Counting upper S upper L left parenthesis 2 comma double struck upper C right parenthesis S L ( 2 , C ) SL(2,C) connections on Seifert-fibered spaces

We study the upper S upper L left parenthesis 2 comma double struck upper C right parenthesis S L ( 2 , C ) SL(2, C) character variety of a Seifert-fibered homology 3-sphere from the point of view of gauge theory. Namely, we introduce a class of perturbations of the upper S upper L left parenthesis 2 comma double struck upper C right parenthesis S L ( 2 , C ) SL(2,C) Chern–Simons functional and prove a localization result: the perturbed critical points either approach a compact subset of the upper S upper L left parenthesis 2 comma double struck upper C right parenthesis S L ( 2 , C ) SL(2, C) character variety or else ‘escape to infinity’. Furthermore, the Euler characteristic and Poincaré polynomial of the stable locus of the character variety are obtained by suitably counting the localizing critical points. As an application, we obtain formulas for the Euler characteristic and Poincaré polynomial of the stable locus of the upper S upper L left parenthesis 2 comma double struck upper C right parenthesis S L ( 2 , C ) SL(2, C) character variety of a Seifert-fibered homology 3-sphere. In particular, we prove that the Euler characteristic equals the Milnor number (divided by 4) of any weighted-homogeneous isolated complete intersection singularity whose link is the given 3-manifold.