Abstract. The first two authors have constructed a gauge-equivariant 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

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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 well-known fact for the torus. Over a non-orientable 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 Yang-Mills theory and gauge theory. 1.