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.

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


 * $$\begin{align}

\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{align}$$

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


 * $$\cdots \rightarrow \tilde{H}_0(A\cap B) \xrightarrow{(i_*,j_*))} \tilde{H}_0(A)\oplus \tilde{H}_0(B)\,\xrightarrow{k_* - l_*}\,\tilde{H}_0(X)\rightarrow\,0.$$

Examples
Consider the cover of $$S^2$$ formed by 2-discs A and B in the figure.



The space $$A\cap B$$ is homotopy equivalent to the circle. We know that the homology groups are preserved by homotopy and so $$H_n(A\cap B)\cong 0$$ for $$n\neq 1$$ and $$H_1(A\cap B)\cong \mathbb{Z}$$. Also note how the homology groups of A and B are trivial since they are both contractable. So we know that


 * $$0 \rightarrow \tilde{H}_2(S^2) \xrightarrow{\partial_*} \mathbb{Z}\rightarrow 0$$

This means that $$\tilde{H}_2(S^2)\cong \mathbb{Z}$$ since $$\partial_*$$ is an isomorphism by exactness.

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



Exercises
(under construction)