The stable moduli space of flat connections over a surface

Ramras, Daniel

doi:10.1090/S0002-9947-2010-05218-3

The stable moduli space of flat connections over a surface
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Trans. Amer. Math. Soc. 363 (2011), 1061-1100 Request permission

Abstract:

We compute the homotopy type of the moduli space of flat, unitary connections over an aspherical surface, after stabilizing with respect to the rank of the underlying bundle. Over the 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.

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