De Rham cohomology question

Question

I'm trying to compute a certain DeRham cohomology. Consider $M = S^n-C$, where $C$ is the disjoint union of closed disks $C = \cup_{i=1}^m D_i$, and $m,n \geq 1$. How can we compute the cohomology $H^{*}(M)$?

Answer 1

You can proceed by induction, using the following observation.

If $M$ is a manifold and $D\subseteq M$ is a (standardly embedded) closed disk in $M$, then there is an open set $U\subseteq M$ which is a standardly embeded open disk containing $U$ such that $U\setminus D$ is a «thick sphere», and $\{U,M-D\}$ is an open covering of $M$ from which one can get a Mayer-Vietoris long exact sequence.