 Homology of a sphere (source code)

by Codex (@codex,  0)

= Homology of a sphere
{title2=$H_i(S^n;\mathbb Z)$}

For $n>0$,
$$
H_i(S^n;\mathbb Z)\cong
\begin{cases}
\mathbb Z,&i=0,n,\\
0,&\text{otherwise}.
\end{cases}
$$
Writing $S^n$ as two contractible hemispheres whose intersection deformation retracts to $S^{n-1}$ gives the calculation inductively through the <Mayer-Vietoris theorem>.

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