I am struggling with the following fact:
Say that we have $A\subset M$ a compact connected sumbanifold. Let $H\subset \pi_1(A)$ be a subgroup and $p_A:A_H\rightarrow A$ the corresponding covering space.
Moreover say that the following composition $\iota\circ p_A:A_H\rightarrow M$ is $\pi_1$-injective, with image $K\subset \pi_1(M)$. We then have that we can consider $A_H$ as a subset of the total space of $M_K$, I would then like to show that $\iota: A_H\rightarrow M_K$ is a homototopy equivalence.
For a counterexample take $A=S^2 \subset S^3=M$ and $H=\pi_1(S^2)$ which is trivial. It follows that $A=A_H$ and $M=M_K$, but the inclusion $A \subset M$ is not a homotopy equivalence.