相关论文: Topological components of spaces of commuting elem…

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…

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…

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…

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,…

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…

In this note we study topological invariants of the spaces of homomorphisms Hom(\pi,G), where \pi\ 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)…

Let G be a connected, compact, semisimple Lie group. It is known that for a compact closed orientable surface $\Sigma$ of genus $l >1$, the order of the group $H^2(\Sigma,\pi_1(G))$ is equal to the number of connected components of the…

Let $G$ be a compact connected Lie group with $\pi_1(G)\cong\mathbb{Z}$. We study the homotopy types of gauge groups of principal $G$-bundles over Riemann surfaces. This can be applied to an explicit computation of the homotopy groups of…

We study the homotopy type of the space $E(L)$ of unparametrised embeddings of a split link $L=L_1\sqcup \ldots \sqcup L_n$ in $\mathbb{R}^3$. Our main result is a simple description of the fundamental group, or motion group, of $E(L)$, and…

The semigroup of the homotopy classes of the self-homotopy maps of a finite complex which induce the trivial homomorphism on homotopy groups is nilpotent. We determine the nilpotency of these semigroups of compact Lie groups and finite Hopf…

Let $\pi$ be a discrete group, and let $G$ be a compact connected Lie group. $\mathrm{Hom}(\pi,G)_0$ denotes the null-component of the space of homomorphisms from $\pi$ to $G$, and $\mathrm{map}_*(B\pi,BG)_0$ denotes the null-component of…

The moduli space for a flat G-bundle over the two-torus is completely determined by its holonomy representation. When G is compact, connected, and simply connected, we show that the moduli space is homeomorphic to a product of two tori mod…

We give an example of a compact connected Lie group of the lowest rank such that the mod 2 cohomology ring of its classifying space has a nonzero nilpotent element.

The problem in question is whether the quotient space of a compact linear group is a topological manifold and whether it is a homological manifold. In the paper, the case of an infinite group with commutative connected component is…

Let $M$ be a smooth manifold. We use Chern-Weil theory to study the characteristic classes of principal $G$-bundles built from continuous families of $\pi_{1}(M)$-representations, where $G$ is a compact Lie group. We then relate these…

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…

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 $G$…

In this paper, we show that every topological group is a strong small loop transfer space at the identity element. This implies that the quasitopological fundamental group of a connected locally path connected topological group is a…

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…

Let P be a principal bundle with semisimple compact simply connected structure group G over a compact simply connected four-manifold M. In this note we give explicit formulas for the rational homotopy groups and cohomology algebra of the…