Four-manifold invariants from higher-rank bundles

P.B. Kronheimer
2005 Journal of differential geometry  
Lemma 34. Suppose N is odd. Then under the same hypothesis as Lemma 33, all the points of the moduli space M w (Z + K ;α 0 ) have the same sign. Proof. We wish to compare the orientations of two points [A 0 ] and [A 1 ] in the moduli space M w (Z + K ;α 0 ). Let [A t ] be a 1-parameter family of connections in Ꮽ (Z + K ;α 0 ) joining A 0 to A 1 . Let D t be the corresponding Fredholm operators on the cylindrical-end manifold Z + K , as in (14) . The operators D 0 and D 1 are invertible, and so
more » ... here are canonical trivializations of the determinant lines det(D 0 ) and det(D 1 ). The lemma asserts that these canonical trivializations can be extended to a trivialization of the determinant line det(D • ) on [0, 1] . Let p : M → M be again the N -fold cyclic cover of the knot complement, and let p : Z K → Z K be the corresponding cyclic cover of Z K = S 1 × M . Let Z + K be the corresponding cylindrical-end manifold. Set These are connections on Z + K , asymptotic to p * (α 0 ) on the cylindrical end. There are corresponding Fredholm operators det(D t ), for t ∈ [0, 1]; we use a weighted
doi:10.4310/jdg/1143572014 fatcat:a3ogilyb3zdovialgqymv5oqta