Let $X$ a connected manifold, $x \in X$ and $V$ a neighborhood of $x$. Assume $i:V \to X$ induce isomorphism between all homology groups.
Does $X-p$ and $V-p$ still have the same homology groups ?
Motivation : I think this is true if $X,V$ have no homology. The first case when it's false is where $X$ is the sphere $\mathbb S^2$ and $V$ a small ball intersected with $X$, containing $p$. We have $H_1(X-p) = 0$ when $H_1(V-p) \cong \mathbb Z$. But here $H_2(X)$ is not isomorphic to $H_2(V)$ ! So my guess is the first condition is strong enough for give me this isomorphism.
Suppose the inclusion $V\to X$ induces isomorphisms in homology. Consider now the inclusion of pairs $(V,V\setminus\{p\})\to(X,X\setminus\{p\})$, fromwhich you can construct a big diagram: the two rows are the the long exact sequences of the pairs $(V,V\setminus\{p\})$ and $(X,X\setminus\{p\})$ and the maps going from the groups in one to the coresponding groups in the other are induced by inclusions.
The hypothesis is that the maps $H_*(V)\to H_*(X)$ are isomorphisms. If you can prove that the maps $H_*(V,V\setminus\{p\})\to H_*(X,X\setminus\{p\})$ are also isomorphism, the $5$-lemma will tell you that the maps $H_*(V\setminus\{p\})\to H_*(X\setminus\{p\})$ are also isomorphisms, which gives you want you want. Can you?