Homology of Subspace vs. Homology of Ambient Space.

Question

Let $M$ be a manifold embedded in $\mathbb R^n$ , so that the manifold has non-trivial $k-th$ homology for some $n \geq k\geq 0$ . How do we identify the fact that while there is a non-trivial cycle in $M$ (i.e.,a cycle $C$that does not bound), that $C$ is not a non-trivial cycle in $\mathbb R^n$ itself (since $\mathbb R^n$ has trivial homology , this obviously cannot happen). I am thinking of $H^n(S^n; \mathbb Z )=\mathbb Z$, but also of other cases. How do we allow for this condition in general where there is a subspace $S$ embedded in $X$ where $S$ is, e.g., orientable, while $S$ is not, say for even-dimensional projective spaces embedded in odd-dimensional projective spaces)? Do we just declare the non-bounding cycle C , as a non-bounding cycle in the subspace S to be a bounding cycle in the ambient space? And, do we declare the cycle that "witnesses" non-orientability in the subspace to not be a cycle in the ambient space?