  We revisit Atiyah and Bott's study of Morse theory for the Yang-Mills
functional over a Riemann surface, and establish new formulas for the minimum
codimension of a (non-semi-stable) stratum. These results yield the exact
connectivity of the natural map (C_{min} E)//G(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
non-orientable surfaces via Ho and Liu's non-orientable Yang-Mills theory.
  A somewhat less detailed version of this paper (titled "On the Yang-Mills
stratification for surfaces") will appear in the Proceedings of the AMS.
