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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 sodoi:10.4310/jdg/1143572014 fatcat:a3ogilyb3zdovialgqymv5oqta