Possibly low-level question here: if one has an embedding $M\hookrightarrow X$, call it $\iota$, then this is of course an immersion, so the differential maps $\iota_{\ast}:T_{p}M\rightarrow T_{\iota(p)}X$ are injective for every $p\in M$. Does this imply that $\iota$ induces an injective map on homology, or cohomology? If not, what are the known conditions on $\iota$ that would guarantee these types of injectivity? I am seeking to select a class of symplectic embeddings $(M,\sigma)\hookrightarrow (X,\omega)$ that are cohomologically injective.
My question is initially rather low, I recognize, but it blooms into something that requires more investigation. In lieu of an answer, I would appreciate some nice sources for more directed investigation. I am seeking topological enlightenment, but 'homologically injective' as search terms returns mostly algebraic research, much to my chagrin.
No. For example, $S^2$ embeds into $\mathbb{R}^3$, and this doesn't induce an injection on homology. Similarly, $\mathbb{R}^2$ embeds into $S^2$, and this doesn't induce an injection on cohomology.
In general it's tricky to guarantee injections or surjections on homology or cohomology. For an embedding $i : A \to X$ to induce an injection on homology is equivalent to the boundary maps $\partial$ in the long exact sequence
$$\cdots \xrightarrow{\partial} H_n(A) \to H_n(X) \to H_n(X, A) \to \cdots$$
in relative homology vanishing, and this is a tricky condition to verify. A sufficient condition is that $A$ is a retract of $X$. (And if you really want an injection on cohomology, as opposed to a surjection, then things are even weirder.)