An official website of the United States government
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.
more »
« less
ON THE COHOMOLOGY OF TORELLI GROUPS
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.
more »
« less
- Award ID(s):
- 1803766
- PAR ID:
- 10159821
- Date Published:
- Journal Name:
- Forum of Mathematics, Pi
- Volume:
- 8
- ISSN:
- 2050-5086
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We prove a 1979 conjecture of Lusztig on the cohomology of semi-infinite Deligne–Lusztig varieties attached to division algebras over local fields. We also prove the two conjectures of Boyarchenko on these varieties. It is known that in this setting, the semi-infinite Deligne–Lusztig varieties are ind-schemes comprised of limits of certain finite-type schemes X h {X_{h}} . Boyarchenko’s two conjectures are on the maximality of X h {X_{h}} and on the behavior of the torus-eigenspaces of their cohomology. Both of these conjectures were known in full generality only for division algebras with Hasse invariant 1 / n {1/n} in the case h = 2 {h=2} (the “lowest level”) by the work of Boyarchenko–Weinstein on the cohomology of a special affinoid in the Lubin–Tate tower. We prove that the number of rational points of X h {X_{h}} attains its Weil–Deligne bound, so that the cohomology of X h {X_{h}} is pure in a very strong sense. We prove that the torus-eigenspaces of the cohomology group H c i ( X h ) {H_{c}^{i}(X_{h})} are irreducible representations and are supported in exactly one cohomological degree. Finally, we give a complete description of the homology groups of the semi-infinite Deligne–Lusztig varieties attached to any division algebra, thus giving a geometric realization of a large class of supercuspidal representations of these groups. Moreover, the correspondence θ ↦ H c i ( X h ) [ θ ] {\theta\mapsto H_{c}^{i}(X_{h})[\theta]} agrees with local Langlands and Jacquet–Langlands correspondences. The techniques developed in this paper should be useful in studying these constructions for p -adic groups in general.more » « less
-
Several well-known open questions (such as: are all groups sofic/hyperlinear?) have a common form: can all groups be approximated by asymptotic homomorphisms into the symmetric groups $$\text{Sym}(n)$$ (in the sofic case) or the finite-dimensional unitary groups $$\text{U}(n)$$ (in the hyperlinear case)? In the case of $$\text{U}(n)$$ , the question can be asked with respect to different metrics and norms. This paper answers, for the first time, one of these versions, showing that there exist finitely presented groups which are not approximated by $$\text{U}(n)$$ with respect to the Frobenius norm $$\Vert T\Vert _{\text{Frob}}=\sqrt{\sum _{i,j=1}^{n}|T_{ij}|^{2}},T=[T_{ij}]_{i,j=1}^{n}\in \text{M}_{n}(\mathbb{C})$$ . Our strategy is to show that some higher dimensional cohomology vanishing phenomena implies stability , that is, every Frobenius-approximate homomorphism into finite-dimensional unitary groups is close to an actual homomorphism. This is combined with existence results of certain nonresidually finite central extensions of lattices in some simple $$p$$ -adic Lie groups. These groups act on high-rank Bruhat–Tits buildings and satisfy the needed vanishing cohomology phenomenon and are thus stable and not Frobenius-approximated.more » « less
-
null (Ed.)Abstract Key message The first cytological characterization of the 2N v S segment in hexaploid wheat; complete de novo assembly and annotation of 2N v S segment; 2N v S frequency is increasing 2N v S and is associated with higher yield. Abstract The Aegilops ventricosa 2N v S translocation segment has been utilized in breeding disease-resistant wheat crops since the early 1990s. This segment is known to possess several important resistance genes against multiple wheat diseases including root knot nematode, stripe rust, leaf rust and stem rust. More recently, this segment has been associated with resistance to wheat blast, an emerging and devastating wheat disease in South America and Asia. To date, full characterization of the segment including its size, gene content and its association with grain yield is lacking. Here, we present a complete cytological and physical characterization of this agronomically important translocation in bread wheat. We de novo assembled the 2N v S segment in two wheat varieties, ‘Jagger’ and ‘CDC Stanley,’ and delineated the segment to be approximately 33 Mb. A total of 535 high-confidence genes were annotated within the 2N v S region, with > 10% belonging to the nucleotide-binding leucine-rich repeat (NLR) gene families. Identification of groups of NLR genes that are potentially N genome-specific and expressed in specific tissues can fast-track testing of candidate genes playing roles in various disease resistances. We also show the increasing frequency of 2N v S among spring and winter wheat breeding programs over two and a half decades, and the positive impact of 2N v S on wheat grain yield based on historical datasets. The significance of the 2N v S segment in wheat breeding due to resistance to multiple diseases and a positive impact on yield highlights the importance of understanding and characterizing the wheat pan-genome for better insights into molecular breeding for wheat improvement.more » « less
-
Williamson's theorem states that for any 2n×2n real positive definite matrix A, there exists a 2n×2n real symplectic matrix S such that STAS=D⊕D, where D is an n×n diagonal matrix with positive diagonal entries which are known as the symplectic eigenvalues of A. Let H be any 2n×2n real symmetric matrix such that the perturbed matrix A+H is also positive definite. In this paper, we show that any symplectic matrix S̃ diagonalizing A+H in Williamson's theorem is of the form S̃ =SQ+(‖H‖), where Q is a 2n×2n real symplectic as well as orthogonal matrix. Moreover, Q is in symplectic block diagonal form with the block sizes given by twice the multiplicities of the symplectic eigenvalues of A. Consequently, we show that S̃ and S can be chosen so that ‖S̃ −S‖=(‖H‖). Our results hold even if A has repeated symplectic eigenvalues. This generalizes the stability result of symplectic matrices for non-repeated symplectic eigenvalues given by Idel, Gaona, and Wolf [Linear Algebra Appl., 525:45-58, 2017].more » « less
- Free Publicly Accessible Full Text
-
Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/fmp.2020.5
