Results 1 
6 of
6
ORIENTABILITY IN YANGMILLS THEORY OVER NONORIENTABLE SURFACES
, 810
"... Abstract. The first two authors have constructed a gaugeequivariant Morse stratification on the space of connections on a principal U(n)bundle over a connected, closed, nonorientable surface Σ. This space can be identified with the real locus of the space of connections on the pullback of this bund ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
Abstract. The first two authors have constructed a gaugeequivariant Morse stratification on the space of connections on a principal U(n)bundle over a connected, closed, nonorientable surface Σ. This space can be identified with the real locus of the space of connections on the pullback of this bundle over the orientable double cover of Σ. In this context, the normal bundles to the Morse strata are real vector bundles. We show that these bundles, and their associated homotopy orbit bundles, are orientable for any n when Σ is not homeomorphic to the Klein bottle, and for n ≤ 3 when Σ is the Klein bottle. We also derive similar
The Yang–Mills stratification for surfaces revisited
, 2008
"... Abstract. We revisit Atiyah and Bott’s study of Morse theory for the Yang– Mills functional over a Riemann surface. We construct gaugeinvariant tubular neighborhoods for the Yang–Mills strata and establish new formulas for the minimum codimension of a (nonsemistable) stratum. These results yield ..."
Abstract

Cited by 3 (2 self)
 Add to MetaCart
Abstract. We revisit Atiyah and Bott’s study of Morse theory for the Yang– Mills functional over a Riemann surface. We construct gaugeinvariant tubular neighborhoods for the Yang–Mills strata and establish new formulas for the minimum codimension of a (nonsemistable) stratum. These results yield the exact connectivity of the natural map (Nss(E)) hG(E) → Map E (M, BU(n)) from the homotopy orbits of the space of central Yang–Mills connections to the classifying space of the gauge group G(E). All of these results carry over to nonorientable surfaces via Ho and Liu’s nonorientable Yang–Mills theory. Our construction of invariant tubular neighborhoods for locally closed submanifolds of infinite dimensional Riemannian manifolds (with an isometric group action) may be of independent interest. 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.
ON THE YANG–MILLS STRATIFICATION FOR SURFACES
"... Abstract. Atiyah and Bott showed that Morse theory for the Yang–Mills functional can be used to study the space of flat, or more generally central, connections on a bundle over a Riemann surface. These methods have recently been extended to nonorientable surfaces by Ho and Liu. In this article, we ..."
Abstract
 Add to MetaCart
Abstract. Atiyah and Bott showed that Morse theory for the Yang–Mills functional can be used to study the space of flat, or more generally central, connections on a bundle over a Riemann surface. These methods have recently been extended to nonorientable surfaces by Ho and Liu. In this article, we use Morse theory to determine the exact connectivity of the natural map from the homotopy orbits of the space of central Yang–Mills connections to the classifying space of the gauge group. The key ingredient in this computation is a combinatorial study of the Morse indices of Yang–Mills critical sets. 1.