Representation spaces for central extensions and almost commuting unitary matrices

On the moduli spaces of commuting elements in the projective unitary groups

We provide descriptions for the moduli spaces $\text{Rep}(Γ, PU(m))$, where $Γ$ is any finitely generated abelian group and $PU(m)$ is the group of $m\times m$ projective unitary matrices. As an application we show that for any connected CW-complex $X$ with $π_1(X)\cong \mathbf{Z}^n$, the natural map $π_0(\text{Rep}(π_1(X), PU(m)))\to [X, BPU(m)]$ is injective, hence providing a complete enumeration of the isomorphism classes of flat principal $PU(m)$-bundles over $X$.

酉群中交换元素空间的稳定分裂

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。

Commuting elements in central products of special unitary groups

In this paper the space of commuting elements in the central product $G_{m,p}$ of $m$ copies of the special unitary group $SU(p)$ is studied, where $p$ is a prime number. In particular, a computation for the number of path connected components of these spaces is given and the geometry of the moduli space $\Rep(\mathbb Z^n, G_{m,p})$ of flat principal $G_{m,p}$--bundles over the $n$--torus is completely described for all values of $n$, $m$ and $p$.

Configuration spaces of commuting elements

In this article we introduce the space of configurations of commuting elements in a topological group and show that it satisfies rational homological stability for the sequences of unitary, special unitary and symplectic groups. We also prove that it satisfies cohomological rational representation stability with respect to the number of elements in the tuple for finite products of such groups, in particular cohomological rational stability for the space of unordered configurations of commuting elements. Finally we present some computations of cohomology in the unstable range.

Commuting matrices and Atiyah's Real K-theory

We describe the $C_2$-equivariant homotopy type of the space of commuting n-tuples in the stable unitary group in terms of Real K-theory. The result is used to give a complete calculation of the homotopy groups of the space of commuting n-tuples in the stable orthogonal group, as well as of the coefficient ring for commutative orthogonal K-theory.

上三角矩阵的交换概形与表示同调

Commuting Schemes of Upper Triangular Matrices and Representation Homology

本研究建立了与上三角幂幺矩阵群概形$U_n$相关的高亏格广义交换概形$C_g(U_n)$和黎曼曲面的$U_n$系数表示同调的联系，并给出了判定$C(U_n)$是否为完全交的数值准则。