@Arnaud I think you misread the question. The OP asking if for a submanifold $N$, does there exist a deformation $N_d$ of $N$ with $f(N_d)$ transverse to $N_d$. This is clearly never exists for $f$ the identity unless dim N= dim M.

Why not replace "intersections" with "crossings"?

@user44651 That looks right. Brieskorn's original paper computes both $b_2(W)$ and $\sigma(W)$ using a variant of Picard-Lefschetz theory for open manifolds and the Serre spectral sequence.

I at some point had some mathematica code to compute the signature of $M(p,q,r)$ and the homotopy type of the plane field associated to the boundary contact structure which I can scrounge up if any one is interested.

Is this true in the case $G= \Bbb Z$?

Is there any easy way to recognize that a knot $K \subset S^1 \times S^2$ with winding number 1 is hyperbolic? i.e. is there some version of the Thurston's hyperbolic/torus/satellite trichotomy for these knots in $S^1 \times S^2$?

@aglearner There is also an argument of Eliashberg which shows given that the form is zero in cohomology, that any weak filling of a contact structure can be modified to a strong filling of the same contact structure. This is a short and straightforward argument in the "A few remarks..." paper (prop 4.1 here arxiv.org/pdf/math/0311459.pdf).

@aglearner Typically a weak symplectic filling is defined so that $\omega$ restricted the contact planes is $>0$ (in the sense that it is non-degenerate on the contact planes and agrees with their preferred orientation), but this is completely equivalent to the taming condition as the contact planes are complex.

What are you using to construct such a $J$? Certainly if such a $J$ exists it satisfies the conditions of the question, but I don't see where the construction is coming from.