arXiv 0905.28952009-05math.ATmath.GR

Commuting elements in central products of special unitary groups

Alejandro Adem, Frederick R. Cohen, Jose Manuel Gomez

备注：: revised version to appear in Proc.Edinburgh Math. Soc. (Isle of Skye 2009 Conference Volume)

英文摘要：

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 $p$.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

阅读原文

阅读 PDF ↗

arXiv 页面 ↗HTML 原文 ↗

本文内容

英文摘要相关论文

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

继续探索

相关论文

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。

2025-01-27math.AT

Representation spaces for central extensions and almost commuting unitary matrices

Let $Γ$ denote a central extension of the form $1\to \mathbb{Z}^r\toΓ\to \mathbb{Z}^n\to 1$. In this paper we describe the topology of the spaces of homomorphisms $\text{Hom}(Γ, U(m))$ and the associated moduli spaces $\text{Rep}(Γ, U(m))$, where $U(m)$ is the group of $m\times m$ unitary matrices.

2016-07-09math.ATmath.RT

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 $X$.

2019-07-02math.ATmath.RT

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.

2022-01-10math.AT

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.

2019-03-22math.ATmath.KT

Presentations of the first homotopy groups of the unitary groups

We describe explicit presentations of all stable and the first nonstable homotopy groups of the unitary groups. In particular, for each n >= 2 we supply n homotopic maps that each represent the (n-1)!-th power of a suitable generator of pi_2n(U(n)) = Z_{n!}. The product of these n commuting maps is the constant map to the identity matrix.

2003-01-17math.ATmath.DG

更多相关论文收起更多论文

Commuting Elements and Spaces of Homomorphisms

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.

2006-05-19math.AT

Examples of quasitoric manifolds as special unitary manifolds

This note shows that for each $n\geq 5$ with only $n\not= 6$, there exists a $2n$-dimensional specially omnioriented quasitoric manifold $M^{2n}$ which represents a nonzero element in $Ω_*^U$. This provides the counterexamples of Buchstaber--Panov--Ray conjecture.

2015-03-03math.AT

On toric generators in the unitary and special unitary bordism rings

We construct a new family of toric manifolds generating the unitary bordism ring. Each manifold in the family is the complex projectivisation of the sum of a line bundle and a trivial bundle over a complex projective space. We also construct a family of special unitary quasitoric manifolds which contains polynomial generators of the special unitary bordism ring with 2 inverted in dimensions >8. Each manifold in the latter family is obtained from an iterated complex projectivisation of a sum of line bundles by amending the complex structure to make the first Chern class vanish.

2016-11-18math.ATmath.GT