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

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

is an exact sequence where $i:A\cap B\hookrightarrow A,j:A\cap B\hookrightarrow B,l:A\hookrightarrow X,k:B\hookrightarrow X$. 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.

$S^{2}$ covered by 2-discs A and B

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.

How we choose A and B.

Exercises

(under construction)