Let $M$ be a compact connected $d$ dimensional manifold, and now I want to compute $H^k(M-\{p\})$ for $k=0,...,d$. So I believe the idea here is to consider the open sets $M^*=M-\{p\}$ and $V$ an open ball around $p$ diffeomorphic to $\mathbb{R}^n$, and then we use mayer-vietoris. Now there is a thing that is bothering me which is the following, we need to know what $H^k(M^*\cap V)$ is and this basically will be the open set $V-\{p\}$, and now I have seen the claim that this is homotopic to $S^{d-1}$ but I am not sure why this would be true, I could see this being true if we have that $V$ was a closed ball and then the homotopy would be clear to me, but for a closed ball we can't consider the Mayer-Vietoris sequence, at least that I am aware of. So does anyone know an explicit homotopy between the open set $V-\{p\}$ and $S^{d-1}$, so that we have that their cohomology groups are the same ?
Also latter one when we have the exact sequence $0\rightarrow H^{d-1}(M)\rightarrow H^{d-1}(M-\{p\})\rightarrow H^{d-1}(S^{d-1})\rightarrow \mathbb{R}\rightarrow H^d(M-\{p\})\rightarrow 0$ to compute these cohomology groups I believe we check that the map $H^{d-1}(M-\{p\})\rightarrow H^{d-1}(S^{d-1})$ is the zero map by using an argument with the Stoke's theorem and again we are using boundarys of manifolds and it seems weird since all the manifolds involved at first don't have boundary, unless I guess we do get that previous homotopy but I am not completely sure how this is done.
Any help with this is appreciated, thanks in advance.