Antiperfect Morse Stratification Nan-kuo Ho and Chiu-chu

For an equivariant Morse stratification which contains a unique open stratum, we introduce the notion of equivariant antiperfection, which means the difference of the equivariant Morse series and the equivariant Poincaré series achieves the maximal possible value (instead of the minimal possible value 0 in the equivariantly perfect case). We also introduce a weaker condition of local equivariant antiperfection. We prove that the Morse stratification of the Yang-Mills functional on the space of connections on a principal G-bundle over a connected, closed, nonorientable surface Σ is locally equivariantly Q-antiperfect when G = U(2), SU(2), U(3), SU(3); we propose that the Morse stratification is actually equivariantly Q-antiperfect in these cases. Our proposal yields formulas of Poincaré series P t (Hom(π1(Σ), G);Q) when G = U(2), SU(2), U(3), SU(3). Our U(2), SU(2) formulas agree with formulas proved by T. Baird, who also verified our conjectural U(3) formula.