Suppose I have a manifold $M$ and a submanifold or a boundary $N\subset M$. By the natural inclusion $\iota:N\hookrightarrow M$ we can easily see that $$\omega\in\mathrm{H}^k(M) \quad\implies\quad \iota^*\omega\in\mathrm{H}^k(N).$$
On the other hand, obviously $$\iota^*\omega\in\mathrm{H}^k(N) \quad\not\kern{-0.5em}\implies\quad \omega\in\mathrm{H}^k(M).$$
Is there something to be said about those forms that are in the cohomology of $N$ but not in that of $M$? Precisely, does either the space $$\mathrm{Hmm}_{(1)}^k(N,M):=\left\lbrace\omega\in\mathrm{H}^k(N)\ \middle|\ \omega\neq\iota^*\eta,\quad \eta\in\mathrm{H}^k(M)\right\rbrace$$ or the space $$ \mathrm{Hmm}_{(2)}^k(N,M) := \mathrm{H}^k(N)\Big/\left\lbrace\omega\in\mathrm{H}^k(N)\ \middle|\ \omega=\iota^*\eta,\quad \eta\in\mathrm{H}^k(M)\right\rbrace$$ have a simple description?1 I have a hunch that it should come from some relative cohomology but I wasn't able to make it precise. Even if there is no simple description, can we say something about its dimension?
I am more interested in the case where $N=\partial M$ (so if there's something to be said there but not in the general case, I'm perfectly happy), but the general case seems interesting too.
1I'm writing both spaces to maximize my chances, see the helpful comments of @Thorgott and @Osama Ghani