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The stable moduli space of flat connections over a surface

Author: Daniel A. Ramras
Journal: Trans. Amer. Math. Soc. 363 (2011), 1061-1100
MSC (2010): Primary 58D27, 55N15; Secondary 53C07, 55P42
Published electronically: September 21, 2010
MathSciNet review: 2728597
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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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Additional Information

Daniel A. Ramras
Affiliation: Department of Mathematics, 1326 Stevenson Center, Vanderbilt University, Nash- ville, Tennessee 37240
Address at time of publication: Department of Mathematical Sciences, New Mexico State University, P.O. Box 30001, Department 3MB, Las Cruces, New Mexico 88003-8001

Received by editor(s): November 1, 2008
Received by editor(s) in revised form: October 7, 2009
Published electronically: September 21, 2010
Additional Notes: This work was partially supported by an NSF graduate fellowship and by NSF grants DMS-0353640 (RTG) and DMS-0804553.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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