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.
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).
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…
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.
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.
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.
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.
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.
In this paper we study the homotopy type of the partially ordered set of left cosets of abelian subgroups in an extraspecial $p$-group. We prove that the universal cover of its nerve is homotopy equivalent to a wedge of $r$-spheres where $2r \geq 4$ is the rank of its Frattini quotient. This determines the homotopy type of the universal cover of the classifying space of transitionally commutative bundles.
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…
Let $G$ be the classical group, and let Hom$(\mathbb{Z}^m,G)$ denote the space of commuting $m$-tuples in $G$. First, we refine the formula for the Poincaré series of Hom$(\mathbb{Z}^m,G)$ due to Ramras and Stafa by assigning (signed) integer partitions to (signed) permutations. Using the refined formula, we determine the top term of the Poincaré series, and apply it to prove the dependence of the topology of Hom$(\mathbb{Z}^m,G)$ on the parity of $m$ and the rational hyperbolicity of Hom$(\mathbb{Z}^m,G)$ for $m\ge 2$. Next, we give a minimal generating set of the cohomology of Hom$(\mathbb{Z…
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 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 $p$.
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.
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 …
We study the image of a transfer homomorphism in the stable homotopy groups of spheres. Actually, we show that an element of order 8 in the 18 dimensional stable stem is in the image of a double transfer homomorphism, which reproves a result due to P J Eccles that the element is represented by a framed hypersurface. Also, we determine the image of the transfer homomorphism in the 16 dimensional stable stem.
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.
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 $X$.
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}^…
