I have this exercise where I'm asked to compute the homology groups of the 4-tori $T^4$ minus a point $p$, where $T^4\simeq (\mathbb{R}/\mathbb{Z})^4$. I'll denote the space $T^4 \setminus \{p\}$ with $X$. The ring $R$ where we are computing the homology is $\mathbb{R}$.
I applied Mayer Vietoris, covering $T^4$ with $X$ and a neighborhood of $p$, so their intersection is homeomorphic to $\mathbb{S}^3$.
I encountered no problems for $k \neq 3,4$. For those cases, I ended up with this sequence: $$ 0 \to H_4(X) \to H_4(T^4) \to H_3(\mathbb{S}^3) \to H_3(X) \to H_3(T^4) \to 0 $$ which translates to $$ 0 \to H_4(X) \to \mathbb{R} \to \mathbb{R} \to H_3(X) \to \mathbb{R}^4 \to 0 $$
Here I'm stuck. I tried to study the maps defined in the M.V. sequence but to no use. I hope you can help me, thanks in advance.
The generator of $H_4(T^4)$ is represented by a triangulation of the whole of $T^4$. When we pull this back to $X \sqcup {\rm int}(B^4)$, we get a sum of triangulations of each piece. When we take the boundary, we get a triangulation of $S^3$ in each component. Finally this pulls back to a triangulation of $S^3$ on the intersection, representing the generator of $H_3(S^3)$.
Thus the "snake" map is an isomorphism $\mathbb{R}\cong \mathbb{R}$, which gives us isomorphisms either side: $$0\cong H_4(X),\qquad H_3(X)\cong \mathbb{R}^4.$$