## Mayer-Vietoris Sequence applied to Spheres

Mayer-Vietoris Sequence
For a pair of subspaces $A,B\subset X$ such that $X=\text{int}(A)\cup\text{int}(B)$, the exact MV sequence has the form
\begin{aligned} \dots&\to H_n(A\cap B)\xrightarrow{\Phi}H_n(A)\oplus H_n(B)\xrightarrow{\Psi}H_n(X)\xrightarrow{\partial}H_{n-1}(A\cap B)\\ &\to\dots\to H_0(X)\to 0. \end{aligned}

Example: $S^n$
Let $X=S^n$ with $A$ and $B$ the northern and southern hemispheres, so that $A\cap B=S^{n-1}$. Then in the reduced Mayer-Vietoris sequence the terms $\tilde{H}_i(A)\oplus\tilde{H}_i(B)$ are zero. So from the reduced Mayer-Vietoris sequence $\displaystyle \dots\to\tilde{H}_i(A)\oplus\tilde{H}_i(B)\to\tilde{H}_i(X)\to\tilde{H}_{i-1}(A\cap B)\to\tilde{H}_{i-1}(A)\oplus\tilde{H}_{i-1}(B)\to\dots$ we get the exact sequence $\displaystyle 0\to\tilde{H}_i(S^n)\to\tilde{H}_{i-1}(S^{n-1})\to 0.$
We obtain isomorphisms $\tilde{H}_i(S^n)\cong\tilde{H}_{i-1}(S^{n-1})$.