I asked similar question before, after some modification, I have a new question. Suppose $M^{n\geq 4}$ is a connected compact smooth manifold with connected nonempty boundary. Suppose $i\_\*: H\_1(\partial M)\rightarrow H\_1(M)$ is not injective, for simplicity, assume that $\dim (\ker(i\_\*))=1$, that means, there is 1-cycle (choose a loop $C$) in $\partial M$ which is nontrivial in $\partial M$ but trivial in $M$, assume it is boundary of 2-dimensional embedded submanifold $D$ (let's say $D$ is a 2-disc) in $M$, now denote the small neighbourhood of $D$ in $M$ by $U\_\epsilon(D)$ ($\epsilon$ can be arbitrarily small), now I am wondering if new homomorphism
$$j\_\*: H\_1(\partial(M\setminus U\_\epsilon(D)))\rightarrow H\_1(M\setminus U\_\epsilon(D))$$
is injective now?


What I have tried: suppose $n=4$, the neighbourhood of $C$ in $\partial M$ is a solid torus, and any circle in the kernel of $i\_\*$ is homologous to $C$, thus homologous to a circle on the boundary of the solid torus, which retracts along the new boundary $\partial (M\setminus U\_\epsilon(D))$.



Injectivity of homomorphism between homology groups of manifold and its boundary
I am not sure whether this is the kind of thing you want to know, but here is a procedure to make $\partial M \rightarrow M$ an isomorphism on $\pi\_1$ if $\text{dim}(M) \geq 5$.


As $M$ is compact, $\pi\_1 M$ is finitely presented. Given one of the finitely many generators of $\pi\_1 M$, represent it by an embedded circle based in $\partial M$. Excising a tubular neighborhood of each such circle corresponing to a generator, yields a new manifold $M'$ and a map $\partial M' \rightarrow M'$ which is surjective on $\pi\_1$. 


There is a lemma (I think it is due to Siebenmann) that states that a surjective homomorphism of finitely presented groups has finitely normally generated kernel. Hence, we find finitely many disjointly embedded circles in $\partial M'$ that normally generate the kernel of $\pi\_1(\partial M') \rightarrow \pi\_1(M')$. Since $\text{dim}(M') \geq 5$, these circles bound **embedded** disks in $M'$. If we excise tubular neighborhoods of these, we obtain the desired manifold whose inclusion of the boundary is a $\pi\_1$-isomorphism.


