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.
Abstract. Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah-Bott's "Yang-Mills over a Riemann Surface" to compute Z/2-Betti numbers of these spaces.
Let Y := Hom(Z n , SU (2)) denote the space of commuting n-tuples in SU (2). We determine the homotopy type of the suspension ΣY , and compute the integral cohomology groups of Y for all positive integers n.
Primary 55R40. Secondary 57S051
We study the topology of the moduli space of flat SU (2)-bundles over a nonorientable surface Σ. This moduli space may be identified with the space of homomorphisms Hom(π1(Σ), SU (2)) modulo conjugation by SU (2). In particular, we compute the (rational) equivariant cohomology ring of Hom(π1(Σ), SU (2)) and use this to compute the ordinary cohomology groups of the quotient Hom(π1(Σ), SU (2))/SU (2). A key property is that the conjugation action is equivariantly formal.
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