The Mayer-Vietoris Theorem

The Mayer-Vietoris theorem is one of the most powerful computational tools in algebraic topology. It provides a long exact sequence relating the homology of a space to the homology of two overlapping subspaces, analogous to the inclusion-exclusion principle in combinatorics.

Statement of the Theorem

Theorem6.6Mayer-Vietoris Theorem

Let $X = A \cup B$ where $A$ and $B$ are open subsets of $X$ (or more generally, the interiors of $A$ and $B$ cover $X$ ). Then there is a long exact sequence $\cdots \to H_n(A \cap B) \xrightarrow{\Phi} H_n(A) \oplus H_n(B) \xrightarrow{\Psi} H_n(X) \xrightarrow{\Delta} H_{n-1}(A \cap B) \to \cdots$ where $\Phi(\alpha) = (i_{A*}(\alpha), -i_{B*}(\alpha))$ , $\Psi(\alpha, \beta) = j_{A*}(\alpha) + j_{B*}(\beta)$ , and $\Delta$ is the connecting homomorphism. Here $i_A, i_B$ are inclusions of $A \cap B$ into $A$ and $B$ , and $j_A, j_B$ are inclusions of $A$ and $B$ into $X$ .

The Mayer-Vietoris sequence arises from the short exact sequence of chain complexes $0 \to C_*(A \cap B) \xrightarrow{\Phi} C_*(A) \oplus C_*(B) \xrightarrow{\Psi} C_*(A + B) \to 0$ where $C_*(A + B) \subseteq C_*(X)$ consists of chains that are sums of chains in $A$ and chains in $B$ . The small chains theorem guarantees that inclusion $C_*(A + B) \hookrightarrow C_*(X)$ induces isomorphisms on homology.

Applications

ExampleHomology of the torus

Write the torus $T^2 = A \cup B$ where $A$ and $B$ are open cylinders overlapping in two disjoint annuli, so $A \cap B \simeq S^1 \sqcup S^1$ . Both $A$ and $B$ deformation retract to $S^1$ . The Mayer-Vietoris sequence gives: $0 \to H_2(T^2) \xrightarrow{\Delta} H_1(S^1 \sqcup S^1) \xrightarrow{\Phi} H_1(S^1) \oplus H_1(S^1) \xrightarrow{\Psi} H_1(T^2) \to 0$ Computing the maps yields $H_2(T^2) \cong \mathbb{Z}$ , $H_1(T^2) \cong \mathbb{Z}^2$ , and $H_0(T^2) \cong \mathbb{Z}$ .

Theorem6.7Mayer-Vietoris for Reduced Homology

For a pointed space $X = A \cup B$ with $A \cap B \neq \emptyset$ , there is a reduced Mayer-Vietoris sequence $\cdots \to \tilde{H}_n(A \cap B) \to \tilde{H}_n(A) \oplus \tilde{H}_n(B) \to \tilde{H}_n(X) \to \tilde{H}_{n-1}(A \cap B) \to \cdots$ This is particularly useful for computations involving wedge sums and suspensions.

Significance

RemarkMayer-Vietoris as a local-to-global principle

The Mayer-Vietoris theorem embodies the local-to-global principle: if we understand the homology of pieces of a space and their overlap, we can reconstruct the homology of the whole space. This makes it the homological analogue of the Seifert-van Kampen theorem for fundamental groups.

The Mayer-Vietoris sequence, together with the long exact sequence of a pair and cellular homology, forms the core toolkit for computing the homology groups of essentially any space built from familiar pieces.