]-=- a relation between the finite type invariants of homology 3spheres and the graded module associated to the lower central series of Ig. Also we learned from J. Murakami that the restriction to Ig of t... |

5 |
On the stable cohomology algebra of extended mapping class groups for surfaces
- Kawazumi
(Show Context)
Citation Context ...over the stable integral cohomology of Mg,1, and certain algebraic combinations of the (modified) twisted Mumford-Morita-Miller classes can serve as its (topologically constructed and new) free basis =-=[33]-=-. Here the Lyndon-Hochschild-Serre spectral sequence for a pair of groups introduced in [34] is used instead of geometric considerations including Hodge theory. Since it has been found out that the tw... |