Results 1 
4 of
4
Yang–Mills theory over surfaces and the AtiyahSegal theorem
, 2008
"... Abstract. In this paper we explain how Morse theory for the YangMills functional can be used to prove an analogue, for surface groups, of the AtiyahSegal theorem. Classically, the AtiyahSegal theorem relates the representation ring R(Γ) of a compact group Γ to the complex Ktheory of the classify ..."
Abstract

Cited by 7 (5 self)
 Add to MetaCart
Abstract. In this paper we explain how Morse theory for the YangMills functional can be used to prove an analogue, for surface groups, of the AtiyahSegal theorem. Classically, the AtiyahSegal theorem relates the representation ring R(Γ) of a compact group Γ to the complex Ktheory of the classifying space BΓ. For infinite discrete groups, it is necessary to take into account deformations of representations, and with this in mind we replace the representation ring by Carlsson’s deformation Ktheory spectrum Kdef(Γ) (the homotopytheoretical analogue of R(Γ)). Our main theorem provides an isomorphism in homotopy K ∗ def (π1Σ) ∼ = K ∗ (Σ) for all compact, aspherical surfaces Σ and all ∗> 0. Combining this result with work of Lawson, we obtain homotopy theoretical information about the stable moduli space of flat connections over surfaces. 1.
THE STABLE MODULI SPACE OF FLAT CONNECTIONS OVER
, 810
"... Abstract. We compute the homotopy type of the moduli space of flat, unitary connections over any aspherical surface, after stabilizing with respect to the rank of the underlying bundle. Over an orientable surface M g, we show that this space has the homotopy type of the infinite symmetric product of ..."
Abstract

Cited by 2 (1 self)
 Add to MetaCart
Abstract. We compute the homotopy type of the moduli space of flat, unitary connections over any aspherical surface, after stabilizing with respect to the rank of the underlying bundle. Over an orientable surface M g, we show that this space has the homotopy type of the infinite symmetric product of M g, generalizing a wellknown fact for the torus. Over a nonorientable surface, we show that this space is homotopy equivalent to a disjoint union of two tori, whose common dimension corresponds to the rank of the first (co)homology group of the surface. Similar calculations are provided for products of surfaces, and show a close analogy with the Quillen–Lichtenbaum conjectures in algebraic K–theory. The proofs utilize Tyler Lawson’s work in deformation K–theory, and rely heavily on YangMills theory and gauge theory. 1.
PERIODICITY IN THE STABLE REPRESENTATION THEORY OF CRYSTALLOGRAPHIC GROUPS
"... Abstract. Deformation K–theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2–periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson’s Bott map is ..."
Abstract
 Add to MetaCart
Abstract. Deformation K–theory associates to each discrete group G a spectrum built from spaces of finite dimensional unitary representations of G. In all known examples, this spectrum is 2–periodic above the rational cohomological dimension of G (minus 2), in the sense that T. Lawson’s Bott map is an isomorphism on homotopy in these dimensions. We establish a periodicity theorem for crystallographic subgroups of the isometries of k–dimensional Euclidean space. For a certain subclass of torsionfree crystallographic groups, we prove a vanishing result for the homotopy groups of the stable moduli space of representations, and we provide examples relating these homotopy groups to the cohomology of G. These results are established as corollaries of the fact that for each n> 0, the onepoint compactification of the moduli space of irreducible n–dimensional representations of G is a CW complex of dimension at most k. This is proven using real algebraic geometry and projective representation theory. 1.