In this paper we study homological stability for spaces Hom(Z n , G) of pairwise commuting n-tuples in a Lie group G. We prove that for each n 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, Comm(G) and BcomG, introduced by Cohen-Stafa and Adem-Cohen-Torres-Giese respectively. In addition, we show that the rational homology of the space of unordered commuting n-tuples in a fixed group G stabilizes as n increases.Our proofs use the theory of representation stability -in particular, J. Wilson's theory of FI W -modules. In all of the these results, we obtain specific bounds on the stable range, and we show that the homology isomorphisms are induced by maps of spaces.1.1. Ordered commuting tuples. Let {G r } r 1 denote one of the classical families of compact, connected Lie groups -namely G r = SU(r), U(r), SO(2r + 1), Sp(r), or SO(2r) -or the complexifications thereof. We have standard inclusions G r ֒→ G r+1 (see Section 6).Theorem 1.1. Fix k, n 0 and let {G r } r 1 be one of the classical sequences of Lie groups listed above. Then the standard inclusions G r ֒→ G r+1 induce isomorphismsfor r k, andThese results are proved in Theorems 7.1 and 10.7, and Corollary 11.3 respectively. We prove similar results for G r -equivariant homology in Section 9. This result, and the others discussed below, also apply to the groups Spin(r) (Example 12.2), and to the projectivizations of the above sequences (Example 12.3), although for the projective groups there are no maps inducing the isomorphisms.

Abstract. 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(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(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 elements in a compact Lie group is homotopy equivalent to an equivariant function space after inverting the order of the Weyl group. In addition, the homology of the stabilized space admits a very simple description in terms of the tensor algebra generated by the reduced homology of a maximal torus in favorable cases. The stabilized space also allows the description of the additive reduced homology of the individual spaces Hom(Z n , G), with the order of the Weyl group inverted.

Communicated by C.A. Weibel MSC: 55U10; 58K10; 13F55; 14F45We study the monodromy representation corresponding to a fibration introduced by G. Denham and A. Suciu [5], which involves polyhedral products given in Definition 2.2. Algebraic and geometric descriptions for these monodromy representations are given. In particular, we study the case of a product of two finite cyclic groups and obtain representations into Out(F n ) and SL n (Z). We give algebraic descriptions of monodromy for the case of a product of any two finite groups. Finally we give a geometric description for monodromy representations of a product of 2 or more finite groups to Out(F n ), as well as some algebraic properties. The geometric description does not rely on choosing a basis for the fundamental group of the fibre in terms of commutators, hence avoids this delicate question.

Abstract. Let K be a finite simplicial complex, and (X, A) be a pair of spaces. The purpose of this article is to study the fundamental group of the polyhedral product denoted Z K (X, A), which denotes the moment-angle complex of Buchstaber-Panov in the case (X, A) = (D 2 , S 1 ), with extension to arbitrary pairs in [2] as given in Definition 2.2 here.For the case of a discrete group G, we give necessary and sufficient conditions on the abstract simplicial complex K such that the polyhedral product denoted by Z K (BG) is an Eilenberg-Mac Lane space. The fundamental group of Z K (BG) is shown to depend only on the 1-skeleton of K. Further special examples of polyhedral products are also investigated.Finally, we use polyhedral products to study an extension problem related to transitively commutative groups, which are given in Definition 5.2.

The purpose of this article is to give an exposition of topological properties of spaces of homomorphisms from certain finitely generated discrete groups to Lie groups G, and to describe their connections to classical representation theory, as well as other structures. Various properties are given when G is replaced by a small category, or the discrete group is given by a right-angled Artin group.