Topology/Mayer-Vietoris Sequence

A Mayer-Vietoris Sequence is a powerful tool used in finding Homology groups for spaces that can be expressed as the unions of simpler spaces from the perspective of Homology theory.

If X is a topological space covered by the interiors of two subspaces A and B, then

${\displaystyle {\begin{aligned}\cdots \rightarrow H_{n+1}(X)\,&{\xrightarrow {\partial _{*}}}\,H_{n}(A\cap B)\,{\xrightarrow {(i_{*},j_{*})}}\,H_{n}(A)\oplus H_{n}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{n}(X){\xrightarrow {\partial _{*}}}\,H_{n-1}(A\cap B)\rightarrow \\&\quad \cdots \rightarrow H_{0}(A)\oplus H_{0}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,H_{0}(X)\rightarrow \,0.\end{aligned}}}$

is an exact sequence where ${\displaystyle i:A\cap B\hookrightarrow A,j:A\cap B\hookrightarrow B,l:A\hookrightarrow X,k:B\hookrightarrow X}$. There is a slight adaptation for the reduced homology where the sequence ends instead

${\displaystyle \cdots \rightarrow {\tilde {H}}_{0}(A\cap B){\xrightarrow {(i_{*},j_{*}))}}{\tilde {H}}_{0}(A)\oplus {\tilde {H}}_{0}(B)\,{\xrightarrow {k_{*}-l_{*}}}\,{\tilde {H}}_{0}(X)\rightarrow \,0.}$

Consider the cover of ${\displaystyle S^{2}}$ formed by 2-discs A and B in the figure.

The space ${\displaystyle A\cap B}$ is homotopy equivalent to the circle. We know that the homology groups are preserved by homotopy and so ${\displaystyle H_{n}(A\cap B)\cong 0}$ for ${\displaystyle n\neq 1}$ and ${\displaystyle H_{1}(A\cap B)\cong \mathbb {Z} }$. Also note how the homology groups of A and B are trivial since they are both contractable. So we know that

${\displaystyle 0\rightarrow {\tilde {H}}_{2}(S^{2})\xrightarrow {\partial _{*}} \mathbb {Z} \rightarrow 0}$

This means that ${\displaystyle {\tilde {H}}_{2}(S^{2})\cong \mathbb {Z} }$ since ${\displaystyle \partial _{*}}$ is an isomorphism by exactness.

Consider the cover of the torus by 2 open ended cylinders A and B.

(under construction)