# A Survey on Spaces of Homomorphisms to Lie Groups

@article{Cohen2016ASO, title={A Survey on Spaces of Homomorphisms to Lie Groups}, author={Frederick R. Cohen and Mentor Stafa}, journal={arXiv: Algebraic Topology}, year={2016}, pages={361-379} }

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.

## 14 Citations

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## References

SHOWING 1-10 OF 42 REFERENCES

Cohomology of the space of commuting n-tuples in a compact Lie group

- Mathematics
- 2007

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…

Commuting elements and spaces of homomorphisms

- Mathematics
- 2007

This article records basic topological, as well as homological properties of the space of homomorphisms Hom(π,G) where π is a finitely generated discrete group, and G is a Lie group, possibly…

On braid groups and homotopy groups

- Mathematics
- 2008

This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2-sphere, as well as Lie algebras attached to the descending central series of pure…

Fundamental groups of commuting elements in Lie groups

- Mathematics
- 2008

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 mod‐2 cohomology of the components of these spaces is…

Commuting elements, simplicial spaces and filtrations of classifying spaces

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2011

Abstract Let G denote a topological group. In this paper the descending central series of free groups are used to construct simplicial spaces of homomorphisms with geometric realizations B(q, G) that…

On the fundamental group of Hom(Z^k,G)

- Mathematics
- 2010

Let G be a compact Lie group, and consider the variety Hom(Z^k,G) of representations of Z^k into G. We view this as a based space by designating the trivial representation to be its base point. We…

Commuting tuples in reductive groups and their maximal compact subgroups

- Mathematics
- 2013

Let G be a reductive algebraic group and K G a maximal compact subgroup. We consider the representation spaces Hom.Z k ;K/ and Hom.Z k ;G/ with the topology induced from an embedding into K k and G k…

A classifying space for commutativity in Lie groups

- Mathematics
- 2015

In this article we consider a space BcomG assembled from commuting elements in a Lie group G first defined by Adem, Cohen and Torres-Giese. We describe homotopy-theoretic properties of these spaces…

On Polyhedral Products and Spaces of Commuting Elements in Lie Groups

- Mathematics
- 2013

This thesis consists of two parts. The first part concentrates on polyhedral products. Certain homotopy theoretic properties of polyhedral products, such as the fundamental group, are investigated,…

On spaces of commuting elements in Lie groups†

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2016

Abstract The main purpose of this paper is to introduce a method to “stabilise” certain spaces of homomorphisms from finitely generated free abelian groups to a Lie group G, namely Hom(ℤ n , G). We…