## Exact sequence (Quotient space)

Exact sequence (Quotient space)
If $X$ is a space and $A$ is a nonempty closed subspace that is a deformation retract of some neighborhood in $X$, then there is an exact sequence
\begin{aligned} \dots\to\widetilde{H}_n(A)\xrightarrow{i_*}\widetilde{H}_n(X)\xrightarrow{j_*}\widetilde{H}_n(X/A)\xrightarrow{\partial}\widetilde{H}_{n-1}(A)&\xrightarrow{i_*}\widetilde{H}_{n-1}(X)\to\dots\\ &\dots\to\widetilde{H}_0(X/A)\to 0 \end{aligned}
where $i$ is the inclusion $A\to X$ and $j$ is the quotient map $X\to X/A$.

Reduced homology of spheres (Proof)
$\widetilde{H}_n(S^n)\cong\mathbb{Z}$ and $\widetilde{H}_i(S^n)=0$ for $i\neq n$.

For $n>0$ take $(X,A)=(D^n,S^{n-1})$ so that $X/A=S^n$. The terms $\widetilde{H}_i(D^n)$ in the long exact sequence are zero since $D^n$ is contractible.

Exactness of the sequence then implies that the maps $\widetilde{H}_i(S^n)\xrightarrow{\partial}\widetilde{H}_{i-1}(S^{n-1})$ are isomorphisms for $i>0$ and that $\widetilde{H}_0(S^n)=0$. Starting with $\widetilde{H}_0(S^0)=\mathbb{Z}$, $\widetilde{H}_i(S^0)=0$ for $i\neq 0$, the result follows by induction on $n$.