example of cohomology and Mayer-Vietoris sequence

Consider n-dimensional sphere $S^n=\{v\in R^{n+1}|\mathit{\hspace{1em}}|v|=1\}$.

Let $A=\{(x_0,\mathrm{\dots },x_n)\in S^n|\mathit{\hspace{1em}}{x}_0>-1/2\}$ and.

Of course both $A$ and $B$ are open (in $S^n$) and their union is $S^n$. Furthermore, it can be easily seen, that their intersection can be contracted into ”big circle”, i.e. $A\cap B$ has homotopy type of $S^{n-1}$. Also both $A$ and $B$ are contractible (they are homeomorphic to $R^n$ via stereographic projection). So, write part of a Meyer-Vietoris sequence (for the cohomology $H^m(X)=H^m(X,G)$, where $G$ is a fixed Abelian group):

$\mathrm{\cdots }\to H^m(A)\oplus H^m(B)\to H^m(A\cap B)\to H^{m+1}(S^n)\to H^{m+1}(A)\oplus H^{m+1}(B)\to \mathrm{\cdots }$

Since both $A$ and $B$ are contractible and $A\cap B$ is homotopic to $S^{n-1}$, we have the following short exact sequence:

$0\to H^m(S^{n-1})\to H^{m+1}(S^n)\to 0$

which shows that $H^m(S^{n-1})$ is isomorphic to $H^m(S^n)$ for every $n>0$ and $m>0$. So, in order to calculate cohomology groups of spheres, we only need to know the cohomology groups of $S^1$. And those can be also calculated, if we once again apply previous schema. Note, that in the case of $S^1$ we have that $A\cap B$ has the homotopy type of a discrete space with two points. Therefore all their cohomology groups are trivial, except for $H^0$ (which can be easily calculated to be equal to $H^0(*)\oplus H^0(*)$, where * is a one-pointed space).

This schema can be used for other spaces like the torus (which can be also calculated from Kunneth’s formula).