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Abstract: For an equivariant Morse stratification which contains a unique open stratum,we introduce the notion of equivariant antiperfection, which means thedifference of the equivariant Morse series and the equivariant Poincare seriesachieves the maximal possible value instead of the minimal possible value 0 inthe equivariantly perfect case. We also introduce a weaker condition of localequivariant antiperfection. We prove that the Morse stratification of theYang-Mills functional on the space of connections on a principal Un-bundleover a connected, closed, nonorientable surface is locally equivariantlyQ-antiperfect when the rank n=2,3; we propose that it is actually equivariantlyQ-antiperfect when n=2,3. Our proposal yields formulas of G-equivariantPoincare series of the representation variety of flat G-connections for thenonorientable surface where G=U2, SU2, U3, SU3. Our rank 2 formulasagree with formulas proved by T. Baird in arXiv:0806.1975.Baird verified our conjectural rank 3 formulas when the nonorientable surfaceis the real projective plane or the Klein bottle arXiv:0901.1604; he provedour conjectural U3 formula for any closed nonorientable surfaces byestablishing equivariant Q-antiperfection in this case arXiv:0902.4581.

Author: Nan-Kuo Ho, Chiu-Chu Melissa Liu


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