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

$\begin{array}{cc}\cdots \to H_{n+1}(X)& \stackrel{{\partial }_{*}}{\to }{H}_n(A\cap B)\stackrel{(i_{*},j_{*})}{\to }{H}_n(A)\oplus H_n(B)\stackrel{k_{*}-l_{*}}{\to }{H}_n(X)\stackrel{{\partial }_{*}}{\to }{H}_{n-1}(A\cap B)\to \\ & \cdots \to H_0(A)\oplus H_0(B)\stackrel{k_{*}-l_{*}}{\to }{H}_0(X)\to 0.\end{array}$

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

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

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 $n\ne 1$ 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

$0\to {\tilde{H}}_2(S^2)\stackrel{{\partial }_{*}}{\to }\mathbb{Z}\to 0$

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)

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