Perfect forms, K-theory and the cohomology of modular groups

Elbaz-Vincent, Philippe; Gangl, Herbert; Soulé, Christophe

doi:10.1016/j.aim.2013.06.014

Cited by 26 publications

(36 citation statements)

References 25 publications

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“…We expect that there should be a close relation between the cohomology of Out F n and that of GL(n, Z). For example, Elbaz-Vincent, Gangl and Soulé [15] recently calculated the rational cohomology of GL(n, Z) for n = 5, 6, 7 and it will be a very interesting problem to compare these results with the known results about H * (Out F 6 ; Q). Also we have a conjectural geometric meaning of the classes µ k ∈ H 4k (Out F 2k+2 ; Q).…”

Section: The Case Of H G1 and The Outer Automorphism Groups Of Free mentioning

confidence: 99%

Computations in formal symplectic geometry and characteristic classes of moduli spaces

Morita

Sakasai

Suzuki³

2015

Quantum Topol.

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We make explicit computations in the formal symplectic geometry of Kontsevich and determine the Euler characteristics of the three cases, namely commutative, Lie and associative ones, up to certain weights. From these, we obtain some nontriviality results in each case. In particular, we determine the integral Euler characteristics of the outer automorphism groups Out F n of free groups for all n ≤ 10 and prove the existence of plenty of rational cohomology classes of odd degrees. We also clarify the relationship of the commutative graph homology with finite type invariants of homology 3-spheres as well as the leaf cohomology classes for transversely symplectic foliations. Furthermore we prove the existence of several new non-trivalent graph homology classes of odd degrees. Based on these computations, we propose a few conjectures and problems on the graph homology and the characteristic classes of the moduli spaces of graphs as well as curves.

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Section: The Case Of H G1 and The Outer Automorphism Groups Of Free mentioning

confidence: 99%

Computations in formal symplectic geometry and characteristic classes of moduli spaces

Morita

Sakasai

Suzuki³

2015

Quantum Topol.

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“…Enumeration of the case d = r for d ≤ 8, respectively d = 9, is done in [Mar01], respectively [KMS12]. Enumeration of cases d ≤ 6, respectively d = 7, is done in [EVGS02], respectively [EVGS13]. The cases (r, d) = (8, 9), (8, 10), (9, 10) are treated in [DSHS15].…”

Section: Appendix Computations By Mathieu Dutour Sikirićmentioning

confidence: 99%

Stable Betti Numbers of (Partial) Toroidal Compactifications of the Moduli Space of Abelian Varieties

Grushevsky¹,

Hulek²,

Tommasi³

et al. 2018

Geometry and Physics: Volume II

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We present an algorithm for explicitly computing the number of generators of the stable cohomology algebra of any rationally smooth partial toroidal compactification of Ag satisfying certain additivity and finiteness properties, in terms of the combinatorics of the corresponding toric fans. In particular the algorithm determines the stable cohomology of the matroidal partial compactification A Matr g , in terms of simple regular matroids that are irreducible with respect to the 1-sum operation, and their automorphism groups. The algorithm also applies to compute the stable Betti numbers in close to top degree for the perfect cone toroidal compactification A Perf g . This suggests the existence of an algebra structure on H top −k stable (A Perf g , Q).

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“…In Section 4 we describe a "mass formula" for the cells in our tessellations that provides a strong computational check on the correctness of our constructions. In Section 5 we give an explicit representative for the nontrivial class in the top cohomological degree; this construction is motivated by a similar construction in [14,13]. Finally, in Section 6 we give the results of our computations.…”

Section: 2mentioning

confidence: 99%

“…This suggests that there should be a canonical generator for this homology group, a fact already explored in [14,Section 5]. An obvious choice is the analogue of the chain presented there, namely…”

Section: Explicit Homology Classesmentioning

confidence: 99%

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On the cohomology of linear groups over imaginary quadratic fields

Sikirić

Gangl

Gunnells

et al. 2016

Journal of Pure and Applied Algebra

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Abstract. Let Γ be the group GL N (O D ), where O D is the ring of integers in the imaginary quadratic field with discriminant D < 0. In this paper we investigate the cohomology of Γ for N = 3, 4 and for a selection of discriminants: D ≥ −24 when N = 3, and D = −3, −4 when N = 4. In particular we compute the integral cohomology of Γ up to p-power torsion for small primes p. Our main tool is the polyhedral reduction theory for Γ developed by Ash [4, Ch. II] and Koecher [18]. Our results extend work of Staffeldt [29], who treated the case n = 3, D = −4. In a sequel [11] to this paper, we will apply some of these results to the computations with the K-groups K 4 (O D ), when D = −3, −4.

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Perfect forms, K-theory and the cohomology of modular groups

Cited by 26 publications

References 25 publications

Computations in formal symplectic geometry and characteristic classes of moduli spaces

Computations in formal symplectic geometry and characteristic classes of moduli spaces

Stable Betti Numbers of (Partial) Toroidal Compactifications of the Moduli Space of Abelian Varieties

On the cohomology of linear groups over imaginary quadratic fields

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