# $\mathbb{Z}/2\mathbb{Z}$ coefficients in gysin sequence

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.

• To apply the Gysin sequence with $\mathbb{Q}$-coefficient you need an orientability condition on your bundle. Maybe they want to circumvent that? Mar 4, 2020 at 16:14
• I do not know. It seems to me that in the article they always have these orientability construction. Maybe since they are working with double covers it is easier to compute the $\mathbb{Z}/2\mathbb{Z}$-homology? Mar 5, 2020 at 10:41