Enrique Torres-Giese, Denis Sjerve
We compute the fundamental group of the spaces of ordered commuting $n$-tuples of elements in the Lie groups SU(2), U(2) and SO(3). For SO(3) the computation of the mod-2 cohomology of the components of these spaces is also obtained.
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Let $G$ be the classical group, and let Hom$(\mathbb{Z}^m,G)$ denote the space of commuting $m$-tuples in $G$. Baird proved that the cohomology of Hom$(\mathbb{Z}^m,G)$ is identified with a certain ring of invariants of the Weyl group of $G$. In this paper by using the result of Baird we give the cohomology ring of Hom$(\mathbb{Z}^2,G)$ for simple Lie group $G$ of rank 2.
In this note we study topological invariants of the spaces of homomorphisms Hom(π,G), where π is a finitely generated abelian group and G is a compact Lie group arising as an arbitrary finite product of the classical groups SU(r), U(q) and Sp(k).
Topological components of spaces of commuting elements in connected nilpotent Lie groups
本文研究了连通 nil 拓扑群中交换元素空间的拓扑分量，通过李代数中的近似交换元素分析，给出了 Hom(ℤ^k,G) 路连通的条件，并计算了其路连通分量的同伦类型。
Consider the space Hom(Z^n,G) of pairwise commuting n-tuples of elements in a compact Lie group G. This forms a real algebraic variety, which is generally singular. In this paper, we construct a desingularization of the generic component of Hom(Z^n,G), which allows us to derive formulas for its ordinary and equivariant cohomology in terms of the Lie algebra of a maximal torus in G and the action of the Weyl group. This is an application of a general theorem concerning G-spaces for which every element is fixed by a maximal torus.
In this paper we study homological stability for spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of pairwise commuting $n$-tuples in a Lie group $G$. We prove that for each $n\geqslant 1$, these spaces satisfy rational homological stability as $G$ ranges through any of the classical sequences of compact, connected Lie groups, or their complexifications. We prove similar results for rational equivariant homology, for character varieties, and for the infinite-dimensional analogues of these spaces, ${\rm Comm}(G)$ and ${\rm B_{com}} G$, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In ad…
The main purpose of this paper is to introduce a method to stabilize certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group $G$, namely $Hom(\mathbb Z^n,G)$. We show that this stabilized space of homomorphisms decomposes after suspending once with summands which can be reassembled, in a sense to be made precise below, into the individual spaces $Hom(\mathbb Z^n,G)$ after suspending once. To prove this decomposition, a stable decomposition of an equivariant function space is also developed. One main result is that the topological space of all commuting elemen…
In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits $\Rep(\Z^n, SU)$, $\Rep(\Z^n,U)$ and $\Rep(\Z^n, Sp)$ are explicitly described as finite products of Eilenberg-MacLane spaces.
In this article we study the homology of spaces ${\rm Hom}(\mathbb{Z}^n,G)$ of ordered pairwise commuting $n$-tuples in a Lie group $G$. We give an explicit formula for the Poincare series of these spaces in terms of invariants of the Weyl group of $G$. By work of Bergeron and Silberman, our results also apply to ${\rm Hom}(F_n/Γ_n^m,G)$, where the subgroups $Γ_n^m$ are the terms in the descending central series of the free group $F_n$. Finally, we show that there is a stable equivalence between the space ${\rm Comm}(G)$ studied by Cohen-Stafa and its nilpotent analogues.
Commuting varieties and the rank filtration of topological K-theory
该研究探讨U(m)李代数中交换元素n元组空间，建立其单点紧化与连通复、实K-理论秩滤过子商的关联。
The space of commuting elements in an exceptional Lie group and maps between classifying spaces
本文研究离散群到紧连通李群的同态空间与分类空间映射空间之间的自然映射，针对 $π=\mathbf{Z}^m$（$m\geq 3$）的情形给出了该映射在有理上同调中是否满射的判定条件。
Torsion primes for spaces of commuting elements in Lie groups
该研究针对复约化群中$m$元交换元素组空间的整奇异上同调，证明其挠素数与外尔群阶数的整除关系，解决了Kishimoto和Takeda的猜想，还推导了紧李群与紧李代数的对应结论。
Let $G$ be a compact connected Lie group and $n\geqslant 1$ an integer. Consider the space of ordered commuting $n$-tuples in $G$, $Hom(\mathbb{Z}^n,G)$, and its quotient under the adjoint action, $Rep(\mathbb{Z}^n,G):=Hom(\mathbb{Z}^n,G)/G$. In this article we study and in many cases compute the homotopy groups $π_2(Hom(\mathbb{Z}^n,G))$. For $G$ simply--connected and simple we show that $π_2(Hom(\mathbb{Z}^2,G))\cong \mathbb{Z}$ and $π_2(Rep(\mathbb{Z}^2,G))\cong \mathbb{Z}$, and that on these groups the quotient map $Hom(\mathbb{Z}^2,G)\to Rep(\mathbb{Z}^2,G)$ induces multiplication by the …
This article records basic topological, as well as homological properties of the space of homomorphisms Hom(L,G) where L is a finitely generated discrete group, and G is a Lie group, possibly non-compact. If L is a free abelian group of rank equal to n, then Hom(L,G) is the space of ordered n-tuples of commuting elements in G. If G=SU(2), a complete calculation of the cohomology of these spaces is given for n=2, 3. An explicit stable splitting of these spaces is also obtained, as a special case of a more general splitting.
A stable splitting for spaces of commuting elements in unitary groups
本文证明了酉群中交换元素空间的稳定分裂，在逆转阶乘后将其分解为偏旗流形上交换簇丛的 Thom-like 空间楔和，并利用 Steenrod 运算证明该分裂在整性上不成立，同时推广至辛群和正交群。
Let $G$ be a compact connected Lie group, and let $\mathrm{Hom}(\mathbb{Z}^m,G)$ be the space of pairwise commuting $m$-tuples in $G$. We study the problem of which primes $p$ $\mathrm{Hom}(\mathbb{Z}^m,G)_1$, the connected component of $\mathrm{Hom}(\mathbb{Z}^m,G)$ containing the element $(1,\ldots,1)$, has $p$-torsion in homology. We will prove that $\mathrm{Hom}(\mathbb{Z}^m,G)_1$ for $m\ge 2$ has $p$-torsion in homology if and only if $p$ divides the order of the Weyl group of $G$ for $G=SU(n)$ and some exceptional groups. We will also compute the top homology of $\mathrm{Hom}(\mathbb{Z}^…
Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups.
The space of commuting elements in a Lie group and maps between classifying spaces
本文研究离散群与紧致连通李群之间同态空间到分类空间映射的有理上同调满射性，证明对自由阿贝尔群和经典群（除SO(2n)外）满射，对高维SO(2n)非满射，并应用于幂零群及余核维数。
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