I am reading the article "Signature of links" by Kauffman and Taylor. Here they show that it is possible to calculate the nullity of a link $L\subset S^3$ by knowing the first betti number of the double cover of the exterior of the link in $S^3$. As they say in pages 354 and 355, to compute such betti number one needs to apply the Gysin sequence with $\mathbb{Z}/2\mathbb{Z}$ coefficients and then the Bockstein spectral sequence.

Why is it not possible to apply the Gysin sequence directly with $\mathbb{Q}$ coefficients? It seems to me that this would simplify the proof of the subsequent Theorem 2.6.