Functoriality of Thurston's norm

Question

Let $M$ be a manifold of dimension $3$ and let $N$ be an embedded submanifold of $M$ (also of dimension $3$).

Then, both second homologies $H_2(M)$ and $H_2(N,\delta N)$ are equipped with a norm (technically a semi-norm if we don't assume more properties, but not relevant here), called Thurston's norm.

I am wondering whether there is any relation between the two normed vector spaces $H_2(M)$ and $H_2(N,\delta N)$. More precisely, if $i \colon N \to M$ is the embedding, is there some induced map $i_* \colon H_2(N, \delta N) \to H_2(M)$ of normed vector spaces, possibly making the construction functorial.

The main issue for me here is that if $x \in H_2(N,\delta N)$ is represented by a surface (possibly not connected) with boundaries, I don't see a canonical way to extend this surface to a closed surface in $M$.

Answer 1

For submanifolds that are link complements, Ken Baker and I recently investigated this question in our paper "Dehn Filling and the Thurston Norm" available at arXiv:1608.02443

The basic idea (as in so many 3-manifold papers!) is to play surfaces of different boundary slopes against each other. It's most natural to think of this in terms of comparing Dehn surgeries, but one can also think of it in terms of using a surface with non-meridional boundary on the link to say something about the situation when the ambient 3-manifold has a class with smaller than expected Thurston norm compared to the corresponding (relative) class in the submanifold.