Thanks! Just for clarification regarding the latter case, do you indeed mean that ``rel boundary'' is the case in which you fix diffs on the boundary pointwise and the $\mathbb Z/2 \times (GL_2(\mathbb Z) \rtimes T^2)$ when you don't fix the boundary pointwise?

Thanks a lot! What does the T^3 stand for in $GL_3(\mathbb Z) \rtimes T^3$? I think I might be misunderstanding the notation but you said that the mapping class group for the 3-torus is just $GL_3(\mathbb Z) $. Also, by ``rel boundary'' do you mean in the case in which you fix diffs on the boundary pointwise?

Yes, for the former case, I am thinking about the diffeomorphisms which fix the boundary point-wise.