Mayer–Vietoris sequence

For an open cover $X=U\cup V$, integral singular homology has the long exact sequenceAll four maps other than $\delta$ are induced by inclusions. The minus sign makes consecutive maps compose to zero. Exactness identifies the homology classes created by gluing $U$ and $V$ with cycles in their overlap.^[ref]

To construct $\delta$, subdivide a representing cycle in singular homology until it is a sum $c_U+c_V$ of chains in the two open sets. Since $\partial c_U=-\partial c_V$, this common boundary lies in the overlap; its class is $\delta [c]$. This chain construction proves exactness and also gives the version for reduced homology.

Cover $S^n$, $n\ge 1$, by two contractible enlarged hemispheres. Their overlap has the homotopy type of $S^{n-1}$, giving ${\tilde{H}}_n(S^n)\cong {\tilde{H}}_{n-1}(S^{n-1})$. Iterating starts from ${\tilde{H}}_0(S^0)\cong \mathbb{Z}$ and computes the top homology of every sphere.