Excision & Mayer–Vietoris

Cut a space in two and let an exact sequence assemble its homology.

The idea

Excision and Mayer-Vietoris are the two theorems that make singular homology computable: each lets us calculate the homology of a space from simpler pieces of it.

Excision concerns the relative group $H_{n}(X, A)$, chains in $X$ counted modulo chains in a subspace $A$.

Theorem (Excision).

Let $Z \subseteq A \subseteq X$ be subspaces such that the closure of $Z$ is contained in the interior of $A$. Then the inclusion induces isomorphisms $H_{n}(X \setminus Z,\, A \setminus Z) \cong H_{n}(X, A) \qquad \text{for every } n.$

The hypothesis says that $Z$ keeps clear of the boundary $\partial A$, and the conclusion says that deleting such a $Z$ from $X$ and from $A$ together changes nothing. The relative group therefore depends only on what happens near the frontier between $A$ and the rest of $X$, not on the deep interior of $A$.

Mayer-Vietoris concerns a decomposition $X = A \cup B$ in which the interiors of $A$ and $B$ cover $X$.

Theorem (Mayer-Vietoris).

Let $X = A \cup B$ with the interiors of $A$ and $B$ covering $X$. Then there is a long exact sequence $\cdots \to H_{n}(A \cap B) \to H_{n}(A) \oplus H_{n}(B) \to H_{n}(X) \to H_{n-1}(A \cap B) \to \cdots \to H_{0}(X) \to 0,$ whose maps are built from the inclusions of $A \cap B$ into $A$ and $B$, and of $A$ and $B$ into $X$.

The sequence runs through $H_{n}(A \cap B)$, then $H_{n}(A) \oplus H_{n}(B)$, then $H_{n}(X)$, then down one degree and around again. Because each map's image is the next map's kernel, knowing the homology of the two pieces and of their overlap usually determines the homology of the union.

Mayer-Vietoris is the homology counterpart of van Kampen's theorem, and easier to apply: an exact sequence of abelian groups is more tractable than a free product modulo relations.

Ways to work on it

• Walkthrough. What excision's interior hypothesis buys, collapsing a good pair, and two Mayer-Vietoris computations.
• Proof. Compute H_(S^n) for every n at once, by covering the sphere with two caps.
• Practice. Decide whether an excision is legal, identify an overlap, and fill gaps in a Mayer-Vietoris sequence.
• Hardest. The Klein bottle from two Möbius bands and the torus from two cylinders, computed end to end.

Not sure where to start? Take the ten-question placement test.