scholarly journals ON THE COHOMOLOGY OF TORELLI GROUPS

2020 ◽  
Vol 8 ◽  
Author(s):  
ALEXANDER KUPERS ◽  
OSCAR RANDAL-WILLIAMS
Keyword(s):  
Stable Range ◽  
Cohomology Groups ◽  
Finite Dimensional ◽  
Rational Cohomology

We completely describe the algebraic part of the rational cohomology of the Torelli groups of the manifolds $\#^{g}S^{n}\times S^{n}$ relative to a disc in a stable range, for $2n\geqslant 6$ . Our calculation is also valid for $2n=2$ assuming that the rational cohomology groups of these Torelli groups are finite-dimensional in a stable range.

scholarly journals The cohomology of Torelli groups is algebraic

2020 ◽  
Vol 8 ◽  
Author(s):  
Alexander Kupers ◽  
Oscar Randal-Williams
Keyword(s):  
Stable Range ◽  
Cohomology Groups ◽  
Torelli Group ◽  
Classifying Space ◽  
Arithmetic Subgroup ◽  
Group Of Diffeomorphisms ◽  
Rational Cohomology ◽  
Algebraic Representations

Abstract The Torelli group of $W_g = \#^g S^n \times S^n$ is the group of diffeomorphisms of $W_g$ fixing a disc that act trivially on $H_n(W_g;\mathbb{Z} )$ . The rational cohomology groups of the Torelli group are representations of an arithmetic subgroup of $\text{Sp}_{2g}(\mathbb{Z} )$ or $\text{O}_{g,g}(\mathbb{Z} )$ . In this article we prove that for $2n \geq 6$ and $g \geq 2$ , they are in fact algebraic representations. Combined with previous work, this determines the rational cohomology of the Torelli group in a stable range. We further prove that the classifying space of the Torelli group is nilpotent.

scholarly journals Holomorphic Cohomology of Complex Analytic Linear Groups

1966 ◽  
Vol 27 (2) ◽  
pp. 531-542 ◽  
Author(s):  
G. Hochschild ◽  
G. D. Mostow
Keyword(s):  
Linear Group ◽  
Structure Theory ◽  
Linear Groups ◽  
Dimensional Representation ◽  
Lie Algebra Cohomology ◽  
Analytic Group ◽  
Finite Dimensional Representation ◽  
Semidirect Products ◽  
Finite Dimensional ◽  
Rational Cohomology

Let G be a complex analytic group, and let A be the representation space of a finite-dimensional complex analytic representation of G. We consider the cohomology for G in A, such as would be obtained in the usual way from the complex of holomorphic cochains for G in A. Actually, we shall use a more conceptual categorical definition, which is equivalent to the explicit one by cochains. In the context of finite-dimensional representation theory, nothing substantial is lost by assuming that G is a linear group. Under this assumption, it is the main purpose of this paper to relate the holomorphic cohomology of G to Lie algebra cohomology, and to the rational cohomology, in the sense of [1], of algebraic hulls of G. This is accomplished by using the known structure theory for complex analytic linear groups in combination with certain easily established results concerning the cohomology of semidirect products. The main results are Theorem 4.1 (whose hypothesis is always satisfied by a complex analytic linear group) and Theorems 5.1 and 5.2. These last two theorems show that the usual abundantly used connections between complex analytic representations of complex analytic groups and rational representations of algebraic groups extend fully to the superstructure of cohomology.

scholarly journals The space of immersed surfaces in a manifold

2013 ◽  
Vol 154 (3) ◽  
pp. 419-438 ◽  
Author(s):  
OSCAR RANDAL–WILLIAMS
Keyword(s):  
Stable Range ◽  
Connected Manifold ◽  
Immersed Surfaces ◽  
Rational Cohomology ◽  
Simply Connected

AbstractWe study the cohomology of the space of immersed genus g surfaces in a simply-connected manifold. We compute the rational cohomology of this space in a stable range which goes to infinity with g. In fact, in this stable range we are also able to obtain information about torsion in the cohomology of this space, as long as we localise away from (g-1).

scholarly journals On finite dimensionality of mixed Tate motives

2008 ◽  
Vol 4 (1) ◽  
pp. 145-161 ◽  
Author(s):  
Shahram Biglari
Keyword(s):  
Grothendieck Group ◽  
Ring Structure ◽  
Triangulated Category ◽  
Cohomology Groups ◽  
Weight Filtration ◽  
Finite Dimensional ◽  
Finite Dimensionality ◽  
Tate Motives

AbstractWe prove a few results concerning the notion of finite dimensionality of mixed Tate motives in the sense of Kimura and O'Sullivan. It is shown that being oddly or evenly finite dimensional is equivalent to vanishing of certain cohomology groups defined by means of the Levine weight filtration. We then explain the relation to the Grothendieck group of the triangulated category D of mixed Tate motives. This naturally gives rise to a λ–ring structure on K0(D).

scholarly journals Bounding the Dimensions of Rational Cohomology Groups

2014 ◽  
pp. 51-69 ◽  
Author(s):  
Christopher P. Bendel ◽  
Brian D. Boe ◽  
Christopher M. Drupieski ◽  
Daniel K. Nakano ◽  
Brian J. Parshall ◽  
...  
Keyword(s):  
Cohomology Groups ◽  
Rational Cohomology

Cohomology of Lie Conformal Algebra Vir⋉Cur g

2021 ◽  
Vol 28 (03) ◽  
pp. 507-520
Author(s):  
Maosen Xu ◽  
Yan Tan ◽  
Zhixiang Wu
Keyword(s):  
Lie Algebra ◽  
Conformal Algebra ◽  
Cohomology Groups ◽  
Simple Lie Algebra ◽  
Finite Dimensional ◽  
Irreducible Modules

In this article, we compute cohomology groups of the semisimple Lie conformal algebra [Formula: see text] with coefficients in its irreducible modules for a finite-dimensional simple Lie algebra [Formula: see text].

scholarly journals On the Hochschild coho-mology groups of endomorphism algebras of exceptional sequences

2004 ◽  
Vol 35 (3) ◽  
pp. 189-196
Author(s):  
Hailou Yao ◽  
Lihong Huang
Keyword(s):  
Associative Algebra ◽  
Hochschild Cohomology ◽  
Closed Field ◽  
Cohomology Groups ◽  
Endomorphism Algebra ◽  
Exceptional Sequence ◽  
Algebraically Closed Field ◽  
Finite Dimensional ◽  
Endomorphism Algebras ◽  
Exceptional Sequences

Let $ A $ be a finite dimensional associative algebra over an algebraically closed field $ k $, and $\mod A$ be the category of finite dimensional left $ A $-module and $ X_1,X_2,\ldots,X_n$ in $\mod A$ be a complete exceptional sequence, then we investigate the Hochschild Cohomology groups of endomorphism algebra of exceptional sequence $ {X_1,X_2, \ldots,X_n}$ in this paper.

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