Some computations of stable twisted homology for mapping class groups

Soulié, Arthur

doi:10.1080/00927872.2020.1716981

Cited by 3 publications

(2 citation statements)

References 28 publications

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“…In turn, by Theorem 11.1, it suffices to prove that the stable homology of MappO k q is rationally acyclic. Using [38,Theorem 1.1] and the Lyndon-Hochschild-Serre spectral sequence, it is enough to show the groups A s n,0 of [38] are stably rationally acyclic; this is proved in [58,Theorem A.2] based on the fact that the automorphism group of free group is rationally acyclic [26].…”

Section: Homology Of Asymptotic Mapping Class Groupsmentioning

confidence: 99%

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Aramayona¹,

Bux²,

Jonas³

et al. 2021

Preprint

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A Cantor manifold C is a non-compact manifold obtained by gluing (holed) copies of a fixed compact manifold Y in a tree-like manner. Generalizing classical families of groups due to Brin [10], Dehornoy [18] and 23,1], we introduce the asymptotic mapping class group B of C, whose elements are proper isotopy classes of selfdiffeomorphisms of C that are eventually trivial. The group B happens to be an extension of a Higman-Thompson group by a direct limit of mapping class groups of compact submanifolds of C.We construct an infinite-dimensional contractible cube complex X on which B acts. For certain well-studied families of manifolds, we prove that B is of type F8 and that X is CATp0q; more concretely, our methods apply for example when Y is diffeomorphic to S 1 ˆS1 , S 2 ˆS1 , or S n ˆSn for n ě 3. In these cases, B contains, respectively, the mapping class group of every compact surface with boundary; the automorphism group of the free group on k generators for all k; and an infinite family of (arithmetic) symplectic or orthogonal groups.In particular, if Y -S 2 or S 1 ˆS1 , our result gives a positive answer to [24, Problem 3] and [3, Question 5.32]. In addition, for Y -S 1 ˆS1 or S 2 ˆS1 , the homology of B coincides with the stable homology of the relevant mapping class groups, as studied by Harer [31] and .

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Section: Homology Of Asymptotic Mapping Class Groupsmentioning

confidence: 99%

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Aramayona¹,

Bux²,

Jonas³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…According to Soulie [1], computing the homology of a group is a fundamental question and can be a very difficult task. From the theory of topological spaces emerged, algebraic topology.…”

Section: Introductionmentioning

confidence: 99%

The Universal Coefficient Theorem for Homology and Cohomology an Enigma of Computations

Nkrumah

Gyampoh

Obeng–Denteh

2019

ACRI

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According to Soulie [1], computing the homology of a group is a fundamental question and can be a very difficult task. In his assertion, a complete understanding of all the homology groups of mapping class groups of surfaces and 3-manifolds remains out of reach at present time. It is imperative that we give the universal coefficient theorem the supposed needed attention. In this article, we study some product topologies as well as the kiinneth formula for computing the (co) homology group of product spaces. The paper begins with study on the algebraic background with specific definitions and extends into four theorems considered as the Universal Coefficient Theorem. Though this article does not proof the theorems, yet much is done on some properties of each of these theorems, which is enough for the calculation of (co) homology groups.

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Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

Kawazumi,

Soulié

2024

Bulletin of London Math Soc

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We compute the stable cohomology groups of the mapping class groups of compact orientable surfaces with one boundary, with twisted coefficients given by the rational homology of the unit tangent bundles of the surfaces. These coefficients define a covariant functor over the classical category associated to mapping class groups, rather than a contravariant one, and are thus out of the scope of the traditional framework to study twisted cohomological stability. A remarkable property is that the computed stable twisted cohomology is not free as a module over the stable cohomology algebra with constant rational coefficients. For comparison, we also compute the stable cohomology group with coefficients in the first rational cohomology of the unit tangent bundle of the surface, which fits into the traditional framework.

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Some computations of stable twisted homology for mapping class groups

Cited by 3 publications

References 28 publications

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

Asymptotic mapping class groups of Cantor manifolds and their finiteness properties

The Universal Coefficient Theorem for Homology and Cohomology an Enigma of Computations

Stable twisted cohomology of the mapping class groups in the unit tangent bundle homology

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