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}).


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