The stable moduli space of flat connections over a surface
Abstract
We compute the homotopy type of the moduli space of flat, unitary connections over aspherical surfaces, 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.
Keywords
Cite
@article{arxiv.0810.1784,
title = {The stable moduli space of flat connections over a surface},
author = {Daniel A. Ramras},
journal= {arXiv preprint arXiv:0810.1784},
year = {2018}
}
Comments
40 pages. V2: Expanded Section 5.2; updated references; various minor revisions. Submitted version. V3: various proofs simplified, esp. in Sections 3, 5.2, and 6. This version includes many revisions based on a referee's comments, especially in Section 6. V4: proof of Lemma 5.7 corrected, other minor revisions